The Ricci Flow in Riemannian Geometry - MSIandrews/book.pdf · The Ricci Flow in Riemannian...

Christopher Hopper · Ben Andrews

The Ricci Flow inRiemannian Geometry

A complete proof of the differentiable1/4-pinching sphere theorem

27 July 2010

Springer

Authors

Christopher HopperMathematical Institute24–29 St Giles’Oxford OX1 [email protected]

Ben AndrewsAustralian National UniversityCanberraACT [email protected]

To be published in Springer’s Lecture Notes in Mathematics book series.The original publication is available at www.springerlink.com/content/

110312/

mailto:[email protected]

mailto:[email protected]

www.springerlink.com/content/110312/

www.springerlink.com/content/110312/

For in the very torrent, tempest, and as I may say,whirlwind of your passion, you must acquire and begeta temperance that may give it smoothness.

— Shakespeare, Hamlet.

Preface

There is a famous theorem by Rauch, Klingenberg and Berger which statesthat a complete simply connected n-dimensional Riemannian manifold, forwhich the sectional curvatures are strictly between 1 and 4, is homeomorphicto a n-sphere. It has been a longstanding open conjecture as to whetheror not the ‘homeomorphism’ conclusion could be strengthened to a ‘diffeo-morphism’.

Since the introduction of the Ricci flow by Hamilton [Ham82b] some twodecades ago, there have been several inroads into this problem — particularlyin dimensions three and four — which have thrown light upon a possible proofof this result. Only recently has this conjecture (and a considerably strongergeneralisation) been proved by Simon Brendle and Richard Schoen. The aimof the present book is to provide a unified expository account of the differen-tiable 1/4-pinching sphere theorem together with the necessary backgroundmaterial and recent convergence theory for the Ricci flow in n-dimensions.This account should be accessible to anyone familiar with enough differentialgeometry to feel comfortable with tensors, covariant derivatives, and normalcoordinates; and enough analysis to follow standard pde arguments. Theproof we present is self-contained (except for the quoted Cheeger-Gromovcompactness theorem for Riemannian metrics), and incorporates several im-provements on what is currently available in the literature.

Broadly speaking, the structure of this book falls into three main top-ics. The first centres around the introduction and analysis the Ricci flowas a geometric heat-type partial differential equation. The second concernsPerel’man’s monotonicity formulæ and the ‘blow up’ analysis of singularitiesassociated with the Ricci flow. The final topic focuses on the recent contri-butions made — particularly by Bohm and Wilking [BW08], and by Brendleand Schoen [BS09a] — in developing the necessary convergence theory for theRicci flow in n-dimensions. These topics are developed over several chapters,the final of which aims to prove the differentiable version of the sphere the-orem.

v

vi Preface

The book begins with an introduction chapter which motivates the pinch-ing problem. A survey of the sphere theorem’s long historical development isdiscussed as well as possible future applications of the Ricci flow.

As with any discussion in differential geometry, there is always a labyrinthof machinery needed before any non-trivial analysis can take place. Wepresent some of the standard and non-standard aspects of this in Chapter1. The chapter’s focus is to set the notational conventions used throughout,as well as provide supplementary material needed for future computations— particular for those in Chapters 2 and 3. Careful attention is paid to theconstruction of the connection and curvature on various bundles togetherwith some nonstandard aspects of the pullback bundle structure. We referthe reader to [Lee02,Lee97,Pet06,dC92,Jos08] as additional references withrespect to this background material.

In Chapter 2 we look at some classical results related to Harmonic mapheat-flow between Riemannian manifolds. The inclusion of this chapter servesas a gently introduction to the techniques of geometric analysis as well asprovides good motivation for the Ricci flow. Within, we present the con-vergence result of Eells and Sampson [ES64] with improvements made byHartman [Har67].

After establishing this, Chapter 3 introduces the Ricci flow as a geometricparabolic equation. Some basic properties of the flow are discussed followed bya detailed derivation of the associated evolution equations for the curvaturetensor and its various traces. Thereafter we give a brief survey of the spheretheorem of Huisken [Hui85], Nishikawa [Nis86] and Margerin [Mar86] togetherwith the algebraic decomposition of the curvature tensor.

Short-time existence for the Ricci flow is discussed in Chapter 4. We followthe approach first outlined by DeTurck [DeT83] which relates Ricci flow toRicci-DeTurck flow via a Lie derivative. A discussion on the ellipticity failureof the Ricci tensor due to the diffeomorphism invariance of the curvature isalso included.

Chapter 5 discusses the so-called Uhlenbeck trick, which simplifies the evol-ution equation of the curvature so that it can be written as a reaction-diffusiontype equation. This will motivate the development of the vector bundle max-imum principle of the next chapter. We present the original method first dis-cussed in [Ham86], and improved in [Ham93], which uses an abstract bundleand constructs an identification with the tangent bundle at each time. There-after we introduce a new method that looks to place a natural connection ona ‘spatial’ vector bundle over the space-time manifold M × R. We will buildupon this space-time construction in subsequent chapters.

In Chapter 6 we discuss the maximum principle for parabolic pde as a verypowerful tool central to our understanding of the Ricci flow. A new generalvector bundle version, for heat-type pde of section u ∈ Γ (E →M ×R) overthe space-time manifold, is discussed here. The stated vector bundle max-imum principle, Theorem 6.15 and the related Corollary 6.16, will providethe main tools for the convergence theory of the Ricci flow discussed in later

Preface vii

chapters. Emphasis is placed on the ‘vector field points into the set’ con-dition as it correctly generalises the null-eigenvector condition of Hamilton[Ham82b]. The related convex analysis necessary for the vector bundle max-imum principle is discussed in Appendix B — where we use the same con-ventions as that of the classic text [Roc70]. The maximum principle for sym-metric 2-tensors is also discussed as well as applications of the Ricci flow for3-manifolds.

The parabolic nature of the Ricci flow is further developed in Chapter 7where regularity and long-time existence is discussed. We see that the Ricciflow enjoys excellent regularity properties by deriving global Shi estimates[Shi89]. They are used to prove long-time existence soon thereafter.

Chapter 8 look at a compactness theorem for sequences of solutions tothe Ricci flow. The result originates in the convergence theory developed byCheeger and Gromov. We use the regularity of previous chapter to give aproof of the compactness theorem for the Ricci flow, given the compactnesstheorem for metrics; however, we will not give a proof of the general Cheeger-Gromov compactness result. It has natural applications in the analysis ofsingularities of the Ricci flow by ‘blow-up’ — which we will employ in theproof of the differentiable sphere theorem.

Chapters 9 and 10 aim to establish Perel’man’s local noncollapsing resultfor the Ricci flow [Per02]. This will provide a positive lower bound on theinjectivity radius for the Ricci flow under blow-up analysis. We also discussthe gradient flow formalism of the Ricci flow and Perel’man’s motivation fromphysics [OSW06,Car10].

The work of Bohm and Wilking [BW08], in which whole families of pre-served convex sets for the Ricci flow are derived from an initial one, is presen-ted in Chapters 11 and 12. Using this we will be able to argue, in conjunctionwith the vector bundle maximum principle, that solutions of the Ricci flowwhich have their curvature in a given initial cone will evolve to have constantcurvature as they approach their limiting time. Chapter 11 focuses on thealgebraic decomposition of the curvature and the required family of scaledtransforms. In particular we use the inherent Lie algebra structure (discussedin Appendix C) related to the curvature to elucidate the nature of the reac-tion term in the evolution equation for the curvature. The key result (The-orem 11.32) is that these transforms induce a change in the reaction termswhich does not depend on the Weyl curvature, and so can be computed en-tirely from the Ricci tensor and expressed in terms of its eigenvalues. Chapter12 uses these explicit eigenvalues to generate a one-parameter family of pre-served convex cones that are parameterised piecewise into two parts; one toaccommodate the initial behaviour of the cone, the other to accommodate therequired limiting behaviour. Thereafter we discuss the formulation of gener-alised pinching sets. The main result of this section, Theorem 12.7, providesthe existence of a pinching set simply from the existence of a suitable familyof cones.

viii Preface

In Chapter 13 we discuss the positive curvature condition on totally iso-tropic 2-planes, first introduced by Micallef and Moore [MM88], as a possibleinitial convex cone. We show, using ideas from [BS09a,Ngu08,Ngu10,AN07],that the positive isotropic curvature (PIC) condition is preserved by the Ricciflow; as is the positive complex sectional curvature (PCSC) condition. We alsogive a simplified proof that PIC is preserved by the Ricci flow by working dir-ectly with the complexification of the tangent bundle. In order to relate thePIC condition to the 1/4-pinching sphere theorem, we present the argumentof Brendle and Schoen [BS09a] that relates the 1/4-pinching condition withthe PIC condition on M ×R2. The result, i.e. Corollary 13.13, shows that Mis a compact manifold with pointwise 1/4-pinched sectional curvature, thenM × R2 has positive isotropic curvature.

Chapter 14 brings the discussion to a climax. Here we finally give a proofof the differentiable 1/4-pinching sphere theorem from the material presentedin earlier chapters. In the final section we outline a general convergence resultdue to Brendle [Bre08] which looks at the weaker condition of PIC on M×R.

A synopsis of the chapter progressions and inter-relationships is summar-ised by the following diagram:

−−→−−→

−−→ −−→−−→

−−→ −−→ −−→ −−→ −−→ −−→−−→ −−→

4 7

9 10 8

3 5 6 11 12 13 14

B C

Here the main argument is represented along the horizontal together withthe supplementary appendices. The regularity, existence theory and blow-upanalysis are shown above this.

This book grew from my honours thesis completed in 2008 at the Aus-tralian National University. I would like to express my deepest gratitudes tomy supervisor, Dr Ben Andrews, for without his supervision, assistance andimmeasurable input this book would not be possible. Finally, I would like tothank my parents for their continuous support and encouragement over theyears.

Christopher HopperJuly, 2009

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Notation and List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Background Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Metric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Connection Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.8 Pullback Bundle Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.9 Integration and Divergence Theorems . . . . . . . . . . . . . . . . . . . . . 31

2 Harmonic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 Global Existence of Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Harmonic Map Heat Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Evolution of the Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1 Introducing the Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 The Laplacian of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Metric Variation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 Evolution of the Curvature Under the Ricci Flow . . . . . . . . . . . 603.5 A Closer Look at the Curvature Tensor . . . . . . . . . . . . . . . . . . . 63

4 Short-Time Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 The Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 The Linearisation of the Ricci Tensor . . . . . . . . . . . . . . . . . . . . . 714.3 Ellipticity and the Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . 72

ix

x Contents

4.4 DeTurck’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Uhlenbeck’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Abstract Bundle Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Orthonormal Frame Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Time-Dependent Metrics and Vector Bundles over M × R . . . 87

6 The Weak Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.1 Elementary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2 Scalar Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3 Maximum Principle for Symmetric 2-Tensors . . . . . . . . . . . . . . . 1016.4 Vector Bundle Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . 1026.5 Applications of the Vector Bundle Maximum Principle . . . . . . 107

7 Regularity and Long-Time Existence . . . . . . . . . . . . . . . . . . . . . 1177.1 Regularity: The Global Shi Estimates . . . . . . . . . . . . . . . . . . . . . 1177.2 Long-Time Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8 The Compactness Theorem for Riemannian Manifolds . . . 1258.1 Different Notions of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 1258.2 Cheeger-Gromov Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.3 Statement of the Compactness Theorem . . . . . . . . . . . . . . . . . . . 1328.4 Proof of the Compactness Theorem for Flows . . . . . . . . . . . . . . 1358.5 Blowing Up of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9 The F-Functional and Gradient Flows . . . . . . . . . . . . . . . . . . . . 1399.1 Introducing the Gradient Flow Formulation . . . . . . . . . . . . . . . . 1399.2 Einstein-Hilbert Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419.3 F-Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1429.4 Gradient Flow of Fm and Associated Coupled Equations . . . . 144

10 The W-Functional and Local Noncollapsing . . . . . . . . . . . . . . . 14910.1 Entropy W-functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14910.2 Gradient Flow of W and Monotonicity . . . . . . . . . . . . . . . . . . . . 15110.3 µ-Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15310.4 Local Noncollapsing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15710.5 The Blow-Up of Singularities and Local Noncollapsing . . . . . . 16210.6 Remarks Concerning Perel’man’s Motivation From Physics . . 164

11 An Algebraic Identity for Curvature Operators . . . . . . . . . . . 16711.1 A Closer Look at Tensor Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 16811.2 Algebraic Curvature Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 17311.3 Decomposition of Algebraic Curvature Operators . . . . . . . . . . . 18111.4 A Family of Transformations for the Ricci flow . . . . . . . . . . . . . 187

Contents xi

12 The Cone Construction of Bohm and Wilking . . . . . . . . . . . . 19512.1 New Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19512.2 Generalised Pinching Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

13 Preserving Positive Isotropic Curvature . . . . . . . . . . . . . . . . . . 20713.1 Positive Isotropic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20813.2 The 1/4-Pinching Condition and PIC . . . . . . . . . . . . . . . . . . . . . 21013.3 PIC is Preserved by the Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . 21513.4 PCSC is Preserved by the Ricci Flow . . . . . . . . . . . . . . . . . . . . . 22213.5 Preserving PIC Using the Complexification . . . . . . . . . . . . . . . . 225

14 The Final Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22914.1 Proof of the Sphere Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22914.2 Refined Convergence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

A Gateaux and Frechet Differentiability . . . . . . . . . . . . . . . . . . . . . 241A.1 Properties of the Gateaux Derivative . . . . . . . . . . . . . . . . . . . . . . 242

B Cones, Convex Sets and Support Functions . . . . . . . . . . . . . . . 245B.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245B.2 Support Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245B.3 The Distance From a Convex Set . . . . . . . . . . . . . . . . . . . . . . . . . 246B.4 Tangent and Normal Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247B.5 Convex Sets Defined by Inequalities . . . . . . . . . . . . . . . . . . . . . . . 247

C Canonically Identifying Tensor Spaces with Lie Algebras . 251C.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251C.2 Tensor Spaces as Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252C.3 The Space of Second Exterior Powers as a Lie Algebra . . . . . . 252

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

xii Contents

Notation and List of Symbols

Γ kij Christoffel symbol of a connection ∇ w.r.t. a local frame (∂i).

(xi) A local coordinate chart for a neighbourhood U with a localchart x = (xi) : U → Rn.

Curv Algebraic curvature operators.I, id The identities on S2(

∧2V ∗) and S2(V ). N.B. I = id ∧ id.

inj Injectivity radius.∆g,h Harmonic map Laplacian w.r.t. the domain metric g and

codomain metric h.∧2V Second exterior power of the vector space V .

LX Lie derivative w.r.t. the vector field X.Met Space of metrics.NxA Normal cone to A at x.ν Unit outward normal.? Kulkarni-Nomizu product.f∇ Pullback connection on f∗E.f∗E Pullback bundle of E by f .ξf , (ξi)f Restriction of ξ, ξi ∈ Γ (E) to f .R, Rijk` Riemannian curvature tensor.Ric, Rij Ricci curvature tensor.Scal Scalar curvature tensor.Γ (E) The space of smooth sections of a vector bundle π : E →M .S Spatial tangent bundle.Sym2 T ∗M Symmetric (2, 0)-bundle over M .S2(U) Symmetric tensor space of U .T k` (V ) The set of all multilinear maps (V ∗)` × V k → R over V .T k` M (k, `)-tensor bundle over a manifold M .T k` (M) The space of (k, `)-tensor fields over M , i.e. Γ (⊗kT ∗M⊗`TM).TxA Tangent cone to A at x.dµ, dµ(g) Volume form with respect to a metric g.dσ Volume form on a hypersurface or boundary of a manifold.X (M) The space of vector fields, i.e. Γ (TM).

xiii

xiv Notation and List of Symbols

Introduction

The relationship between curvature and topology has traditionally been oneof the most popular and highly developed topics in Riemannian geometry. Inthis area, a central issue of concern is that of determining global topologicalstructures from local metric properties. Of particular interest to us the so-called pinching problem and related sphere theorems in geometry. We beginwith a brief overview of this problem, from Hopf’s inspiration to the latestdevelopments in Hamilton’s Ricci flow.

Manifolds with Constant Sectional Curvature

One of the earliest insights into the relationship between curvature and to-pology is the problem of classifying complete Riemannian manifolds withconstant sectional curvature, referred to as space forms. In the late 1920sHeinz Hopf studied the global properties of such space forms and proved, inhis PhD dissertation [Hop25] (see also [Hop26]), the following:

Theorem (Uniqueness of Constant Curvature Metrics). Let M be acomplete, simply-connected, n-dimensional Riemannian manifold with con-stant sectional curvature. Then M is isometric to either Rn, Sn or Hn.

Furthermore, if the manifold is compact then the space forms are compactquotients of the either Rn, Sn or Hn. Placing this result on solid ground wasone of Hopf’s tasks during the 1930s, however the classification is still incom-plete (the categorisation of hyperbolic space quotients has been extremelyproblematic).

Given these developments, curiosity permits one to ask if a similar resultwould hold under a relaxation of the curvature hypothesis. In other words, as-suming a compact manifold has a sectional curvature ‘varying not too much’(we will later say the manifold is ‘pinched’), can one deduce that the under-lying manifold is topologically (one would even hope differentiably) identicalto one of the above space forms? After rescaling the metric there are three

xv

xvi Introduction

cases: the pinching problem around κ0 = +1, 0,−1. Therefore if the sectionalcurvature K satisfies |K−κ0| ≤ ε, the question becomes one of finding an op-timal ε > 0 in which the manifold is identical (in some sense) to a particularspace form.

For our purposes, the question of interest is that of positive pinchingaround κ0 = 1; our subsequent discussion will focus entirely on this case. Theproblem has enjoyed a great deal interest over the years due to its historicalimportance both in triggering new results and as motivation for creating newmathematical tools.

The Topological Sphere Theorem

The only simply connected manifold of constant positive sectional curvatureis, by the above theorem, the sphere. A heuristic sense of continuity leadsone to hope that if the sectional curvature of a manifold is close to a positiveconstant, then the underlying manifold will still be a sphere. Hopf himselfrepeatedly put forward this problem, in particular when Harry Rauch (ananalyst and expert in Riemannian surfaces) visited him in Zurich through-out 1948-49. Rauch was so enthusiastic about Hopf’s pinching that, back atthe Institute for Advanced Study in Princeton, he finally managed to proveHopf’s conjecture with a pinching constant of roughly δ ≈ 3/4. SpecificallyRauch [Rau51] proved:

Theorem (Rauch, 1951). Let (Mn, g) be a complete Riemannian manifoldwith n ≥ 2. If the sectional curvature K(p,Π) (where p ∈ M and Π is a 2-dimensional plane through p) satisfies

δk ≤ K(p,Π) ≤ k − ε

for some constant k > 0, some ε > 0, all p ∈M ; and all planes Π, where δ ≈3/4 is the root of the equation sinπ

√δ =√δ/2. Then the simply connected

covering space of M is homeomorphic to the n-dimensional sphere Sn. Inparticular if M is simply connected, then M is homeomorphic to Sn.

Rauch’s paper is seminal in two respects. First of all, he was able to controlthe metric on both sides. Secondly, to get a global result, he made a subtlegeometric study in which he proved that (under the pinching assumption)one can build a covering of the manifold from the sphere.

Thereafter Klingenberg [Kli59] sharpened the constant — in the even di-mensional case — to the solution of sinπ

√δ =√δ (i.e. δ ≈ 0.54). Crucially,

the precise notion of an injectivity radius was introduced to the pinchingproblem. Using this, Berger [Ber60a] improved Klingenbergs result, underthe even dimension assumption, with δ = 1/4. Finally, Klingenberg [Kli61]extended the injectivity radius lower estimate to odd dimensions. This provedthe sphere theorem in its entirely. The resulting quarter pinched sphere the-orem is stated as follows:

Introduction xvii

Theorem (Rauch-Klingenberg-Berger Sphere Theorem). If a simplyconnected, complete Riemannian manifold has sectional curvature K satis-fying

1

4< K ≤ 1,

then it is homeomorphic to a sphere.1

The theorem’s pinching constant is optimal since the conclusion is false ifthe inequality is no longer strict. The standard counterexample is complexprojective space with the Fubini-Study metric (sectional curvatures of thismetric take on values between 1/4 and 1 inclusive). Other counterexamplesmay be found among the rank one symmetric spaces (see e.g. [Hel62]).

As a matter of convention, we say a manifold is strictly δ-pinched in theglobal sense if 0 < δ < K ≤ 1 and weakly δ-pinched in the global sense if0 < δ ≤ K ≤ 1. In which case the sphere theorem can be reformulated as:‘Any complete, simply-connected, strictly 1/4-pinched Riemannian manifoldis homeomorphic to the sphere.’

Remarks on the Classical Proof. The proof from the 1960s consistedin arguing that the manifold can be covered with only two topological balls(e.g. see [AM97, Sect. 1]). The proof relies heavily on classical comparisontechniques.

Idea of Proof. Choose points p and q in such a way that the distance d(p, q)is maximal, that is d(p, q) = diam(M). The key property of such a pair ofpoints is that for any unit tangent vector v ∈ TqM , there exists a minimisinggeodesic γ from q to p making an acute angle with v. From this one can applyToponogov’s triangle comparison theorem [Top58] (see also [CE08, Chap. 2])and Klingenberg’s injectivity radius estimates to show that

M = B(p, r) ∪B(q, r)

for π/2√δ < r < π and 1/4 < δ ≤ K ≤ 1. In other words, under the

hypothesis of the sphere theorem, we can write M as a union of two balls. Onethen concludes, by classical topological arguments, that M is homeomorphicto the sphere.

Manifolds with Positive Isotropic Curvature. One cannot overlookthe contributions made by Micallef and Moore [MM88] in generalising theclassical Rauch-Klingenberg-Berger sphere theorem. By introducing the so-called positive isotropic curvature condition to the problem together withharmonic map theory, they managed to prove the following:

1 In the literature some quote the sectional curvature as taking values in the interval(1, 4]. This is only a matter of scaling. For instance, if the metric is scaled by a factorλ then the sectional curvatures are scaled by 1/λ. Thus one can adjust the maximumprincipal curvature to 1, say. What is important about the condition is that the ratioof minimum to maximum curvature remains greater than 1/4.

xviii Introduction

Theorem (Micallef and Moore, 1988). Let M be a compact simplyconnected n-dimensional Riemannian manifold which has positive curvatureon totally isotropic 2-planes, where n ≥ 4. Then M is homeomorphic to asphere.

This is achieved as follows: Firstly, for a n-dimensional Riemannian man-ifold M with positive curvature on totally isotropic two-planes one can showthat any nonconstant conformal harmonic map f : S2 →M has index at leastn−3

2 . The proof uses classical results of Grothendieck on the decompositionof holomorphic bundles over S2. Secondly, one can show: If M is a compactRiemannian manifold such that πk(M) 6= 0, where k ≥ 2, then there ex-ists a nonconstant harmonic 2-sphere in M of index ≤ k − 2. This fact is amodification of the Sacks-Uhlenbeck theory of minimal 2-spheres in Rieman-nian manifolds. From this, the above theorem easily follows by using dualitywith these two results, and using the higher-dimensional Poincare conjecture,which was proved for n ≥ 5 by Smale and for n = 4 by Freedman.

A refined version of the classical sphere theorem now follows by repla-cing the global pinching hypotheses on the curvature by a pointwise one.In particular we say that a manifold M is strictly δ-pinched in the point-wise sense if 0 < δK(Π1) < K(Π2) for all points p ∈ M and all 2-planesΠ1, Π2 ⊂ TpM . Furthermore, we say M is weakly δ-pinched in the point-wise sense if 0 ≤ δK(Π1) ≤ K(Π2) for all point p ∈ M and all 2-planesΠ1, Π2 ⊂ TpM . In which case it follows from Berger’s Lemma (see Sec-tion 1.7.7) that any manifold which is strictly 1/4-pinched in the pointwisesense has positive isotropic curvature (and so is homeomorphic to Sn).

A Question of Optimality. Given the above stated Rauch-Klingenberg-Berger sphere theorem, a natural question to ask is: Can the homeomorphiccondition can be replaced by a diffeomorphic one? In answering this, thereare some dramatic provisos. The biggest concerns the method used in theabove classical proof — as the argument presents M as a union of two ballsglued along their common boundary so that M is homeomorphic to Sn.However, Milnor [Mil56] famously showed that there are ‘exotic’ structureson S7, namely:

Theorem (Milnor, 1956). There are 7-manifolds that are homeomorphicto, but not diffeomorphic to, the 7-sphere.2

Moreover, some exotic spheres are precisely obtained by gluing two halfspheres along their equator with a weird identification. Hence the above clas-sical proof cannot do better since it does not allow one to obtain a diffeo-morphism between M and Sn in general. As a result, it remained an openquestion as to whether or not the sphere theorem’s conclusion was optimalin this sense.

2 In fact Milnor and Kervaire [KM63] showed that S7 has exactly 28 non-diffeomorphic smooth structures.

Introduction xix

To add to matters, it has also been a long-standing open problem as towhether there actually are any concrete examples of exotic spheres with pos-itive curvature. This has now apparently been resolved by Petersen and Wil-helm [PW08] who show there is a metric on the Gromoll-Meyer Sphere withpositive sectional curvature.

The Differentiable Sphere Theorem

There have been many attempts at proving the differentiable version, most ofwhich have been under sub-optimal pinching assumptions. The first attemptwas by Gromoll [Gro66] with a pinching constant δ = δ(n) that dependedon the dimension n and converged to 1 as n goes to infinity. The result for δindependent of n was obtained by Sugimoto, Shiohama, and Karcher [SSK71]with δ = 0.87. The pinching constant was subsequently improved by Ruh[Ruh71,Ruh73] with δ = 0.80 and by Grove, Karcher and Ruh [GKR74] withδ = 0.76. Furthermore, Ruh [Ruh82] proved the differentiable sphere theoremunder a pointwise pinching condition with a pinching constant converging to1 as n→∞.

The Ricci Flow. In 1982 Hamilton introduced fundamental new ideasto the differentiable pinching problem. His seminal work [Ham82b] studiedthe evolution of a heat-type geometric evolution equation:

∂

∂tg(t) = −2 Ric(g(t)) g(0) = g0,

herein referred to as the Ricci flow. This intrinsic geometric flow has, over theyears, served to be an invaluable tool in obtaining global results within thedifferentiable category. The first of which is following result due to Hamilton[Ham82b].

Theorem (Hamilton, 1982). Suppose M is a simply-connected compactRiemannian 3-manifold with strictly positive Ricci curvature. Then M isdiffeomorphic to S3.

Following this, Hamilton [Ham86] developed powerful techniques to analysethe global behaviour of the Ricci flow. This enabled him to extend his previousresult to 4-manifolds by showing:

Theorem (Hamilton, 1986). A compact 4-manifold M with a positivecurvature operator is diffeomorphic to the sphere S4 or the real projectivespace RP 4.

Note that we say a manifold has a positive curvature operator if the eigen-values if R are all positive, or alternatively R(φ, φ) > 0 for all 2-forms φ 6= 0(when considering the curvature operator as a self-adjoint operator on two-forms).

xx Introduction

Following this, Chen [Che91] show that the conclusion of Hamilton’s holdsunder the weaker assumption that the manifold has a 2-positive curvatureoperator. Specifically, he proved that: If M is a compact 4-manifold with a2-positive curvature operator, then M is diffeomorphic to either S4 or RP 4.Moreover, Chen showed that the 2-positive curvature condition is implied bypointwise 1/4-pinching. In which case he managed to show:

Theorem (Chen, 1991). If M4 is a compact pointwise 1/4-pinched 4-manifold, then M is either diffeomorphic to the sphere S4 or the real pro-jective space RP 4, or isometric to the complex projective space CP 2.

Note that we say a manifold has a 2-positive curvature operator if the sumof the first two eigenvalues of R are positive.3 It is easy to see that whendimM = 3, the condition of positive Ricci and 2-positive curvature operatorare equivalent (in fact, 2-positive implies positive Ricci in n-dimensions).

However despite this progress it still remained an open problem for overa decade as to whether or not these results could be extended for manifoldsM with dimM ≥ 4.

Ricci Flow in Higher Dimensions. Huisken [Hui85] was one of the firstto study the Ricci flow on manifolds of dimension n ≥ 4. His analysis focuseson decomposing the curvature tensor into Rijk` = Uijk` + Vijk` + Wijk`,where Uijk` denotes the part of the curvature tensor associated with thescalar curvature, Vijk` is the part of the curvature associated with the trace-free Ricci curvature, and Wijk` denotes the Weyl tensor. By following thetechniques outlined in [Ham82b], Huisken showed that if the scalar-curvature-free part of the sectional curvature tensor is small compared to the scalarcurvature, then the same evolution equation for the curvature yields the sameresult — a deformation to a constant positive sectional curvature metric.

Theorem (Huisken, 1985). Let n ≥ 4. If the curvature tensor of a smoothcompact n-dimensional Riemannian manifold of positive scalar curvature sat-isfies

|W |2 + |V |2 < δn|U |2

with δ4 = 15 , δ5 = 1

10 , and δn = 2(n−2)(n+1) for n ≥ 6, then the Ricci flow has

a solution g(t) for all times 0 ≤ t <∞ and g(t) converges to a smooth metricof constant positive curvature in the C∞-topology as t→∞.

There are also similar results by Nishikawa [Nis86] and Margerin [Mar86]with more recent sharp results by Margerin [Mar94a,Mar94b] as well.

In 2006, Bohm and Wilking [BW08] managed to generalise Chen’s work infour dimensions by constructed a new family of cones (in the space of algebraiccurvature operators) which is invariant under Ricci flow. This allowed themto prove:

3 Equivalently one could also say that R(φ, φ) +R(ψ,ψ) > 0 for all 2-forms φ and ψsatisfying |φ|2 = |ψ|2 and 〈φ, ψ〉 = 0.

Introduction xxi

Theorem (Bohm and Wilking, 2006). On a compact manifold Mn theRicci flow evolves a Riemannian metric with 2-positive curvature operator toa limit metric with constant sectional curvature.

Their paper [BW08] overcomes some major technical problems, particularthose related to controlling the Weyl part of the curvature.

Soon thereafter, Brendle and Schoen [BS09a] finally managed to prove:

Theorem (Brendle and Schoen, 2007). LetM be a compact Riemannianmanifold of dimension n ≥ 4 with sectional curvature strictly 1/4-pinched inthe pointwise sense. Then M admits a metric of constant curvature andtherefore is diffeomorphic to a spherical space form.

The key novel step in the proof is to show that nonnegative isotropic curvatureis invariant under the Ricci flow. This was also independently proved byNguyen [Ngu08,Ngu10]. Brendle and Schoen obtain the abovementioned res-ult by working with the nonnegative isotropic curvature condition on M×R2,which is implied by pointwise 1/4-pinching. We will examine this proof byBrendle and Schoen [BS09a] in Chapter 13. In Chapters 11 and 12 we willlook at the methods of Bohm and Wilking [BW08] in detail with the specificaim to proving the differentiable pointwise 1/4-pinching sphere theorem forn ≥ 4.

Finally, in Chapter 14 we survey the following result of Brendle [Bre08]which generalises the earlier results of [Ham82b,Hui85,Che91,BW08,BS09a].

Theorem (Brendle, 2008). Let (M, g0) be a compact Riemannian man-ifold of dimension n ≥ 4 such that M × R has positive isotropic curvature.Then there is a unique maximal solution g(t), t ∈ [0, T ) to the Ricci flowwith initial metric g0 which converges to a metric of constant curvature. Inparticular, M is diffeomorphic to a spherical space form.

Where to Next?

There is a big problem crying out for an application of Ricci flow:

If (M, g0) is a simply connected compact Riemannian manifold with pos-itive curvature on isotropic 2-planes, is M diffeomorphic to a sphericalspace form? 4

The difficulties are manifold: First, it is very difficult to bring in the conditionof simple-connectedness (a global condition) to the analysis of the Ricci flow(which is locally defined). If the assumption is dropped, then there are manymore manifolds which can appear and which have positive isotropic curvature,notably the product Sn−1×S1, for which the universal cover is not compact

4 Recall that Micallef and Moore [MM88] proved that such manifolds are homeo-morphic to Sn.

xxii Introduction

(and is not a sphere!). Second, singularities can happen in the Ricci flow ofsuch metrics where the curvature operators do not approach those of constantsectional curvature (in the above example, the Sn−1 factor shrinks while theS1 factor does not).

Hamilton [Ham97] used a surgery argument to analyse the situation infour dimensions. His method is to show that blow-ups at singularities lookclose to either a shrinking cylinder or a shrinking sphere. He stops the Ricciflow just before a singularity, and cuts all the cylindrical ‘necks’, pasting insmooth caps on the ‘stumps’. Then he continues the Ricci flow again.5 In thisway he proves that the original manifold must have been a connected sum ofspheres with S3 × S1 (or quotients of these).

The question is: Can the algebraic methods of Bohm and Wilking [BW08]be used — with the preservation of positive isotropic curvature as presentedin Chapter 13 — to prove a similar result for manifolds with positive isotropiccurvature in higher dimensions? Doing so would in the best scenario provethe following conjecture of Schoen:

Conjecture (Schoen, [CTZ08,Sch07]). For n ≥ 4, letM be an n-dimensionalcompact Riemannian manifold with positive isotropic curvature. Then a fi-nite cover of M is diffeomorphic to Sn, Sn−1 × S1 or a connected sum ofthese. In particular, the fundamental group of M is virtually free.

5 This argument was an inspiration for Perel’man’s later use of Ricci flow to provethe Poincare and Geometrisation conjectures. A correction to Hamilton’s argumentwas later published by Zhu and Chen [CZ06].

Chapter 1

Background Material

This chapter establishes the notational conventions used throughout whilealso providing results and computations needed for later analyses. Readersfamiliar with differential geometry may wish to skip this chapter and referback when necessary.

1.1 Smooth Manifolds

Suppose M is a topological space. We say M is a n-dimensional topologicalmanifold if it is Hausdorff, second countable and is ‘locally Euclidean ofdimension n’ (i.e. every point p ∈M has a neighbourhood U homeomorphicto an open subset of Rn). A coordinate chart is a pair (U, φ) where U ⊂ Mis open and φ : U → φ(U) ⊂ Rn is a homeomorphism. If (U, φ) and (V, ψ)are two charts, the composition ψ φ−1 : φ(U ∩ V )→ ψ(U ∩ V ) is called thetransition map from φ to ψ. It is a homeomorphism since both φ and ψ are.

In order for calculus ideas to pass to the setting of manifolds we needto impose an extra smoothness condition on the chart structure. We say twocharts (U, φ) and (V, ψ) are smoothly compatible if the transition map ψφ−1,as a map between open sets of Rn, is a diffeomorphism.

We define a smooth atlas for M to be a collection of smoothly compatiblecharts whose domains cover M . We say two smooth atlases are compatible iftheir union is also a smooth atlas. As compatibility is an equivalence relation,we define a differentiable structure for M to be an equivalence class of smoothatlases. Thus a smooth manifold is a pair (M,A ) where M is a topologicalmanifold and A is a smooth differentiable structure for M . When there isno ambiguity, we usually abuse notation and simply refer to a ‘differentiablemanifold M ’ without reference to the atlas. From here on, manifolds willalways be of the differentiable kind.

1.1.1 Tangent Space. There are various equivalent ways of defining thetangent space of a manifold. For our purposes, we emphasise the constructionof the tangent space as derivations on the algebra C∞(M).

1

2 1 Background Material

Definition 1.1. Let M be a smooth manifold with a point p. A R-linear mapX : C∞(M) → R is called a derivation at p if it satisfies the Leibniz rule:X(fg) = f(p)Xg+ g(p)Xf . The tangent space at p, denoted by TpM , is theset of all derivations at p.

The tangent space TpM is clearly a vector space under the canonical op-erations (X + Y )f = Xf + Y f and (λX)f = λ (Xf) where λ ∈ R. In factTpM is of finite dimension and isomorphic to Rn. By removing the pointwisedependence in the above definition, we define:

Definition 1.2. A derivation is an R-linear map Y : C∞(M) → C∞(M)which satisfies the Leibniz rule: Y (fg) = f Y g + g Y f .

We identify such derivations with vector fields on M (see Remark 1.10 below).

Remark 1.3. In the setting of abstract algebra, a derivation is a functionon an algebra which generalises certain features of the derivative operator.Specifically, given an associative algebra A over a ring or field R, a R-derivation is a R-linear map D : A → A that satisfies the product rule:D(ab) = (Da)b + a(Db) for a, b ∈ A . In our case, the algebra A = C∞(M)and the field is R.

1.2 Vector Bundles

Definition 1.4. Let F and M be smooth manifolds. A fibre bundle over Mwith fibre F is a smooth manifold E, together with a surjective submersionπ : E → M satisfying a local triviality condition: For any p ∈ M thereexists an open set U in M containing p, and a diffeomorphism φ : π−1(U)→U × F (called a local trivialization) such that π = π1 φ on π−1(U), whereπ1(x, y) = x is the projection onto the first factor. The fibre at p, denotedEp, is the set π−1(p), which is diffeomorphic to F for each p.

Although a fibre bundle E is locally a product U×F , this may not be trueglobally. The space E is called the total space, M the base space and π theprojection. Occasionally we refer to the bundle by saying: ‘let π : E → Mbe a (smooth) fibre bundle’. In most cases the fibre bundles we considerwill be vector bundles in which the fibre F is a vector space and the localtrivializations induce a well-defined linear structure on Ep for each p:

Definition 1.5. Let M be a differentiable manifold. A smooth vector bundleof rank k over M is a fibre bundle π : E →M with fibre Rk, such that

1. The fibres Ep = π−1(p) have a k-dimensional vector space structure.2. The local trivializations φ : π−1(U)→ U ×Rk are such that π2 φ|Ep is

a linear isomorphism for each p ∈ U , where π2(x, y) = y.

1.2 Vector Bundles 3

One of the most fundamental vector bundles over a manifold M is thetangent bundle TM =

⋃p∈M TpM . It is a vector bundle of rank equal to

dimM . Other examples include the tensor bundles constructed from TM(see Section 1.3.3).

Definition 1.6. A section of a fibre bundle π : E → M is a smooth mapX : M → E, written p 7→ Xp, such that π X = idM . If E is a vector bundlethen the collection of all smooth sections over M , denoted by Γ (E), is a realvector space under pointwise addition and scalar multiplication.

Definition 1.7. A local frame for a vector bundle E of rank k is a k-tuple(ξi) of pointwise linearly independent sections of E over open U ⊂ M , thatis ξ1, . . . , ξk ∈ Γ (E

∣∣U

).

Given such a local frame, any section α of E over U can be written in theform

∑ki=1 α

iξi, where αi ∈ C∞(U). Local frames correspond naturally to

local trivializations, since the map (p,∑ki=1 α

iξi(p)) 7→ (p, α1(p), . . . , αk(p))is a local trivialization, while the inverse images of a standard basis in alocal trivialization defines a local frame. Moreover we recall the local framecriterion for smoothness of sections.

Proposition 1.8. A section α ∈ Γ (E|U ) is a smooth if and only if its com-ponent functions αi, with respect to (ξi), on U are smooth.

Remark 1.9. In fact Γ (E) is a module over the ring C∞(M) since for eachX ∈ Γ (E) we define fX ∈ Γ (E) by (fX)(p) = f(p)X(p). For instance thespace of sections of any tensor bundle is a module over C∞(M).1

Remark 1.10. There is an important identification between derivations (as inDefinition 1.2) and smooth sections of the tangent bundle:

Proposition 1.11. Smooth sections of TM → M are in one-to-one corres-pondence with derivations of C∞(M).

Proof. If X ∈ Γ (TM) is a smooth section, define a derivation X : f → Xfby (Xf)(p) = Xp(f). Conversely, given a derivation Y : C∞(M) → C∞(M)define a section Y of TM by Yp f = (Yf)(p). An easy exercise shows thatthis section is smooth.

As a result, we can either think of a smooth section X ∈ Γ (TM) as a smoothmap X : M → TM with X π = idM or as a derivation — that is, a R-linearmap X : C∞(M) → C∞(M) that satisfies the Leibniz rule. We call such anX a vector field and let the set of vector fields be denoted by X (M).

1 Recall that a module over a ring generalises the notion of a vector space. Instead ofrequiring the scalars to lie in a field, the ‘scalars’ may lie in an arbitrary ring. Formallya left R-module over a ring R is an Abelian group (G,+) with scalar multiplication· : R×G→ G that is associative and distributive.


1.2.1 Subbundles.

Definition 1.12. For a vector bundle π : E → M , a subbundle of E is avector bundle E′ over M with an injective vector bundle homomorphismi : E′ → E covering the identity map on M (so that πE i = πE′ , where πEand πE′ are the projections on E and E′ respectively).

The essential idea of a subbundle of a vector bundle E → M is that itshould be a smoothly varying family of linear subspaces E′p of the fibres Epthat constitutes a vector bundle in their own right. However it is convenientto distinguish sections of the subbundle from sections of the larger bundle,and for this reason we use the definition above. One can think of the map ias an inclusion of E′ into E.

Example 1.13. Let f : M → N be a smooth immersion between manifolds.The pushforward f∗ : TM → TN (ref. Section 1.8.2) over f : M → N inducesa vector bundle mapping i : TM → f∗(TN) over M . On fibres over p ∈ Mthis is the map f∗|p : TpM → Tf(p)N = (f∗TN)p which is injective since fis an immersion. Hence, i exhibits TM as a subbundle of f∗TN over M .

It is also useful to note, using a rank type theorem, that:

Proposition 1.14. If f : E → E′ is a smooth bundle surjection over M .Then there exists a subbundle j : E0 → E such that j(E0(p)) = ker(f |p) foreach p ∈M .

In which case we have a well-defined subbundle E0 = ker f inside E.

1.2.2 Frame Bundles. For a vector bundle π : E →M of rank k, thereis an associated fibre bundle over M with fibre GL(k) called the generallinear frame bundle F (E). The fibre F (E)x over x ∈M consists of all linearisomorphisms Y : Rk → Ex, or equivalently the set of all ordered bases forEx (by identifying the map Y with the basis (Ya), where Ya = Y (ea) fora = 1, . . . , k). The group GL(k) acts on each fibre by composition, so thatA ∈ GL(k) acts on a frame Y : Rk → Ex to give Y A = Y A : Rk →Ex (alternatively, the basis (Ya) ∈ F (E)x maps to the basis (AabYa)). Fromstandard linear algebra, this action

GL(k)× F (E)→ F (E); (A, Y ) 7→ Y A

is simply transitive on each fibre (that is, for any Y,Z ∈ F (E)x there existsa unique A ∈ GL(k) such that Y A = Z).

Note that a local trivialization φ : π−1(U) → U × Rk for E, togetherwith a local chart η : U → Rn for M , produces a chart for E compat-ible with the bundle structure: We take (x, v) 7→ φ(x, v) = (x, π2φ(x, v)) 7→(η(x), π2φ(x, v)) ∈ Rn × Rk. Any such chart also produces a chart forF (E) giving it the structure of a manifold of dimension n + k2: We take

(x, Y ) 7→ (η(x), π2 φx Y ) ∈ Rn × GL(k) ⊂ Rn+k2

, where φx(·) = φ(x, ·).

1.3 Tensors 5

Similarly a local trivialization of F (E) is defined by (x, Y ) 7→ (x, π2 φx Y ),giving F (E) the structure of a fibre bundle with fibre GL(k) as claimed.

If the bundle E is equipped with a metric g (ref. Section 1.4.4) — so thatEx is an inner product space — then one can introduce the orthonormalframe bundle O(E). Specifically O(E) is the subset of F (E) defined by

O(E) = Y ∈ F (E) : g(Ya, Yb) = δab.

The orthogonal group O(k) acts on O(E) by

O(k)×O(E)→ O(E); (O, Y ) 7→ Y O

where Y O(u) = Y (Ou) for each u ∈ Rk and O(E) is a fibre bundle over Mwith fibre O(k).

Remark 1.15. A local frame for E consists of k pointwise linearly independentsmooth sections of E over an open set U , say p 7→ ξi(p) ∈ Ep for i = 1, . . . , k.This corresponds to a section Y of F (E) over U , defined by Yp(u

iei) = uiξi(p)for each p ∈ U and u ∈ Rk.

1.3 Tensors

Let V be a finite dimensional vector space. A covariant k-tensor on V isa multilinear map F : V k → R. Similarly, a contravariant `-tensor is amultilinear map F : (V ∗)` → R. A mixed tensor of type

(k`

)or (k, `) is a

multilinear map

F : V ∗ × · · · × V ∗︸︷︷︸` times

×V × · · · × V︸︷︷︸k times

→ R.

We denote the space of all k-tensors on V by T k(V ), the space of contravariant`-tensors by T`(V ), and the space of all mixed (k, `) tensors by T k` (V ).

The following canonical isomorphism is frequently useful:

Lemma 1.16. The tensor space T 11 (V ) is canonically isomorphic to End(V ),

where the (bases independent) isomorphism Φ : End(V )→ T 11 (V ) is given by

Φ(A) : (ω,X) 7−→ ω(A(X))

for all A ∈ End(V ), ω ∈ V ∗, and X ∈ V .

Remark 1.17. Alternatively one could state this lemma by saying: If V andW are vector spaces then V ⊗ W ∗ ' End(W,V ) where the isomorphismΨ : V ⊗W ∗ → End(W,V ) is given by Ψ(v ⊗ ξ) : w → ξ(w)v.

A general version of this identification is expressed as follows.


Lemma 1.18. The tensor space T k`+1(V ) is canonically isomorphic to the

space Mult((V ∗)`×V k, V ), where the isomorphism Φ : Mult((V ∗)`×V k, V )→T k`+1(V ) is given by

Φ(A) : (ω0, ω1, . . . , ω`, X1, . . . , Xk) 7−→ ω0(A(ω1, . . . , ω`, X1, . . . , Xk))

for all A ∈ Mult((V ∗)` × V k, V ), ωi ∈ V ∗, and Xj ∈ V .2

1.3.1 Tensor Products. There is a natural product that links the varioustensor spaces over V . If F ∈ T k` (V ) and G ∈ T pq (V ), then the tensor product

F ⊗G ∈ T k+p`+q (V ) is defined to be

(F ⊗G)(ω1, . . . , ω`+q, X1, . . . , Xk+p)

= F (ω1, . . . , ω`, X1, . . . , Xk)G(ω`+1, . . . , ω`+q, Xk+1, . . . , Xk+p).

Moreover, if (e1, . . . , en) is a basis for V and (ϕ1, . . . , ϕn) is the correspondingdual basis, defined by ϕi(ej) = δij , then it can be shown that a basis for T k` (V )takes the form

ej1 ⊗ · · · ⊗ ej` ⊗ ϕi1 ⊗ · · · ⊗ ϕik ,

where

ej1 ⊗ · · · ⊗ ej` ⊗ ϕi1 ⊗ · · · ⊗ ϕik(ϕs1 , . . . , ϕs` , er1 , . . . , erk)

= δs1j1 · · · δs`j`δi1r1 · · · δ

ikrk.

Therefore any tensor F ∈ T k` (V ) can be written, with respect to this basis,as

F = F j1,...,j`i1,...,ikej1 ⊗ · · · ⊗ ej` ⊗ ϕi1 ⊗ · · · ⊗ ϕik

where F j1,...,j`i1,...,ik = F (ϕj1 , . . . , ϕj` , ei1 , . . . , eik).

1.3.2 Tensor Contractions. A tensor contraction is an operation on oneor more tensors that arises from the natural pairing of a (finite-dimensional)vector space with its dual.

Intuitively there is a natural notion of ‘the trace of a matrix’ A = (Aij) ∈Matn×n(R) given by trA =

∑iA

ii. It is R-linear and commutative in the

sense that trAB = trBA. From the latter property, the trace is also cyclic(in the sense that trABC = trBCA = trCAB). Therefore the trace issimilarity-invariant, which means for any P ∈ GL(n) the trace trP−1AP =trPP−1A = trA. Whence we can extend tr over End(V ) by taking thetrace of a matrix representation — this definition is basis independent sincedifferent bases give rise to similar matrices — and so by Lemma 1.16 tr canact on tensors as well.

2 Here Mult((V ∗)` × V k, V ) is the set of multilinear maps from (V ∗)` × V k to V .

1.3 Tensors 7

Naturally, we define the contraction of any F ∈ T 11 (V ) by taking the trace

of F as a linear map in End(V ). In which case tr : T 11 (V ) → R is given

by trF = F (ϕi, ei) =∑i F

ii , since Φ−1(F ) = (F (ϕi, ej))

ni,j=1 ∈ End(V ). In

general we definetr : T k+1

`+1 (V )→ T k` (V )

by

(trF )(ω1, . . . , ω`, X1, . . . , Xk) = tr(F (ω1, . . . , ω`, · , X1, . . . , Xk, · )

).

That is, we define (trF )(ω1, . . . , ω`, X1, . . . , Xk) to be the trace of the endo-morphism F (ω1, . . . , ω`, · , X1, . . . , Xk, · ) ∈ T 1

1 (V ) ' End(V ). In compon-ents this is equivalent to

(trF )j1...j`i1...ik = F j1...j`mi1...ikm.

It is clear that the contraction is linear and lowers the rank of a tensor by 2.Unfortunately there is no general notation for this operation! So it is best toexplicitly describe the contraction in words each time it arises. We give somesimple examples of how this might occur.

There are several variations of tr: Firstly, there is nothing special aboutwhich particular component pairs the contraction is taken over. For instanceif F = F j

i kϕi ⊗ ej ⊗ ϕk ∈ T 2

1 (V ) then one could take the contraction of F

over the first two components: (tr12 F )k = trF (·, ·, ek) = F ii k or the last two

components: (tr23 F )k = trF (ek, ·, ·) = F ik i.

Another variation occurs if one wants to take the contraction over multiplecomponent pairs. For example if F = F k`

ij ϕi ⊗ ϕj ⊗ ek ⊗ e` ∈ T 22 (V ), one

could take the trace over the 1st and 3rd with the 2nd and 4th so thattrF = tr13 tr24 F = trF (?, ·, ?, ·) = F pq

pq .Furthermore, if one has a metric g then it is possible to take contractions

over two indices that are either both vectors or covectors. This is done bytaking a tensor product with the metric tensor (or its inverse) and contractingeach of the two indices with one of the indices of the metric. This operationis known as metric contraction (see Section 1.4.3 for further details).

Finally, one of the most important applications arises when F = ω ⊗X ∈T 1

1 (V ) for some (fixed) vector X and covector ω. In this case

trF = F (∂i, dxi) = ω(∂i)X(dxi) = ωiX

i = ω(X).

The idea can be extended as follows: If F ∈ T k` (V ) with ω1, . . . , ω` ∈ V ∗ andX1, . . . , Xk ∈ V (fixed) then

F ⊗ ω1 ⊗ · · · ⊗ ω` ⊗X1 ⊗ · · · ⊗Xk ∈ T 2k2` (V ).

So the contraction of this tensor over all indexes becomes


tr (F ⊗ ω1 ⊗ · · · ⊗ ω` ⊗X1 ⊗ · · · ⊗Xk)

= ωj1 · · ·ωj`Fj1···j`

i1···ikXi1 · · ·Xik

= F (ω1, . . . , ω`, X1, . . . , Xk). (1.1)

1.3.3 Tensor Bundles and Tensor Fields. For a manifold M we canapply the above tensor construction pointwise on each tangent space TpM .In which case a (k, `)-tensor at p ∈M is an element T k` (TpM). We define thebundle of (k, `)-tensors on M by

T k` M =⋃p∈M

T k` (TpM) =⋃p∈M⊗kT ∗pM ⊗` TpM.

In particular, T1M = TM and T 1M = T ∗M . An important subbundle ofT 2M is Sym2 T ∗M , the space of all symmetric (2, 0)-tensors on M . A (k, `)-tensor field is an element of Γ (T k` M) = Γ (⊗kT ∗M ⊗` TM) — we sometimesuse the notation T k

` (M) as a synonym for Γ (T k` M).To check that T k` M is a vector bundle, let π : T k` M → M send F ∈

T k` (TpM) to the base point p. If (xi) is a local chart on open U ⊂M aroundpoint p, then any tensor F ∈ T k` (TpM) can be expressed as

F = F j1,...,j`i1,...,ik∂j1 ⊗ · · · ⊗ ∂j` ⊗ dxi1 ⊗ · · · ⊗ dxik .

The local trivialisation φ : π−1(U)→ U × Rnk+`

is given by

φ : T k` (TpM) 3 F 7−→ (p, F j1,...,j`i1,...,ik).

1.3.4 Dual Bundles. If E is a vector bundle over M , the dual bundleE∗ is the bundle whose fibres are the dual spaces of the fibres of E:

E∗ = (p, ω) : ω ∈ E∗p.

If (ξi) is a local frame for E over an open set U ⊂ M , then the map φ :π−1E∗(U) → U × Rk defined by (p, ω) 7→ (p, ω(ξ1(p)), . . . , ω(ξk(p))) is a local

trivialisation of E∗ over U . The corresponding local frame for E∗ is given bythe sections θi defined by θi(ξj) = δij .

1.3.5 Tensor Products of Bundles. If E1, . . . , Ek are vector bundlesover M , the tensor product E1 ⊗ · · · ⊗ Ek is the vector bundle whose fibresare the tensor products (E1)p ⊗ · · · ⊗ (Ek)p. If U is an open set in M and

ξji : 1 ≤ i ≤ nj is a local frame for Ej over U for j = 1, . . . , k, thenξ1i1⊗ · · · ⊗ ξkik : 1 ≤ ij ≤ nj , 1 ≤ j ≤ k

forms a local frame for E1 ⊗ · · · ⊗

Ek. Taking tensor products commutes with taking duals (in the sense ofSection 1.3.4) and is associative. That is, E∗1 ⊗ E∗2 ' (E1 ⊗ E2)∗ and (E1 ⊗E2)⊗ E3 ' E1 ⊗ (E2 ⊗ E3).

1.4 Metric Tensors 9

1.3.6 A Test for Tensorality. Let E1, . . . , Ek be vector bundles overM . Given a tensor field F ∈ Γ (E∗1 ⊗ · · · ⊗ E∗k) and sections Xi ∈ Γ (Ei),Proposition 1.8 implies that the function on U defined by

F (X1, . . . , Xk) : p 7→ Fp(X1

∣∣p, . . . , Xk

∣∣p),

is smooth, so that F induces a mapping F : Γ (E1)×· · ·×Γ (Ek)→ C∞(M).It can easily be seen that this map is multilinear over C∞(M) in the sensethat

F (f1X1, . . . , fkXk) = f1 · · · fkF (X1, . . . , Xk)

for any fi ∈ C∞(M) and Xi ∈ Γ (Ei). In fact the the converse holds as well.

Proposition 1.19 (Tensor Test). For vector bundles E1, . . . , Ek over M ,the mapping F : Γ (E1) × · · · × Γ (Ek) → C∞(M) is a tensor field, i.e. F ∈Γ (E∗1 ⊗ · · · ⊗ E∗k), if and only if F is multilinear over C∞(M).

By Lemma 1.18 we also have:

Proposition 1.20 (Bundle Valued Tensor Test). For vector bundles E0,E1, . . . , Ek over M , the mapping F : Γ (E1)×· · ·×Γ (Ek)→ Γ (E0) is a tensorfield, i.e. F ∈ Γ (E∗1 ⊗ · · · ⊗ E∗k ⊗ E0), if and only if F is multilinear overC∞(M).

Remark 1.21. This proposition leaves one to interpret

F ∈ Γ (E∗1 ⊗ · · · ⊗ E∗k ⊗ E0)

as an E0-valued tensor acting on E1 ⊗ · · · ⊗ Ek.

The importance of Propositions 1.19 and 1.20 is that it allows one to workwith tensors without referring to their pointwise attributes. For example, themetric g on M (as we shall see) can be consider as a pointwise inner productgp : TpM × TpM → R that smoothly depends on its base point. By ouridentification we can also think of this tensor as a map

g : X (M)×X (M)→ C∞(M).

Therefore if X,Y and Z are vector fields, g(X,Y ) ∈ C∞(M) and so byRemark 1.10 we also have Zg(X,Y ) ∈ C∞(M).

1.4 Metric Tensors

An inner product on a vector space allows one to define lengths of vectorsand angles between them. Riemannian metrics bring this structure onto thetangent space of a manifold.


1.4.1 Riemannian Metrics. A Riemannian metric g on a manifold Mis a symmetric positive definite (2, 0)-tensor field (i.e. g ∈ Γ (Sym2T ∗M) andgp is an inner product for each p ∈M). A manifold M together with a givenRiemannian metric g is called a Riemannian manifold (M, g).

In local coordinates (xi), g = gijdxi ⊗ dxj . The archetypical Riemannian

manifold is (Rn, δij). As the name suggests, the concept of the metric wasfirst introduced by Bernhard Riemann in his 1854 habilitation dissertation.

1.4.1.1 Geodesics. We want to think of geodesics as length minimisingcurves. From this point of view, we seek to minimise the length functional

L(γ) =

ˆ 1

0

‖γ(t)‖gdt

amongst all curves γ : [0, 1]→M . There is also a natural ‘energy’ functional:

E(γ) =1

2

ˆ 1

0

‖γ(t)‖2gdt.

As L(γ)2 ≤ 2E(γ), for any smooth curve γ : [0, 1] → M , the problem ofminimising L(γ) amongst all smooth curves γ is equivalent to minimisingE(γ). By doing so we find:

Theorem 1.22. The Euler-Lagrange equations for the energy functional are

γi(t) + Γ ijk(γ(t))γj(t)γk(t) = 0, (1.2)

where the connection coefficients Γ ijk are given by (1.9).

Hence any smooth curve γ : [0, 1] → M satisfying (1.2) is called a geodesic.By definition they are critical points of the energy functional. Moreover, bythe Picard-Lindelof theorem we recall:

Lemma 1.23 (Short-Time Existence of Geodesics). Suppose (M, g) isa Riemannian manifold. Let p ∈M and v ∈ TpM be given. Then there existsε > 0 and precisely one geodesic γ : [0, ε] → M with γ(0) = p, γ(0) = v andγ depends smoothly on p and v.

1.4.2 The Product Metric. If (M1, g(1)) and (M2, g

(2)) are twoRiemannian manifolds then, by the natural identification T(p1,p2)M1×M2 'Tp1

M1 ⊕ Tp2M2, there is a canonical Riemannian metric g = g(1) ⊕ g(2) on

M1 ×M2 defined by

g(p1,p2)(u1 + u2, v1 + v2) = g(1)p1

(u1, u2) + g(2)p2

(v1, v2),

where u1, u2 ∈ Tp1M1 and v1, v2 ∈ Tp2

M2. If dimM1 = n and dimM2 = m,the product metric, in local coordinates (x1, . . . , xn+m) about (p1, p2), is theblock diagonal matrix:

1.4 Metric Tensors 11

(gij) =

g(1)ij

g(2)ij

where (g

(1)ij ) is an n× n block and (g

(2)ij ) is an m×m block.

1.4.3 Metric Contractions. As the Riemannian metric g is non-degenerate, there is a canonical g-dependent isomorphism between TM andT ∗M .3 By using this it is possible to take tensor contractions over two indicesthat are either both vectors or covectors.

For example, if h is a symmetric (2, 0)-tensor on a Riemannian manifoldthen h] is a (1, 1)-tensor. In which case the trace of h with respect to g,denoted by trgh, is

trg h = trh] = h ii = gijhij .

Equivalently one could also write

trg h = tr13 tr24 g−1 ⊗ h = (g−1 ⊗ h)(dxi, dxj , ∂i, ∂j) = gijhij .

1.4.4 Metrics on Bundles. A metric g on a vector bundle π : E →M isa section of E∗⊗E∗ such that at each point p of M , gp is an inner product onEp (that is, gp is symmetric and positive definite for each p: gp(ξ, η) = gp(η, ξ)for all ξ, η ∈ Ep; gp(ξ, ξ) ≥ 0 for all ξ ∈ Ep, and gp(ξ, ξ) = 0⇒ ξ = 0).

A metric on E defines a bundle isomorphism ιg : E → E∗ given by ιg(ξ) :η 7→ gp(ξ, η) for all ξ, η ∈ Ep.

1.4.5 Metric on Dual Bundles. If g is a metric on E, there is a uniquemetric on E∗ (also denoted g) such that ιg is a bundle isometry:

g(ιg(ξ), ιg(v)) = g(ξ, η)

for all ξ, η ∈ Ep; or equivalently g(ω, σ) = g(ι−1g ω, ι−1

g σ) for all ω, σ ∈ E∗p =(Ep)

∗.

1.4.6 Metric on Tensor Product Bundles. If g1 is a metric on E1

and g2 is a metric on E2, then

g = g1 ⊗ g2 ∈ Γ ((E∗1 ⊗ E∗1 )⊗ (E∗2 ⊗ E∗2 )) ' Γ ((E1 ⊗ E2)∗ ⊗ (E1 ⊗ E2)∗)

is the unique metric on E1 ⊗ E2 such that

g(ξ1 ⊗ η1, ξ2 ⊗ η2) = g1(ξ1, ξ2)g2(η1, η2).

3 Specifically, the isomorphism ] : T∗M → TM sends a covector ω to ω] = ωi∂i =gijωj∂i, and [ : TM → T∗M sends a vector X to X[ = Xidxi = gijXjdxi.


The construction of metrics on tensor bundles now follows. It is well-definedsince the metric constructed on a tensor product of dual bundles agrees withthat constructed on the dual bundle of a tensor product.

Example 1.24. Given tensors S, T ∈ T k` (M), the inner product, denoted by

〈·, ·〉, at p is

〈S, T 〉 = ga1b1 · · · gakbkgi1j1 · · · gi`j`S i1...i`a1...ak

T j1...j`b1...bk

. (1.3)

1.5 Connections

Connections provide a coordinate invariant way of taking directional deriv-atives of vector fields. In Rn, the derivative of a vector field X = Xiei indirection v is given by DvX = v(Xi)ei. Simply put, Dv differentiates thecoefficient functions Xi and thinks of the basis vectors ei as being held con-stant. However there is no canonical way to compare vectors from differentvector spaces, hence there is no natural coordinate invariant analogy applic-able to abstract manifolds.4 To circumnavigate this, we impose an additionalstructure — in the form of a connection operator — that provides a way to‘connect’ these tangent spaces.

Our method here is to directly specify how a connection acts on elementsof Γ (E) as a module over C∞(M). They are of central importance in moderngeometry largely because they allow a comparison between the local geometryat one point and the local geometry at another point.

Definition 1.25. A connection ∇ on a vector bundle E over M is a map

∇ : X (M)× Γ (E)→ Γ (E),

written as (X,σ) 7→ ∇Xσ, that satisfies the following properties:

1. ∇ is C∞(M)-linear in X:

∇f1X1+f2X2σ = f1∇X1

σ + f1∇X1σ

2. ∇ is R-linear in σ:

∇X(λ1σ1 + λ2σ2) = λ1∇Xσ1 + λ2∇Xσ2

and ∇ satisfies the product rule:

4 There is however another generalisation of directional derivatives which is canonical:the Lie derivative. The Lie derivative evaluates the change of one vector field alongthe flow of another vector field. Thus, one must know both vector fields in an openneighbourhood. The covariant derivative on the other hand only depends on the vectordirection at a single point, rather than a vector field in an open neighbourhood of apoint. In other words, the covariant derivative is linear over C∞(M) in the directionargument, while the Lie derivative is C∞(M)-linear in neither argument.

1.5 Connections 13

∇X(fσ) = (Xf)σ + f∇Xσ

We say ∇Xσ is the covariant derivative of σ in the direction X.

Remark 1.26. Equivalently, we could take the connection to be

∇ : Γ (E)→ Γ (T ∗M ⊗ E)

which is linear and satisfies the product rule. For if σ ∈ Γ (E) then ∇σ ∈Γ (T ∗M ⊗ E), where (∇σ)(X) = ∇Xσ is C∞(M)-linear in X by Property 1of Definition 1.25. Thus Proposition 1.20 implies that ∇σ : Γ (TM)→ Γ (E)is an E-valued tensor acting on TM .

For a connection ∇ on the tangent bundle, we define the connection coeffi-cients or Christoffel symbols of ∇ in a given set of local coordinates (xi) bydefining

Γijk = dxk (∇∂i∂j)

or equivalently,∇∂i∂j = Γij

k∂k.

More generally, the connection coefficients of a connection ∇ on a bundle Ecan be defined with respect to a given local frame ξα for E by the equation

∇∂iξα = Γiαβξβ .

1.5.1 Covariant Derivative of Tensor Fields. In applications one isoften interested in computing the covariant derivative on the tensor bundlesT k` M . This is a special case of a more general construction (see Section 1.5.3).

Proposition 1.27. Given a connection ∇ on TM , there is a unique con-nection on the tensor bundle, also denoted by ∇, that satisfies the followingproperties:

1. On TM , ∇ agrees with the given connection.2. On C∞(M) = T 0M , ∇ is the action of a vector as a derivation:

∇Xf = Xf,

for any smooth function f .3. ∇ obeys the product rule with respect to tensor products:

∇X(F ⊗G) = (∇XF )⊗G+ F ⊗ (∇XG),

for any tensors F and G.4. ∇ commutes with all contractions:

∇X(trF ) = tr (∇XF ),

for any tensor F .


Example 1.28. We compute the covariant derivative of a 1-form ω with re-spect to the vector field X. Since tr dxj ⊗ ∂i = δji , ∇X(tr dxj ⊗ ∂i) = 0. Thus(∇Xdxj)(∂i) = −dxj(∇X∂i) and so

(∇Xω)(∂k) = ∇Xωk + ωj (∇Xdxj)(∂k)

= ∇Xωk − ωjdxj(∇X∂k)

= Xi∂iωk − ωjXi Γ jik.

Therefore ∇Xω = (Xi∂iωk − ωjXi Γ jik)dxk.

In general we have the following useful formulæ.

Proposition 1.29. For any tensor field F ∈ T k` (M), vector fields Yi and

1-forms ωj we have

(∇XF )(ω1, . . . , ω`, Y1, . . . , Yk) = X(F (ω1, . . . , ω`, Y1, . . . , Yk))

−∑j=1

F (ω1, . . . ,∇Xωj , . . . , ω`, Y1, . . . , Yk)

−k∑i=1

F (ω1, . . . , ω`, Y1, . . . ,∇XYi, . . . , Yk).

Proof. By (1.1) we have

tr (F ⊗ ω1 ⊗ · · · ⊗ ω` ⊗ Y1 ⊗ · · · ⊗ Yk) = ωi1 · · ·ωikFi1···ik

j1···jkYj1 · · ·Y jk

= F (ω1, . . . , ω`, Y1, . . . , Yk).

Thus by Proposition 1.27,

∇X(F (ω1, . . . , ω`, Y1, . . . , Yk))

= tr[(∇XF )⊗ ω1 ⊗ · · · ⊗ Yk + F ⊗ (∇Xω1)⊗ · · · ⊗ Yk

+ · · ·+ F ⊗ ω1 ⊗ · · · ⊗ (∇XYk)]

= (∇XF )(ω1, . . . , ω`, Y1, . . . , Yk) + F (∇Xω1, . . . , ω`, Y1, . . . , Yk)

+ · · ·+ F (ω1, . . . , ω`, Y1, . . . ,∇XYk).

As the covariant derivative is C∞(M)-linear over X, we define ∇F ∈Γ (⊗k+1T ∗M ⊗` TM) by

(∇F )(X,Y1, . . . , Yk, ω1, . . . , ω`) = ∇XF (Y1, . . . , Yk, ω

1, . . . , ω`),

for any F ∈ T k` (M). Hence (in this case)∇ is an R-linear map∇ : T k

` (M)→T k+1` (M) that takes a (k, `)-tensor field and gives a (k + 1, `)-tensor field.

1.5 Connections 15

1.5.2 The Second Covariant Derivative of Tensor Fields. By util-ising the results of the previous section, we can make sense of the secondcovariant derivative ∇2.

To do this, suppose the vector bundle E = T k` M with associated con-nection ∇. By Remark 1.26, if F ∈ Γ (E) then ∇F ∈ Γ (T ∗M ⊗ E) and so∇2F ∈ Γ (T ∗M ⊗ T ∗M ⊗ E). Therefore, for any vector fields X,Y we findthat

(∇2F )(X,Y ) =(∇X(∇F )

)(Y )

= ∇X((∇F )(Y )

)− (∇F )(∇XY )

= ∇X(∇Y F ))− (∇∇XY F ). (1.4)

Example 1.30. If f ∈ C∞(M) is a (0, 0)-tensor, then ∇2f is a (2, 0)-tensor.In local coordinates:

∇2∂i,∂jf = (∇∂i(∇f))(∂j)

= ∂i((∇f)(∂j))− (∇f)(∇∂i∂j)= ∂i(∂jf)− (∇f)(Γ kij∂k)

= ∂i ∂jf − Γ kij∇∂kf

=∂2f

∂xi∂xj− Γ kij

∂f

∂xk. (1.5)

In general equation (1.4) amounts to the following useful formula:

Proposition 1.31. If ∇ is a connection on TM , then

∇2Y,X = ∇Y ∇X −∇∇YX : T k

` (M)→ T k` (M) (1.6)

where X,Y ∈X (M) are given vector fields.

1.5.2.1 Notational Convention. It is important to note that we interpret

∇X∇Y F = ∇2X,Y F = (∇∇F )(X,Y, · · · ) = (∇X(∇F ))(Y, . . .),

whenever no brackets are specified; this is differs from ∇X(∇Y F ) with brack-ets (ref. [KN96, pp. 124-5]). Furthermore, for notational simplicity we of-ten write ∇∂p as just ∇p and (∇∂pF )(dxi1 , . . . , dxi` , ∂j1 , . . . , ∂jk) simply as

∇pF i1···i`j1···jk .

1.5.2.2 The Hessian. We define the Hessian of f ∈ C∞(M) to be

Hess(f) = ∇df.

When applied to vector fields X,Y ∈ X (M), we find that Hess(f)(X,Y ) =(∇df)(X,Y ) = ∇2

X,Y f . Also note that the Hessian is symmetric preciselywhen the connection is symmetric (see Example 1.30).


1.5.3 Connections on Dual and Tensor Product Bundles. So farwe have looked at the covariant derivative on the tensor bundle

⊗kT ∗M ⊗⊗`

TM . In fact much of the same structure works on a general vector bundleas well.

Proposition 1.32. If ∇ is a connection on E, then there is a unique con-nection on E∗, also denoted by ∇, such that

X(ω(ξ)) = (∇Xω)(ξ) + ω(∇Xξ)

for any ξ ∈ Γ (E), ω ∈ Γ (E∗) and X ∈X (M).

Proposition 1.33. If ∇(i) is a connection on Ei for i = 1, 2, then there is aunique connection ∇ on E1 ⊗ E2 such that

∇X(ξ1 ⊗ ξ2) = (∇(1)X ξ1)⊗ ξ2 + ξ1 ⊗ (∇(2)

X ξ2)

for all X ∈X (M) and ξi ∈ Γ (Ei).

Propositions 1.32 and 1.33 define a canonical connection on any tensorbundle constructed from E by taking duals and tensor products. In particular,if S ∈ Γ (E∗1 ⊗E2) is an E2-valued tensor acting on E1, then ∇S ∈ Γ (T ∗M ⊗E∗1 ⊗ E2) is given by

(∇XS)(ξ) = E2∇X(S(ξ)

)− S

(E1∇X ξ

)(1.7)

where ξ ∈ Γ (E1) and X ∈X (M).

Moreover if we also have a connection ∇ on TM , then ∇2S ∈ Γ (T ∗M ⊗T ∗M ⊗E∗1 ⊗E2) — since we can construct this connection from the connec-tions on TM , E1 and E2 by taking duals and tensor products. Explicitly,

(∇2S)(X,Y, ξ) = E2∇X((∇Y S)(ξ)

)− (∇∇XY S)(ξ)− (∇Y S)(E1∇Xξ)

=(∇X(∇Y S)

)(ξ)− (∇∇XY S)(ξ) (1.8)

where X,Y ∈X (M) and ξ ∈ Γ (E1).

1.5.4 The Levi-Civita Connection. When working on a Riemannianmanifold, it is desirable to work with a particular connection that reflectsthe geometric properties of the metric. To do so, one needs the notions ofcompatibility and symmetry.

Definition 1.34. A connection ∇ on a vector bundle E is said to be com-patible with a metric g on E if for any ξ, η ∈ Γ (E) and X ∈X (M),

X(g(ξ, η)) = g(∇Xξ, η) + g(ξ,∇Xη).

Moreover, if ∇ is compatible with a metric g on E, then the induced con-nection on E∗ is compatible with the induced metric on E∗. Also, if the

1.6 Connection Laplacian 17

connections on two vector bundles are compatible with given metrics, thenthe connection on the tensor product is compatible with the tensor productmetric.

Unfortunately compatibility by itself is not enough to determine a uniqueconnection. To get uniqueness we also need the connection to be symmetric.

Definition 1.35. A connection ∇ on TM is symmetric if its torsion van-ishes.5 That is, if ∇XY −∇YX = [X,Y ] or equivalently Γ kij = Γ kji.

We can now state the fundamental theorem of Riemannian geometry.

Theorem 1.36. Let (M, g) be a Riemannian manifold. There exists a uniqueconnection ∇ on TM which is symmetric and compatible with g. This con-nection is referred to as the Levi-Civita connection of g.

The reason why this connection has been anointed the Riemannian con-nection is that the symmetry and compatibility conditions are invariantlydefined natural properties that force the connection to coincide with the tan-gential connection, whenever M is realised as a submanifold of Rn with theinduced metric (which is always possible by the Nash embedding).

Proposition 1.37. In local coordinate (xi), the Christoffel symbols of theLevi-Civita connection are given by

Γ kij =1

2gk`(∂jgi` + ∂igj` − ∂`gij). (1.9)

1.6 Connection Laplacian

In its simplest form, the Laplacian ∆ of f ∈ C∞(M) is defined by ∆f =div grad f . In fact, the Laplacian can be extended to act on tensor bundlesover a Riemannian manifold (M, g). The resulting differential operator isreferred to as the connection Laplacian. Note that there are a number ofother second-order, linear, elliptic differential operators bearing the nameLaplacian which have alternative definitions.

Definition 1.38. For any tensor field F ∈ T k` (M), the connection Laplacian

∆F = trg∇2F (1.10)

is the trace of the second covariant derivative with the metric g.

Explicitly,

(∆F )j1...j`i1...ik = (trg∇2F )j1...j`i1...ik

=(tr13 tr24 g

−1 ⊗∇2F)j1...j`

i1...ik

= gpq(∇∂p∇∂qF )(∂j1 , . . . , ∂j` , dxi1 , . . . , dxik).

5 The torsion τ of∇ is defined by τ(X,Y ) = ∇XY −∇YX−[X,Y ]. τ is a (2, 1)-tensorfield since ∇fX(gY )−∇gY fX − [fX, gY ] = fg(∇XY −∇YX − [X,Y ]).


Example 1.39. If the tensor bundle is T 0M = C∞(M), then (1.5) impliesthat

∆f = gij∇∂i∇∂jf

= gij(

∂2f

∂xi∂xj− Γ kij

∂f

∂xk

).

1.7 Curvature

We introduce the curvature tensor as a purely algebraic object that arisesfrom a connection on a vector bundle. From this we will look at the curvatureon specific bundle structures.

1.7.1 Curvature on Vector Bundles.

Definition 1.40. Let E be a vector bundle over M . If ∇ is a connection onE, then the curvature of the connection ∇ on the bundle E is the sectionR∇ ∈ Γ (T ∗M ⊗ T ∗M ⊗ E∗ ⊗ E) defined by

R∇(X,Y )ξ = ∇Y (∇Xξ)−∇X (∇Y ξ) +∇[X,Y ]ξ. (1.11)

In the literature, there is much variation in the sign convention; somedefine the curvature to be of opposite sign to ours.

1.7.2 Curvature on Dual and Tensor Product Bundles. Thecurvature on a dual bundle E∗, with respect to the dual connection, is char-acterised by the formula

0 = (R(X,Y )ω)(ξ) + ω(R(X,Y )ξ)

for all X,Y ∈X (M), ω ∈ Γ (E∗) and ξ ∈ Γ (E).The curvature on a tensor product bundle E1 ⊗ E2, with connection ∇

given by Proposition 1.33, can be computed in terms of the curvatures oneach of the factors by the formula

R∇(X,Y )(ξ1 ⊗ ξ2) = (R∇(1)(X,Y )ξ1)⊗ ξ2 + ξ1 ⊗ (R∇(2)(X,Y )ξ2) ,

where X,Y ∈X (M) and ξi ∈ Γ (Ei), i = 1, 2.

Example 1.41. Of particular interest, the curvature on E∗1 ⊗ E2 (E2-valuedtensors acting on E1) is given by

(R(X,Y )S)(ξ) = R∇(2)(X,Y )(S(ξ)

)− S

(R∇(1)(X,Y )ξ

), (1.12)

where S ∈ Γ (E∗1 ⊗ E2), ξ ∈ Γ (E1) and X,Y ∈X (M).

1.7 Curvature 19

1.7.3 Curvature on the Tensor Bundle. One of the most importantapplications is the curvature of the tensor bundle. By (1.11) and Section 1.5.1,we have the following:

Proposition 1.42. Let R be the curvature on the (k, `)-tensor bundle. IfF,G ∈ T k

` (M) are tensors, then

R(X,Y )(trF ) = tr (R(X,Y )F )

R(X,Y )(F ⊗G) = (R(X,Y )F )⊗G+ F ⊗ (R(X,Y )G)

for any vector fields X and Y .

Moreover we also have the following important formulæ.

Proposition 1.43. Let R be the curvature on the (k, `)-tensor bundle. IfF ∈ T k

` (M), then

R(X,Y )(F (ω1, . . . , ω`, Z1, . . . , Zk)

)= (R(X,Y )F )(ω1, . . . , ω`, Z1, . . . , Zk)

+∑j=1

F (ω1, . . . , R(X,Y )ωj , . . . , ω`, Z1, . . . , Zk)

+

k∑i=1

F (ω1, . . . , ω`, Z1, . . . , R(X,Y )Zi, . . . , Zk)

for any vector fields X,Y, Zi and 1-forms ωj.

Proof. Let the vector bundle E = T k` M , so for any ξ ∈ Γ (E) we find that

R(X,Y )(F (ξ)

)= R(X,Y )

(trF ⊗ ξ

)= tr

[(R(X,Y )F )⊗ ξ + F ⊗ (R(X,Y )ξ)

]= (R(X,Y )F )(ξ) + F (R(X,Y )ξ).

As ξ takes the form

ξ = ω1 ⊗ · · · ⊗ ω` ⊗ Z1 ⊗ · · · ⊗ Zk,

a similar argument shows that F (R(X,Y )ξ) = F (R(X,Y )ω1, . . . , Zk) + · · ·+F (ω1, . . . , R(X,Y )Zk) from which the result follows.

Proposition 1.44. Let R be the curvature on the (k, `)-tensor bundle. If theconnection ∇ on TM is symmetric, then

R(X,Y ) = ∇2Y,X −∇2

X,Y (1.13)

and so R(X,Y ) : T k` (M)→ T k

` (M).


Proof. For any (k, `)-tensor F , we see by (1.6) that

∇2Y,XF −∇2

X,Y F = ∇Y (∇XF )−∇∇YXF −∇X(∇Y F ) +∇∇XY F= ∇Y (∇XF )−∇X(∇Y F ) +∇[X,Y ]F.

Example 1.45. The curvature of C∞(M) = T 0M vanishes since

R(∂i, ∂j)f = ∇∂i(∇∂jf)−∇∂j (∇∂if) +∇[∂i,∂j ]f = ∂i∂jf − ∂j∂if = 0

for any f ∈ C∞(M).

1.7.4 Riemannian Curvature. If (M, g) is a Riemannian manifold,the curvature R ∈ Γ (⊗3T ∗M⊗TM) of the Levi-Civita connection ∇ on TMis a (3, 1)-tensor field that, in local coordinates (xi), takes the form

R = R ìjk dxi ⊗ dxj ⊗ dxk ⊗ ∂`,

where R(∂i, ∂j)∂k = R ìjk ∂`. Accompanying this is the Riemann curvature

tensor, also denoted by R. It is a covariant (4, 0)-tensor field defined by

R(X,Y, Z,W ) = g(R(X,Y )Z,W )

for all W,X, Y, Z ∈X (M). In local coordinates (xi) it can be expressed as

R = Rijk` dxi ⊗ dxj ⊗ dxk ⊗ dx`,

where Rijk` = g`pRp

ijk .

Lemma 1.46. In local coordinates (xi), the curvature of the Levi-Civita con-nection can be expressed as follows:

R ìjk = ∂jΓ

ìk − ∂iΓ `jk + Γmik Γ

`jm − ΓmjkΓ ìm

Rijk` =1

2

(∂j∂kgi` + ∂i∂`gjk − ∂i∂kgj` − ∂j∂`gik

)+ g`p(Γ

mik Γ

pjm − Γ

mjkΓ

pim)

1.7.4.1 Symmetries of the Curvature Tensor. The curvature tensor pos-sesses a number of important symmetry properties. They are:

(i) Antisymmetric in first two arguments: Rijk` +Rjik` = 0(ii) Antisymmetric in last two arguments: Rijk` +Rji`k = 0

(iii) Symmetry between the first and last pair of arguments: Rijk` = Rkìj

In addition to this, there are also the ‘cyclic’ Bianchi identities:

(iv) First Bianchi identity: Rijk` +Rjki` +Rkij` = 0(v) Second Bianchi identity: ∇mRijk` +∇kRij`m +∇`Rijmk = 0

1.7 Curvature 21

1.7.5 Ricci and Scalar Curvature. As the curvature tensor can bequite complicated, it is useful to consider various contractions that summarisesome of the information contained in the curvature tensor. The first of thesecontractions is the Ricci tensor, denoted by Ric. It is defined as

Ric(X,Y ) = trg R(X, ·, Y, ·) = (tr14tr26 g−1 ⊗R)(X,Y ).

In component form,

Ric(∂i, ∂j) = Rij = R kikj = gpqRipjq.

From the symmetry properties of R it is clear that Ric is symmetric.A further trace of the Ricci tensor gives a scalar quantity called the scalar

curvature, denoted by Scal:

Scal = trg Ric = Ric ii = gijRij .

It is important to note that if the curvature tensor is defined with oppositesign, the contraction is defined so that the Ricci tensor matches the onegiven here. Hence the Ricci tensor has the same meaning for everyone. ByLemma 1.46, the Ricci tensor can be expressed locally as follows.

Lemma 1.47. In local coordinates (xi), the Ricci tensor takes the form

Rik =1

2gj`(

∂2gi`∂xj∂xk

+∂2gjk∂xi∂x`

− ∂2gj`∂xi∂xk

− ∂2gik∂xj∂x`

)+ Γmik Γ

jjm − Γ

mjkΓ

jim.

1.7.5.1 Contraction Commuting with Covariant Derivative. As we areworking with a compatible connection, ∇g ≡ 0. Thus one can commutecovariant derivatives with metric contractions.

Proposition 1.48. If ∇ is the Levi-Civita connection, then

∇kRij = gpq∇kRipjq (1.14)

∇2k,`Rij = gpq∇2

k,`Ripjq. (1.15)

Proof. To show (1.14), let X,Y, Z ∈X (M) so that

(∇ZRic)(X,Y ) = (∇Z (tr g−1 ⊗R))(X,Y )

=(tr∇Z(g−1 ⊗R)

)(X,Y )

=(tr

∇Zg−1 ⊗R+ tr g−1 ⊗∇ZR)(X,Y )

= (tr14tr26 g−1 ⊗∇ZR)(X,Y )

= (trg∇ZR)(X, ·, Y, ·).

Similarly, to show (1.15) note that


∇2Ric = ∇2(tr g−1 ⊗R)

= tr(∇2g−1 ⊗R+ 2∇g−1 ⊗∇R+ g−1 ⊗∇2R

)= tr g−1 ⊗∇2R.

In later applications we will need the contracted second Bianchi identity :

gjk∇kRicij =1

2∇iScal. (1.16)

This follows easily from (1.14) and the second Bianchi identity, since

0 = gamgbn(∇`Rabmn +∇mRabn` +∇nRab`m)

= gam(∇`Ricam −∇mRica`) + gamgbn∇nRab`m= ∇`Ric a

a − gam∇mRica` − gbn∇nR m`mb

= ∇`Scal− gam∇mRica` − gbn∇nRic`b

from which (1.16) now follows.

1.7.6 Sectional Curvature. Suppose (M, g) is a Riemannian manifold.If Π is a two-dimensional subspace of TpM , we define the sectional curvatureK of Π to be

K(Π) = R(e1, e2, e1, e2),

where e1, e2 is an orthonormal basis for Π. By a rotation or reflection inthe plane, one can show K is independent of the choice of basis. We referto the oriented plane generated from ei and ej by the notation ei ∧ ej (cf.Section C.3). Furthermore if u, v is any basis for the 2-plane Π, one has

K(u ∧ v) =R(u, v, u, v)

|u|2|v|2 − g(u, v)2.

If U ⊂ TpM is a neighbourhood of zero on which expp is a diffeomorphism,then SΠ := expp(Π ∩ U) is a 2-dimensional submanifold of M containing p,called the plane of section determined by Π. That is, it is the surface sweptout by geodesics whose initial tangent vectors lie in Π. One can geometricallyinterpret the sectional curvature of M associated to Π to be the Gaussiancurvature of the surface SΠ at p with the induced metric.

By computing the sectional curvature of the plane 12 (ei + ek) ∧ (ej + e`)

one can show:

1.7 Curvature 23

Proposition 1.49. The curvature tensor R is completely determined by thesectional curvature. In particular,

Rijk` =1

3K( (ei + ek) ∧ (ej + e`)

2

)+

1

3K( (ei − ek) ∧ (ej − e`)

2

)− 1

3K( (ej + ek) ∧ (ei + e`)

2

)− 1

3K( (ej − ek) ∧ (ei − e`)

2

)− 1

6K(ej ∧ e`)−

1

6K(ei ∧ ek) +

1

6K(ei ∧ e`) +

1

6K(ej ∧ ek).

One can also show that the scalar curvature Scal = Ric jj =

∑j 6=kK(ej ∧ek),

where (ei) is orthonormal basis for TpM .Each of the model spaces Rn, Sn and Hn has an isometry group that acts

transitively on orthonormal frames, and so acts transitively on 2-planes in thetangent bundle. Therefore each has a constant sectional curvature — in thesense that the sectional curvatures are the same for all planes at all points.

It is well known that the Euclidean space Rn has constant zero sec-tional curvature (this is geometrically intuitive as each 2-plane section haszero Gaussian curvature). The sphere Sn of radius 1 has constant sectionalcurvature equal to 1 and the hyperbolic space Hn has constant sectionalcurvature equal to −1.

1.7.7 Berger’s Lemma. A simple but important result is the so-calledlemma of Berger [Ber60b, Sect. 6]. Following [Kar70], we show the curvaturecan be bounded whenever the sectional curvature is bounded from above andbelow. That is, if a Riemannian manifold (M, 〈·, ·〉) has sectional curvaturebounds

δ = minu,v∈TpM

K(u ∧ v) and ∆ = maxu,v∈TpM

K(u ∧ v),

with the assumption that δ ≥ 0, we prove the following bounds on thecurvature tensor:

Lemma 1.50 (Berger). For orthonormal u, v, w, x ∈ TpM , one can boundthe curvature tensor by

|R(u, v, w, v)| ≤ 1

2(∆− δ) (1.17)

|R(u, v, w, x)| ≤ 2

3(∆− δ). (1.18)

Proof. From the symmetries of the curvature tensor we find that:

4R(u, v, w, v) = R(u+ w, v, u+ w, v)−R(u− w, v, u− w, v)

6R(u, v, w, x) = R(u, v + x,w, v + x)−R(u, v − x,w, v − x)

−R(v, u+ x,w, u+ x) +R(v, u− x,w, u− x).


Using the definition of the sectional curvature together with the first identitygives

|R(u, v, w, v)| = 1

4

∣∣R(u+ w, v, u+ w, v)−R(u− w, v, u− w, v)∣∣

=1

4

∣∣∣K((u+ w) ∧ v)(|u+ w|2|v|2 −

〈u+ w, v〉)

−K((u− w) ∧ v

)(|u− w|2|v|2 −

〈u− w, v〉)∣∣∣

=1

2

∣∣K((u+ w) ∧ v)−K

((u− w) ∧ v

)∣∣≤ 1

2(∆− δ)

which is identity (1.17). To prove (1.18), use the second identity and apply(1.17) to the four terms — whilst taking into consideration |v ± x|2 = |u ±x|2 = 2, for orthonormal u, v, w and x.

1.8 Pullback Bundle Structure

Let M and N be smooth manifolds, let E be a vector bundle over N and fbe a smooth map from M to N .

Definition 1.51. The pullback bundle of E by f , denoted f∗E, is the smoothvector bundle over M defined by f∗E = (p, ξ) : p ∈M, ξ ∈ E, π(ξ) = f(p).If ξ1, . . . , ξk are a local frame for E near f(p) ∈ N , then Ξi(p) = ξi(f(p)) area local frame for f∗(E) near p.

Lemma 1.52. Pullbacks commute with taking duals and tensor products:

(f∗E)∗ = f∗(E∗) and (f∗E1)⊗ (f∗E2) = f∗(E1 ⊗ E2).

1.8.1 Restrictions. The restriction ξf ∈ Γ (f∗E) of ξ ∈ Γ (E) to f isdefined by

ξf (p) = ξ(f(p)) ∈ Ef(p) = (f∗E)p,

for all points p ∈M .

Example 1.53. Suppose g is a metric on E. Then g ∈ Γ (E∗ ⊗ E∗), and byrestriction we obtain gf ∈ Γ ((f∗E)∗ ⊗ (f∗E)∗), which is a metric on f∗E(the ‘restriction of g to f ’): If ξ, η ∈ (f∗E)p = Ef(p), then (gf (p))(ξ, η) =(g(f(p))(ξ, η).

Remark 1.54. In using this terminology one wants to distinguish the restric-tion of a tensor field on E (which is a section of a tensor bundle over f∗E)with the pullback of a tensor on the tangent bundle, which is discussed be-low. Thus the metric in the above example should not be called the ‘pullbackmetric’. Notice that we can restrict both covariant and contravariant tensors,in contrast to the situation with pullbacks.

1.8 Pullback Bundle Structure 25

1.8.2 Pushforwards. If f : M → N is smooth, then for each p ∈ M ,we have the linear map f∗(p) : TpM → Tf(p)N = (f∗TN)p. That is, f∗(p) ∈T ∗pM ⊗ (f∗TN)p, so f∗ is a smooth section of T ∗M ⊗ f∗TN . Given a sectionX ∈ Γ (TM) = X (M), the pushforward of X is the section f∗X ∈ Γ (f∗TN)given by applying f∗ to X.

1.8.3 Pullbacks of Tensors. By duality (combined with restriction)we can define an operation taking (k, 0)-tensors on N to (k, 0)-tensors onM , which we call the pullback operation: If S is a (k, 0)-tensor on N (i.e.S ∈ Γ (⊗kT ∗N)), then by restriction we have Sf ∈ Γ (⊗k(f∗T ∗N)), and wedefine f∗S ∈ Γ (⊗kT ∗M) by

f∗S(X1, . . . , Xk) = Sf (f∗X1, . . . , f∗Xk) (1.19)

where Xi are vector fields on M .

Example 1.55. If f is an embedding and g is a Riemannian metric on N , thenf∗g is the pullback metric on M (often called the ‘induced metric’).

This definition can be extended a little to include bundle-valued tensors:Suppose S ∈ Γ (⊗kT ∗N ⊗ E) is an E-valued (k, 0)-tensor field on N . Thenrestriction gives Sf ∈ Γ (⊗k(f∗T ∗N)⊗ f∗E), and we will denote by f∗S thef∗E-valued k-tensor on M defined by

f∗S(X1, . . . , Xk) = Sf (f∗X1, . . . , f∗Xk).

That is, the same formula as before except now both sides are f∗E-valued.

1.8.4 The Pullback Connection. Let ∇ be a connection on E over N ,and f : M → N a smooth map.

Theorem 1.56. There is a unique connection f∇ on f∗E, referred to as thepullback connection, such that

f∇v(ξf ) = ∇f∗vξ

for any v ∈ TM and ξ ∈ Γ (E).

Remark 1.57. To justify the term ‘pullback connection’: If ξ ∈ Γ (E), then∇ξ ∈ Γ (T ∗N ⊗ E), so the pullback gives

f∇ξf := f∗(∇ξ) ∈ Γ (T ∗M ⊗ f∗E).

To define f∇vξ for arbitrary ξ ∈ Γ (f∗E), we fix p ∈ M and choose a

local frame σ1, . . . , σk about f(p) for E. Then we can write ξ =∑ki=1 ξ

i (σi)fwith each ξi a smooth function defined near p, so the rules for a connectiontogether with the pullback connection condition give


f∇vξ = f∇v(ξi (σi)f )

= ξi f∇v(σi)f + v(ξi) (σi)f

= ξi∇f∗vσi + v(ξi) (σi)f . (1.20)

Note that pullback connections on duals and tensor products of pullbackbundles agree with those obtained by applying the previous constructions tothe pullback bundles on the factors. There are two other important propertiesof the pullback connection:

Proposition 1.58. If g is a metric on E and ∇ is a connection on E com-patible with g, then f∇ is compatible with the restriction metric gf .

Proof. As ∇ is compatible with g if and only if ∇g = 0, we therefore mustshow that f∇gf = 0 if ∇g = 0. However this is immediate, since f∇v(gf ) =∇f∗vg = 0.

Proposition 1.59. The curvature of the pullback connection is the pullbackof the curvature of the original connection. That is,

Rf∇(X,Y )ξf = (f∗R∇)(X,Y )ξ

where X,Y ∈X (M) and ξ ∈ Γ (E). Note that R∇ ∈ Γ (T ∗N⊗T ∗N⊗E∗⊗E)here, so that

f∗(R∇) ∈ Γ (T ∗M ⊗ T ∗M ⊗ f∗(E∗ ⊗ E))

= Γ (T ∗M ⊗ T ∗M ⊗ (f∗E)∗ ⊗ f∗E).

Proof. Since curvature is tensorial, it is enough to check the formula for abasis. Choose a local frame σpkp=1 for E, so that (σp)f is a local frame

for f∗E. Also choose local coordinates yα for N near f(p) and xi for Mnear p, and write fα = yα f . Then

Rf∇(∂i, ∂j)(σp)f = f∇∂j(f∇∂i(σp)f

)− (i↔ j)

= f∇j (∇f∗∂iσp)− (i↔ j)

= f∇j (∂ifα∇ασp)− (i↔ j)

= (∂j∂ifα)∇ασp + ∂if

α f∇j ((∇ασp)f )− (i↔ j)

= ∂ifα∇f∗∂j (∇ασp)− (i↔ j)

= ∂ifα∂jf

β (∇β (∇ασp)− (α↔ β))

= ∂ifα∂jf

βR∇(∂α, ∂β)σp

= R∇(f∗∂i, f∗∂j)σp.

When pulling back a tangent bundle, there is another important property:


Proposition 1.60. If ∇ is a symmetric connection on TN , then the pull-back connection f∇ on f∗TN is symmetric in the sense that

f∇U (f∗V )− f∇V (f∗U) = f∗([U, V ])

for any U, V ∈ Γ (TM).

Proof. As before choose local coordinates xi for M near p, and yα for N nearf(p). By writing U = U i∂i and V = V j∂j we find that

f∇U (f∗V )− f∇V (f∗U)

= f∇U(V j∂jf

α∂α)− (U ↔ V )

= U i∂i(V j∂jf

α)∂α + V j∂jf

α f∇U∂α − (U ↔ V )

= (U i∂iVj − V i∂iU j)∂jfα∂α + U iV j(∂i∂jf

α − ∂j∂ifα)∂α

+ V jU i∂jfα∂if

β (∇β∂α −∇α∂β)

= f∗ ([U, V ]) .

1.8.5 Parallel Transport. Parallel transport is a way of using a con-nection to compare geometrical data at different points along smooth curves:

Let ∇ be a connection on the bundle π : E →M . If γ : I →M is a smoothcurve, then a smooth section along γ is a section of γ∗E. Associated to thisis the pullback connection γ∇. A section V along a curve γ is parallel alongγ if γ∇∂tV ≡ 0. In a local frame (ej) over a neighbourhood of γ(t0):

γ∇∂tV (t0) =(V k(t0) + Γ kij(γ(t0))V j(t0)γi(t0)

)ek,

where V (t) = V j(t)ej and Γ kij is the Christoffel symbol of ∇ in this frame.

Theorem 1.61. Given a curve γ : I → M and a vector V0 ∈ Eγ(0), thereexists a unique parallel section V along γ such that V (0) = V0. Such a V iscalled the parallel translate of V0 along γ.

Moreover, for such a curve γ there exists a unique family of linear isomorph-isms Pt : Eγ(0) → Eγ(t) such that a vector field V along γ is parallel if andonly if V (t) = Pt(V0) for all t.

1.8.6 Product Manifolds’ Tangent Space Decomposition. Givenmanifolds M1 and M2, let πj : M1 × M2 → Mj be the standard smoothprojection maps. The pushforward (πj)∗ : T (M1 ×M2) → TMj over πj :M1×M2 →Mj induces, via the pullback bundle, a smooth bundle morphismΠj : T (M1×M2)→ π∗j (TMj) over M1×M2. In which case one has the bundlemorphism

Π1 ⊕Π2 : T (M1 ×M2)→ π∗1(TM1)⊕ π∗2(TM2)


over M1×M2. On fibres over (x1, x2) this is simply the pointwise isomorphismT(x1,x2)(M1×M2) ' Tx1

M1⊕Tx2M2, so Π1⊕Π2 is in fact a smooth bundle

isomorphism. Furthermore,Πj : T (M1×M2)→ π∗j (TMj) is bundle surjectionas πj is a projection. Thus by Proposition 1.14 there exists a well-definedsubbundle E1 inside T (M1 ×M2) given by

E1 = kerΠ1 = v ∈ T (M1 ×M2) : Π1(v) = 0.

We observe that this is in fact equal to π∗2(TM2), since E1 consists of allvectors such that Π1 projection vanishes. Therefore one has isomorphic vectorbundles

T (M1 ×M2) ' kerΠ1 ⊕ kerΠ2.

Example 1.62. If we let M1 = M and M2 = R with π1 = π projection fromM × R onto M and π2 = t be the projection from M × R onto R, then onfibres over (x1, x2) = (x0, t0) we have that

Π2(v) = t∗(v) = dt(v)

since t is a R-valued function on M × R. So by letting S = ker dt = v ∈T (M × R) : dt(v) = 0 we have that

T (M × R) ' S⊕ R ∂t,

since (π∗2TR)(x0,t0) = (t∗TR)(x0,t0) = Tt0R = R ∂t|t0 where ∂t|t0 is the stand-ard coordinate basis for Tt0R.

1.8.7 Connections and Metrics on Subbundles. Suppose F is asubbundle of a vector bundle E over a manifold M , as defined in Definition1.12. If E is equipped with a metric g, then there is a natural metric inducedon F by the inclusion: If ι : F → E is the inclusion of F in E, then theinduced metric on F is defined by

gF (ξ, η) = g(ι(ξ), ι(η)).

There is not in general any natural way to induce a connection on F froma connection ∇ on E. We will consider only the following special case:6

Definition 1.63. A subbundle F of a vector bundle E is called parallel if F isinvariant under parallel transport, i.e. for any smooth curve σ : [0, 1]→M ,and any parallel section ξ of σ∗E over [0, 1] with ξ(0) ∈ Fσ(0), we haveξ(t) ∈ Fσ(t) for all t ∈ [0, 1].

One can check that a subbundle F is parallel if and only if the connectionon E maps sections of F to F , i.e. ∇u(ιξ) ∈ ι(Fp) for any u ∈ TpM andξ ∈ Γ (F ). If F is parallel there is a unique connection ∇F on F such that

6 Although natural constructions can be done much more generally, for example whena pair of complementary subbundles is supplied.


ι∇Fu ξ = ∇u (ιξ)

for every u ∈ TM and ξ ∈ Γ (F ). Note also that if ∇ is compatible with ametric g on E, and F is a parallel subbundle of E, then ∇F is compatiblewith the induced metric gF .

Example 1.64. An important example which will reappear later (cf. Sec-tion 6.5) is the following: Let E be a vector bundle with connection ∇. Thenthe bundle of symmetric 2-tensors on E is a parallel subbundle of the bundleof 2-tensors on E. To prove this we need to check that ∇UT is symmetricwhenever T is a symmetric 2-tensor. But this is immediate: We have for anyX,Y ∈ Γ (E) and U ∈X (M),

(∇UT ) (X,Y ) = U(T (X,Y ))− T (∇UX,Y )− T (X,∇UY )

= U(T (Y,X))− T (∇UY,X)− T (Y,∇UX)

= (∇UT ) (Y,X),

so ∇UT is symmetric as required.

1.8.8 The Taylor Expansion of a Riemannian Metric. As an ap-plication of the pullback structure seen in this section, we compute the Taylorexpansion of the Riemannian metric in exponential normal coordinates. Inparticular the curvature tensor is an obstruction to the existence of localcoordinates in which the second derivatives of the metric tensor vanish.

Theorem 1.65. Let (M, g) be a Riemannian manifold. With respect to ageodesic normal coordinates system about p ∈ M , the metric gij may beexpressed as:

gij(u1, . . . , un) = δij −

1

3Rikj`u

ku` +O(‖u‖3).

Remark 1.66. The proof produces a complete Taylor expansion about u = 0(see also [LP87, pp. 60-1]).

Proof. Consider ϕ : R2 →M defined by

ϕ(s, t) = expp(tV (s))

where V (s) ∈ Sn−1 ⊂ TpM . Let V (0) = u, V ′(0) = v.As ϕ∗ : T(s,t)R2 → (ϕ∗TM)(s,t), ϕ∗∂s = (expp)∗(tV

′(s)) = t ∂sVi(s)(∂i)ϕ.

Thus we find that the pullback metric of g via ϕ is

(ϕ∗g)(∂s, ∂s)∣∣(s,t)

= gϕ(ϕ∗∂s, ϕ∗∂s)∣∣(s,t)

= t2 ∂sVi∂sV

jgij(ϕ(s, t)). (1.21)

Note that ϕ∇gϕ ∈ Γ (TR2 ⊗ (ϕ∗TM)∗ ⊗ (ϕ∗TM)∗), so by Proposition 1.58:


0 = (ϕ∇gϕ)(∂t, ϕ∗∂s, ϕ∗∂s)

= ∂tgϕ(ϕ∗∂s, ϕ∗∂s)− gϕ(ϕ∇∂tϕ∗∂s, ϕ∗∂s)− gϕ(ϕ∗∂s,ϕ∇∂tϕ∗∂s).

So by taking (∂t)k derivatives of (1.21), we find that

(∂t)kgϕ(ϕ∗∂s, ϕ∗∂s)

=

k∑`=0

(k

`

)gϕ(ϕ∇(k−`)

∂tϕ∗∂s,

ϕ∇(`)∂tϕ∗∂s)

=(k(k − 1)g

(k−2)ij (t) + 2k tg

(k−1)ij (t) + t2g

(k)ij (t)

)∂sV

i∂sVj .

Evaluating this expression at (s, t) = (0, 0) gives

k(k − 1)g(k−2)ij (0)vivj =

k∑`=0

(k

`

)gϕ(ϕ∇(k−`)

∂tϕ∗∂s,

ϕ∇(`)∂tϕ∗∂s). (1.22)

Note that ϕ∗∂s∣∣(0,0)

= 0, ϕ∗∂t∣∣(0,0)

= u, ϕ∇∂tϕ∗∂t∣∣(0,0)

= 0 and

ϕ∇∂tϕ∗∂s∣∣(0,0)

= (∂t(ϕ∗∂s)i)(∂i)ϕ

∣∣(0,0)

= v.

We now claim:

Claim 1.67. Under the assumption ϕ∇∂tϕ∗∂t ≡ 0,

ϕ∇(k)∂tϕ∗∂s =

k−2∑`=0

(k − 2

`

)(∇(k−2−`)R)ϕ

(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸k−2−` times

, ϕ∇(`)∂t

(ϕ∗∂s) , ϕ∗∂t)ϕ∗∂t.

Proof of Claim. The case k = 2 is proved as follows:

ϕ∇(2)∂tϕ∗∂s = ϕ∇∂t (ϕ∇∂sϕ∗∂t) by Proposition 1.60

= ϕ∇∂s (ϕ∇∂tϕ∗∂t) +ϕ∇R(∂s, ∂t)(ϕ∗∂t) by definition of

ϕ∇R

=ϕ∇R(∂s, ∂t)(ϕ∗∂t) by assumption

= ∇R(ϕ∗∂s, ϕ∗∂t)(ϕ∗∂t) by Proposition 1.59

as required. For the inductive step we suppose the identity is true for k = j,and differentiate. Since ϕ∇∂tϕ∗∂t = 0, we find:

1.9 Integration and Divergence Theorems 31

ϕ∇∂t

(j−2∑`=0

(j − 2

`

)(∇(j−2−`)R)ϕ

(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸j−2−` times

, ϕ∇(`)∂tϕ∗∂s, ϕ∗∂t

)ϕ∗∂t

)

=

j−2∑`=0

(j − 2

`

)ϕ∇∂t(∇(j−2−`)R)ϕ

(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸j−2−` times

, ϕ∇(`)∂tϕ∗∂s, ϕ∗∂t

)ϕ∗∂t

+

j−2∑`=0

(j − 2

`

)(∇(j−2−`)R)ϕ

(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸j−2−` times

, ϕ∇(`+1)∂t

ϕ∗∂s, ϕ∗∂t)ϕ∗∂t

=

j−2∑`=0

(j − 2

`

)(∇(j−1−`)R)ϕ

(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸j−1−` times

, ϕ∇(`)∂tϕ∗∂s, ϕ∗∂t

)ϕ∗∂t

+

j−2∑`=0

(j − 2

`

)(∇(j−2−`)R)ϕ(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸

j−2−` times

, ϕ∇(`+1)∂t

ϕ∗∂s, ϕ∗∂t)ϕ∗∂t

=

j−1∑`=0

((j − 2

`

)+

(j − 2

`− 1

))(∇(j−1−`)R)ϕ

(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸j−1−` times

,ϕ∇(`)∂tϕ∗∂s, ϕ∗∂t

)ϕ∗∂t

=

j−1∑`=0

(j − 1

`

)(∇(j−1−`)R)ϕ

(ϕ∗∂t, . . . , ϕ∗∂t︸︷︷︸j−1−` times

, ϕ∇(`)∂tϕ∗∂s, ϕ∗∂t

)ϕ∗∂t

completing the induction. Here we used the identity ϕ∇∂t(∇(j−2−`)R)ϕ =∇ϕ∗∂t

(∇(j−2−`)R

)to get from the first equality to the second — this is the

characterisation of the pullback connection in Theorem 1.56.

Finally, we compute the Taylor expansion of t 7→ gij(ϕ(0, t)) around

t = 0 (so that gij(γu(t)) = gij(0) + g(1)ij (0) + 1

2g(2)ij (0) + . . .). The 0-

order term gϕ(∂i, ∂j)∣∣(0,0)

= gij(γu(0)) = gij(p) = δij as we are working

in normal coordinates. The 1st order vanishes and by (1.22) we find that

12 g(2)ij (0)vivj = 8 gϕ(Rϕ(v, u)u, v), so g

(2)ij (0) = − 2

3Rikj`uku`. The theorem

now follows.

1.9 Integration and Divergence Theorems

If (M, g) is an oriented Riemannian manifold with boundary and g is theinduced Riemannian metric on ∂M , then we define the volume form of gby dσg = ινdµg|∂M . In particular if X is a smooth vector field, we haveιXdµg|∂M = 〈X, ν〉g dσg. In light of this, we define the divergence divX tobe the quantity that satisfies:

d(ιXdµ) = divXdµ. (1.23)


Theorem 1.68 (Divergence theorem). Let (M, g) be a compact orientedRiemannian manifold. If X is a vector field, then

ˆM

divXdµ =

ˆ∂M

〈X, ν〉g dσ.

In particular, if M is closed then´M

divXdµ = 0.

Proof. Define the (n− 1)-form α by α = ιXdµ. So by Stokes’ theorem,

ˆM

divXdµ =

ˆM

dα =

ˆ∂M

α =

ˆ∂M

ιXdµ =

ˆ∂M

〈X, ν〉 dσ.

From the divergence theorem we have the following useful formulæ:

Proposition 1.69 (Integration by parts). On a Riemannian manifold(M, g) with u, v ∈ C∞(M) the following holds:

(a) On a closed manifold, ˆM

∆udµ = 0.

(b) On a compact manifold,

ˆM

(u∆v − v∆u)dµ =

ˆ∂M

(u∂u

∂ν− v ∂u

∂ν

)dσ.

In particular, on a closed manifold´Mu∆v dµ =

´Mv∆udµ.

(c) On a compact manifold,

ˆM

u∆v dµ+

ˆM

〈∇u,∇v〉 dµ =

ˆ∂M

∂v

∂νu dσ.

In particular, on a closed manifold´M〈∇u,∇v〉 dµ = −

´Mu∆v dµ.

1.9.1 Remarks on the Divergence Expression. We seek a localexpression for the divergence, defined by (1.23), and show it is equivalent tothe trace of the covariant derivative. That is,

divX = tr∇X = tr (∇X)( · , · ) = (∇iX)(dxi) (1.24)

Lemma 1.70. The divergence divX of a vector field X, defined by (1.23),can be expressed in local coordinates by

div(Xi∂i) =1√

det g∂i(X

i√

det g). (1.25)

Proof. By Cartan’s formula7 and (1.23) we have

7 Which states that LXω = ιX(dω) +d(ιXω), for any smooth vector field X and anysmooth differential form ω.

1.9 Integration and Divergence Theorems 33

div(X)dµ = d ιXdµ = (d ιX + ιX d)dµ = LXdµ.

From the left-hand side we find that (divXdµ)(∂1, . . . , ∂n) = divX√

det gand from the right-hand side we find that

(LXdµ)(∂1, . . . , ∂n) = LX(√

det g)− dµ(. . . ,LX∂i, . . .)

= X(√

det g) + dµ(. . . , (∂iXj)∂j , . . .)

= X(√

det g) + (∂iXj)δij

√det g

= ∂i(Xi√

det g)

Claim 1.71. The definitions of divX given by (1.23) and (1.24) coincide.

Proof. As (∇Xdxj)(∂i) = −dxj(∇X∂i), equation (1.24) implies that

(∇iX)(dxi) = ∂iXi −X(∇idxi) = ∂iX

i + Γ iijXj (1.26)

On the other hand, (1.25) implies that

divX = ∂iXi +

1√det g

Xi∂i(√

det g),

where by the chain rule

∂i(√

det g) =1

2

1√det g

∂ det g

∂gpq

∂gpq∂xi

=1

2

√det ggpq

∂gpq∂xi

= gpqΓ `ipg`q√

det g

= Γ pip√

det g.

Chapter 2

Harmonic Mappings

When considering maps between Riemannian manifolds it is possible to asso-ciate a variety of invariantly defined ‘energy’ functionals that are of geomet-rical and physical interest. The core problem is that of finding maps which are‘optimal’ in the sense of minimising the energy functional in some class; oneof the techniques for finding minimisers (or more generally critical points) isto use a gradient descent flow to deform a given map to an extremal of theenergy.

The first major study of Harmonic mappings between Riemannian mani-folds was made by Eells and Sampson [ES64]. They showed, under suitablemetric and curvature assumptions on the target manifold, gradient lines doindeed lead to extremals.

We motivate the study of Harmonic maps by considering a simple problemrelated to geodesics. Following this we discuss the convergence result of Eellsand Sampson. The techniques and ideas used for Harmonic maps providesome motivation for those use later for Ricci flow, and will appear againexplicitly when we discuss the short-time existence for Ricci flow.

2.1 Global Existence of Geodesics

By Lemma 1.23, geodesics enjoy local existence and uniqueness as solutionsof an initial value problem. In this section we examine a global existenceproblem:

Theorem 2.1 (Global Existence). On a compact Riemannian manifoldM , every homotopy class of closed curves contains a curve which is geodesic.

To prove this, we will use methods from Geometric analysis. Specifically,we want to start with some curve (in the homotopy class under consideration)and let it evolve according to a particular pde that decreases its energyuntil, in the limit, the curve becomes energy minimising. This is achieved bygradient descent techniques commonly known as the heat flow method.

35

36 2 Harmonic Mappings

Firstly, we say two closed curves1 γ1, γ2 : S1 → M are equivalent if thereexists a homotopy between them. That is, they are equivalent if there existsa continuous map c : S1× [0, 1]→M such that c(t, 0) = γ0(t), c(t, 1) = γ1(t)for all t ∈ S1.

Now consider a mapping u = u(s, t) : S1 × [0,∞) → M that evolve (inlocal coordinates) according to the pde:

∂ui

∂t=∂2ui

∂s2+ Γ ijk(u(s, t))

∂uj

∂s

∂uk

∂ss ∈ S1, t ≥ 0 (2.1)

u(s, t) = γ(s) s ∈ S1

where S1 is parametrised by s, t denotes the ‘time’ and γ : S1 → M isarbitrary curve in the given homotopy class. Note that the correspondingsteady state equation is the geodesic equation (1.2). Although the equationis written in local coordinates, we will see in Section 2.2 that it does in fact— in a more general context — make sense independent of the choice of thosecoordinates.

Proof (Theorem 2.1, sketch only). We focus our interests on the geometricaspects of the proof and where appropriate, quote the necessary pde results.

Step 1: By general existence and uniqueness theory, there exists a solutionto (2.1), at least on a short time interval [0, T ). Consequently, the maximumtime interval of existence is nonempty and open.

Step 2: Using pde regularity theory, we show that a solution u(s, t) hasbounded spatial derivatives independent of time t. To do this compute(∂2s − ∂t

)gij∂su

i∂suj = gij,k(∂2

suk − ∂tuk)∂su

i∂suj + 4gij,k∂su

k∂2su

i∂suj

+ 2gij∂2su

i∂2su

j + 2gij(∂3su

i − ∂s∂tui)∂suj

+ gij,k`∂sui∂su

j∂suk∂su

`.

From (2.1) note that:

∂2su

k − ∂tuk = −Γ kij∂sui∂suj (2.2a)

∂3su

i − ∂s∂tui = −Γ ijk,`∂su`∂suj∂suk − 2Γ ijk∂2su

j∂suk. (2.2b)

By working in normal coordinates at a point (so that the first derivatives ofthe metric gij and the Christoffel symbol Γ kij vanish), insert equations (2.2a)and (2.2b) (with the help of equation (1.9)) into the above computation, sothat (

∂2s − ∂t

)gij∂su

i∂suj = 2gij∂

2su

i∂2su

j +XXXgij,k`∂sui∂su

j∂suk∂su

`

− (XXXgij,k` +gik,j` −gjk,i`)∂sui∂su

j∂suk∂su

`

= 2gij∂2su

i∂2su

j . (2.3)

1 For technical convenience, we parametrise closed curves on S1 as it is unnecessaryto stipulate end conditions, like γ(0) = γ(1), for closed loops.

2.1 Global Existence of Geodesics 37

Hence (∂2

∂s2− ∂

∂t

)gij∂ui

∂s

∂uj

∂s≥ 0.

Therefore ‖∂su‖2 = gij∂sui∂su

j is a subsolution of (2.1). By the parabolicmaximum principle

sups∈S1

gij(u(s, t))∂ui

∂s(s, t)

∂uj

∂s(s, t)

is a non-increasing function of t. In particular gij∂sui∂su

j is bounded bysome constant C independent of t and s. From this, regularity theory tellsus that ∂su

i stays bounded. So by standard bootstrapping methods we gaintime-independent control of higher derivatives.

Step 3: When a solution exists on [0, T ), u(s, t) will converge to a smoothcurve u(s, T ) as t→ T . By regularity, this curve can be taken as a new initialvalue. In which case the solution can be continued beyond T . Therefore themaximum interval is closed and so, in conjunction with Step 1, the solutionexists for all times t > 0.

Step 4: We directly show t 7→ E(u(·, t)) is a decreasing function. To dothis, note that:

d

dtE(u(·, t)) =

1

2

∂

∂t

ˆS1

gij∂sui∂su

j

=1

2

ˆS1

(gij,k∂tu

k∂sui∂su

j + 2gij∂s∂tui∂su

j)

=1

2

ˆS1

((gij,k − 2gik,j)∂su

i∂suj∂tu

k − 2gij∂tui∂2su

j),

where the last equality follows from integration by parts. Using (2.1) observethat

gip∂tui = gip∂

2su

i +1

2(gjp,k + gpk,j − gjk,p)∂suj∂suk.

Multiply this by ∂tup on both sides and rearrange to get

(gjk,p − 2gjp,k)∂suj∂su

k∂tup = gip∂

2su

i∂tup − 2gip∂tu

i∂tup.

Thusd

dtE(u(·, t)) = −

ˆS1

gij∂tui∂tu

j = −ˆS1

‖∂tu‖2 ≤ 0.

Since E is also non-negative (i.e. bounded below) we can find a sequencetn →∞ for which u(·, tn) will converge to a curve that satisfies the geodesicequation (1.2).

Step 5: Finally, by a similar computation (again in normal coordinates),we find that


d2

dt2E(u(·, t)) = 2

ˆS1

gij∂s∂tui∂s∂tu

j ≥ 0.

Thus the energy E(u(·, t)) is a convex function in t. As we already haveddtE(u(·, tn))→ 0 for some sequence tn →∞, we conclude that d

dtE(u(·, t))→0 for t → ∞. Thus, by our pointwise estimate ∂tu(s, t) → 0 as t → ∞, thecurve u(s) = limt→∞ u(s, t) exists and is a geodesic.

It is a rather natural step to generalise the above problem to maps betweenany Riemannian manifolds. To do so, one needs an energy functional that de-pends only on the intrinsic geometry of the domain manifold, target manifoldand the map between them. Critical points of such an energy are called har-monic maps. Upon such a abstraction, we would like to know if the higherdimensional analogue to Theorem 2.1 remains true. That is:

Given Riemannian manifolds (M, g), (N,h) and a homotopy class ofmaps between them, does there exist a harmonic map in that homotopyclass?

If M and N are compact, there are positive and negative answers to thisquestion. There is a positive answer due to [ES64] if we assume the targetmanifold N has non-positive sectional curvature. In contrast, [EW76] showedthe answer to be negative for maps of degree ±1 from the 2-torus to the 2-sphere (there are of course many other counterexamples).

As Harmonic mappings seem to be one of the most natural problems onecan pose, there are (not surprisingly) many varied examples. For instance:

- Identity and constant maps are harmonic.- Geodesics as maps S1 →M are harmonic.- Every minimal isometric immersion is a harmonic map.- If the target manifold N = Rn, then it follows from the Dirichlet principle

that f is a harmonic map if and only if it is a harmonic function in theusual sense (i.e. a solution of the Laplace equation).

- Holomorphic maps between Kahler manifolds are harmonic.- Minimal submanifolds (or more generally submanifolds with parallel

mean curvature vector) in Euclidean spaces have harmonic Gauss maps.The Gauss map takes a point in the submanifold to its tangent plane atthat point, thought of as a point in the Grassmannian of subspaces ofthat dimension.

- Harmonic maps from surfaces depend only on the conformal structure ofthe source manifold — thus by the uniformisation theorem we can workwith a constant curvature metric on the source manifold (or locally witha flat metric). The resulting equations have many nice properties — inparticular harmonic maps from surfaces into symmetric spaces have anintegrable structure which leads to many explicit solutions.

- In theoretical physics, harmonic maps are also known as σ-models (cf.Section 10.6).

2.2 Harmonic Map Heat Flow 39

In what follows, we give a proof of the convergence result of Eells andSampson [ES64] (Theorem 2.9, here) with improvements by Hartman [Har67].

2.2 Harmonic Map Heat Flow

Consider a C∞-map f : (M, g) → (N,h) between Riemannian manifolds(M, g) and (N,h). Let (xi) be a local chart on M about p ∈ M and let(yα) be a local chart on N about f(p) with fα = yα f . By Section 1.8.2,

f∗ ∈ Γ (T ∗M ⊗ f∗TN) so f∗ = (f∗)αi dx

i ⊗ (∂α)f = ∂fα

∂xi dxi ⊗ ( ∂

∂yα )f . Let

〈·, ·〉 be the inner product on the bundle T ∗M ⊗ f∗TN (in accordance withSection 1.4.6) so that

⟨f∗, f∗

⟩T∗M⊗f∗TN =

∂fα

∂xi∂fβ

∂xj⟨dxi ⊗ (∂α)f , dx

j ⊗ (∂β)f⟩

= gij(hαβ)f∂fα

∂xi∂fβ

∂xj.

In light of this, define the energy density e(f) of the map f to be

e(f) =1

2‖f∗‖2 =

1

2gij(hαβ)f

∂fα

∂xi∂fβ

∂xj(2.4)

and the energy E to be

E(f) =

ˆM

e(f)dµ(g), (2.5)

where dµ(g) is the volume form on M with respect to the metric g. The energycan be considered as a generalisation of the classical integral of Dirichlet; forif N = R (so that f : M → R) then E corresponds to the Dirichlet’s energy12

´M|∇f |2dµ(g).

As we have the connections ∇M on TM and f∇N on f∗TN , there is a con-nection (denoted simply by ∇) on each tensor bundle constructed from these(i.e. on T ∗M ⊗ f∗TN). Moreover, equation (1.12) gives a useful expressionfor the curvature of the connection constructed on T ∗M ⊗ f∗TN :

(R(U, V )f∗) (W ) = Rf∇N (U, V )(f∗W )− f∗(R∇M (U, V )W

)= RN (f∗U, f∗V ) (f∗W )− f∗

(RM (U, V )W

)(2.6)

where the second equality follows from Proposition 1.59.

2.2.1 Gradient Flow of E. For this we take a variation of the map f ,i.e. a smooth map f : M × I → N where I ⊂ R is an interval of time.

We wish to compute on TM at each time, so we define the ‘spatial tangentbundle’ to be the vector subbundle S ⊂ T (M × I) consisting of vectorstangent to the M factor, that is S = v ∈ T (M × I) : dt(v) = 0 (ref.


Example 1.62). On this bundle S → M × I we place the time-independentmetric and connection ∇ given by g and its Levi-Civita connection M∇ —which must be augmented to include the time direction by defining ∇∂tu =[∂t, u], for any u ∈ Γ (S). Further to this, we also extend the connection ∇to T (M × I) by zero so that ∇∂t ≡ 0.2

The map f : M × I → N induces the pullback bundle f∗TN over M × I,on which we can place the restriction metric and the pullback connection.We now look to compute

d

dtE(f) =

1

2

ˆM

d

dt〈f∗, f∗〉 dµ(g) =

ˆM

〈f∗,∇∂tf∗〉 dµ(g),

where ∇ is the connection on S∗ ⊗ f∗TN over M × I. By (1.7) we have

(∇∂tf∗) (∂i) = f∇∂t (f∗∂i)− f∗(∇∂t∂i

).

Using this equation with Proposition 1.60 gives

(∇∂tf∗) (∂i) = f∇∂t (f∗∂i) = f∇∂i(f∗∂t), (2.7)

since [∂t, ∂i] = 0. We denote by fαj the components of f∗ in a local frame(for instance f∗∂t = fα0 (∂α)f ), and by ∇ifαj the components of ∇f∗ (cf.Section 1.5.2.1). Since ∇g = 0 and ∇h = 0 by compatibility, we have that

∇j(gij(hαβ)ff

αi f

β0

)= gij(hαβ)f∇jfαi f

β0 + gij(hαβ)ff

αi ∇jf

β0

=⟨gij∇jfi, f∗∂t

⟩+ 〈f∗,∇∂tf∗〉

where we used (2.7) in the last term and the fact that ∇j∂t = 0 by extension.So by the Divergence theorem with (1.24), we find that

d

dtE(f) =

ˆM

〈f∗,∇∂tf∗〉 dµ

=

ˆM

∇j(gijhf (f∗∂i, f∗∂t)

)dµ−

ˆM

⟨f∗∂t, g

ij∇ifj⟩dµ

= −ˆM

〈f∗∂t, ∆g,hf〉 dµ,

where we define:

Definition 2.2. The harmonic map Laplacian

∆g,hf = trg∇f∗ := gij(∇∂if∗)(∂j).

Note that f∗∂t is the ‘variation of f ’. Hence the gradient of E, with respectto the inner product on f∗TN , is −∆g,hf and the gradient descent flow is:

2 This kind of construction is a natural one when dealing with variations of geometricstructures; we will develop this in a more general context in Section 5.3.1.


f∗∂t = ∆g,hf. (2.8)

Furthermore we say f is harmonic if ∆g,hf = 0, in which case f is a steadystate solution of (2.8). We also note that an immediate consequence of (2.8)is that

d

dtE(f) = −

ˆM

|∆g,hf |2dµ = −‖∆g,hf‖2L2(M) ≤ 0, (2.9)

so the energy decays.

Remark 2.3. The idea now is quite simple: We want to deform a given f0 :M → N to a harmonic map by evolving it along the gradient flow (2.8) sothat f(·, t) converges to a harmonic map homotopic to f0 as t→∞. This isthe so-called heat flow method.

2.2.2 Evolution of the Energy Density. The key computation is theformula for the evolution of the energy density e(f) under harmonic map heatflow (2.8). Here we compute this using the machinery of pullback connections.

First we compute an evolution equation for f∗:

(∇∂tf∗) (∂i) = f∇∂t (f∗∂i)− f∗ (∇∂t∂i)= f∇∂i

(gk`∇kf`

)= ∇∂i

(gk`∇kf`

)= gk`(∇i∇kf∗)(∂`)= gk`

((∇k∇if∗)(∂`) + (R(∂k, ∂i)f∗)(∂`)

).

In the first term we observe by symmetry (i.e. Proposition 1.60) that(∇if∗)(∂l) = f∇`(f∗∂i) − f∗(∇`∂i) = (∇lf∗)(∂i). Using this together withthe formula (2.6) for the second term implies that

(∇∂tf∗) (∂i) = (∆f∗)(∂i) + gk`RN (f∗∂k, f∗∂i)(f∗∂`)− gk`f∗(RM (∂k, ∂i)∂`

)where ∆ = gk`∇k∇`. It follows (noting that the metric on T ∗M ⊗ f∗TN isparallel since g and h are) that

∂

∂te = 〈f∗, ∆f∗〉+ gk`gijRN (f∗∂k, f∗∂i, f∗∂`, f∗∂j)

− gk`hf(f∗((MRic) p

k ∂p), f∗∂`

)= ∆e− ‖∇f∗‖2 + gk`gijRN (f∗∂k, f∗∂i, f∗∂`, f∗∂j)

− gk`hf(f∗((MRic) p

k ∂p), f∗∂`

). (2.10)

Remark 2.4. We note that if M is compact3 then gk`hf (f∗(RicM (∂k), f∗∂`) ≥−Ce, and if the target manifold N has non-positive curvature then the term

3 In which case RicM is bounded, so RicM ≥ −C2g for some constant C.


gk`gijRN (f∗∂k, f∗∂i, f∗∂`, f∗∂j) ≤ 0. Thus under these assumptions we have

∂e

∂t≤ ∆e+ Ce (2.11)

for some constant C.

2.2.3 Energy Density Bounds. The problem here is to derive a boundon the energy density e = e(f), having deduced the inequality (2.11). Theidea is to use this inequality together with the bound on the Dirichlet energy(that is, a bound on the integral of e) to deduce the bound on e.

The argument is a variant on a commonly used technique called Moseriteration [Mos60], which uses the Sobolev inequality to ‘bootstrap’ up froman Lp bound to a sup bound. The argument is a little simpler in the ellipticcase (for instance see [PRS08, p. 118] or [GT83]).

To do this we need the following two ingredients. First, we need a compu-tation for the time derivative of an Lp norm of e:

d

dt

ˆM

e2p ≤ 2p

ˆM

e2p−1 (∆e+ Ce)

≤ −2p(2p− 1)

ˆM

e2p−2 |∇e|2 + 2pC

ˆM

e2p

= −2(2p− 1)

p

ˆM

|∇(ep)|2 + 2pC

ˆM

e2p

≤ −2 ‖ep‖2W 1,2 + 2pC

ˆM

e2p.

The ‘problem’ term on the right can be absorbed by multiplying by an ex-ponential, yielding

d

dt

(e−Ct‖e‖2p

)≤ −1

pe−Ct‖e‖1−2p

2p ‖ep‖2W 1,2 . (2.12)

The second ingredient is the Gagliardo-Nirenberg inequality. It is a standardpde result (for instance see [Eva98, p. 263]) which says that there exists aconstant C1 such that for any W 1,2 function f on M ,

‖f‖2 ≤ C1‖f‖nn+2

W 1,2‖f‖2

n+2

1 . (2.13)

The idea here is that if the L1 norm is controlled, then we can use the goodW 1,2 norm to yield a power bigger than 1 of the L2 norm. Substituting thisinto (2.12) gives

d

dt

(e−Ct‖e‖2p

)≤ −C

21

pe−Ct‖e‖1+ 4p

n2p ‖e‖−

4pn

p ,


which on rearrangement yields

d

dt

(e−Ct‖e‖2p

)− 4pn ≥ 4C2

1

n

(e−Ct‖e‖p

)− 4pn

. (2.14)

From here we will use an induction argument to bound higher and higherLp norms with a bound independent of p. This will imply a sup bound since‖e‖∞ = limp→∞ ‖e‖p.

The first application of this is straightforward, since we have the energybound ‖e‖1 ≤ E. Plugging this into (2.14) with p = 1 gives

d

dt

(e−Ct‖e‖2

)− 4n ≥ 4C2

1

n

(e−CtE

)− 4n

≥ 4C21

nE−4/nt,

since the exponential term is always at least 1. This is equivalent to a L2

bound on e decaying like t−4/n. This is the initial step, now we use inductionto prove the following:

Theorem 2.5. For each k ∈ N,

(e−Ct‖e‖2k

)− 4n ≥

(4C2

1 t

n

)2(1−2−k)

B−2k E−4/n,

where B1 = 1 and Bk+1 ≤ 2k+1

2k Bk.

The induction is easy from (2.14) with p = 2k:

d

dt

(e−Ct‖e‖2k+1

)− 2k+2

n ≥ 4C21

n

(e−Ct‖e‖2k

)− 2k+2

n

≥ 4C21

n

(4C2

1

n

)2(2k−1)

B−2k+1

k E−2k+2

n t2(2k−1).

Integration gives

(e−Ct‖e‖2k+1

)− 2k+2

n ≥ 4C21

n

(4C2

1

n

)2(2k−1)

B−2k+1

k E−2k+2

nt2k+1−1

2k+1 − 1

≥(

4C21 t

n

)2k+1−1

B−2k+1

k E−2k+2

n 2−(k+1).

Taking the power 2−k gives


(e−Ct‖e‖2k+1

)− 4n ≥

(4C2

1 t

n

)2(1−2−(k+1)

B−2k E−4/n2−

k+1

2k

≥(

4C21 t

n

)2(1−2−k)

B−2k+1E

−4/n

for Bk+1 = 2k+1

2k . This completes the induction and the bound for e.

2.2.4 Higher Regularity. Once e is controlled, so are all higher deriv-atives of f . This is a general result needing no special curvature assumptions:

Proposition 2.6. Let (M, g) and (N,h) be compact Riemannian manifolds.Let f : M × [0, T )→ N be a smooth solution of the harmonic map heat flow(2.8) with bounded energy density e. Then there exist constants Ck, k ≥ 1depending on supM×[0,T ) e, |RM |Ck and |RN |Ck such that

‖∇(k)f∗‖ ≤ Ck(

1 + t−k/2).

For example, the evolution of ∇f∗ is given by

∇t (∇f∗) = ∆∇f∗ +RN ∗ (f∗)2 ∗ ∇f∗

+RM ∗ ∇f∗ +∇RN ∗ (f∗)4 +∇RM ∗ f∗,

so that

∂

∂t‖∇f∗‖2 ≤ ∆‖∇f∗‖2 − 2‖∇2f∗‖2 + C1‖∇f∗‖2 + C2,

where C1 and C2 depend on |RM |, |RN |, |∇RM |, |∇RN | and e. This gives

∂

∂t

(t‖∇f∗‖2 + ‖f∗‖2

)≤ ∆

(t‖∇f∗‖2 + ‖f∗‖2

)+ (tC1 − 1)‖∇f∗‖2 + (tC2 + C3),

which gives t‖∇f∗‖2 ≤ C for 0 < t < 1/C1. The same argument appliedon later time intervals gives a bound for any positive time, proving the casek = 1. For a similar argument applied to the Ricci flow see Theorem 7.1.

2.2.5 Stability Lemma of Hartman. Let f(x, t, s) smooth family ofsolutions to (2.8) depending on a parameter s and a ‘time’ t ∈ I. We wantto prove to following result by Hartman [Har67, p. 677].

Lemma 2.7 (Hartman). Let F (x, s) : M × [0, s0] → N be of class C1

and for fixed s, let f(x, t, s) be a solution of (2.8) on 0 ≤ t ≤ T such thatf(x, 0, s) = F (x, s). Then for all s ∈ [0, s0],


supM×t×s

hf (f∗∂s, f∗∂s)

is non-increasing in t.

Proof. Since we now have a map with two parameters (s representing thevariation through the family of harmonic map heat flows, and t representingthe time parameter of the heat flows) we are now working on bundles overM × I1 × I2, where I1 and I2 are real intervals. The bundles we need areagain the spatial tangent bundle S, defined by v ∈ T (M×I1×I2) : dt(v) =ds(v) = 0, and the pullback bundle f∗TN . On the former we have the ‘time-independent’ metric g, and a connection given by the Levi-Civita connectionin spatial directions, together with the prescription ∇∂t∂i = 0 and ∇s∂i = 0(note that this choice gives a compatible metric and connection on S, forwhich ∇∂s and ∇∂t commute with each other and with ∇∂i). On f∗TN wehave as usual the pullback metric and connection.

We compute an evolution equation for f∗∂s:

∇∂t(f∗∂s) = ∇∂s(f∗∂t)= ∇s

(gk`(∇kf∗)(∂`)

)= gk`

((∇k∇sf∗)(∂`) + (R(∂k, ∂s)f∗)(∂`)

)= gk`(∇k∇`f∗)(∂s) + gk`RN (f∗∂k, f∗∂s)(f∗∂`)

= ∆(f∗∂s) + gk`RN (f∗∂k, f∗∂s)(f∗∂`)

since RM (∂k, ∂s) = 0 and ∇`∂s = 0. Therefore the evolution of ‖f∗∂s‖2 isgiven by

∂

∂t‖f∗∂s‖2 = 2〈f∗∂s, ∆(f∗∂s)〉+ 2gk`RN (f∗∂k, f∗∂s, f∗∂`, f∗∂s)

≤ ∆‖f∗∂s‖2 − 2‖∇f∗∂s‖2,

since the curvature term can be written as a sum of sectional curvatures (cf.Proposition 1.49).

By letting Q = ‖f∗∂s‖2 we note, for fixed s, that Q(x, s, t) satisfies theparabolic differential inequality ∂tQ−∆Q ≤ 0. Hence the maximum principleimplies that if 0 ≤ τ ≤ t then maxxQ(x, s, t) ≤ maxxQ(x, s, τ) for everyfixed s ∈ [0, s0]. Consequently maxx,sQ(x, s, t) ≤ maxx,sQ(x, s, τ). Hencethe desired quantity is non-increasing.

An important application of Hartman’s Lemma is to prove that the dis-tance between homotopic solutions of the flow cannot increase. We definethe distance between two homotopic maps f0 and f1 as follows: If H :M × [0, 1]→ N is a smooth homotopy from f0 to f1, so that H(x, 0) = f0(x)and H(x, 1) = f1(x), then the length of H (in analogy to Section 1.4.1.1) isdefined to be


L(H) = supx∈M

ˆ 1

0

∥∥∥∂H∂s

(x, s)∥∥∥ ds.

We define d(f0, f1) to be the infimum of the lengths over all homotopiesfrom f0 to f1. When N is non-positively curved the infimum is attained bya smooth homotopy H in which s 7→ H(x, s) is a geodesic for each x ∈ M ,and in this case L(H) = sup|∂sH(x, s)| : x ∈M for each s ∈ [0, 1].

Corollary 2.8 ([Har67, Sect. 8]). If N is non-positively curved and f0(x, t)and f1(x, t) are solutions of the harmonic map heat flow with homotopicinitial data, then t 7→ d(f0(·, t), f1(·, t)) is non-increasing.

Proof. Fix t0 ≥ 0 and let H be the minimising homotopy from f0(·, t0) tof1(·, t0). There exists δ > 0 such that the flows f(x, t, s) with f(x, t0, s) =H(x, s) exist for t0 ≤ t ≤ t0 + δ. By Lemma 2.7, sup|∂sf(x, t, s)| : x ∈Mis non-increasing in t for each s, and therefore for t0 ≤ t ≤ t0 + δ,

d(f0(t), f1(t)) ≤ L[H(·, t, ·)] = supx∈M

ˆ 1

0

∥∥∥∂H∂s

(x, t, s)∥∥∥ ds

≤ˆ 1

0

supx∈M

∥∥∥∂H∂s

(x, t, s)∥∥∥ ds

≤ˆ 1

0

supx∈M

∥∥∥∂H∂s

(x, t0, s)∥∥∥ ds

= d(f0(t0), f1(t0)).

2.2.6 Convergence to a Harmonic Map.

Theorem 2.9 (Eells and Sampson). If N is a non-positively curved com-pact manifold, then f(·, t) converges in C∞(M,N) to a limit f in the samehomotopy class as f0 with ∆g,hf = 0.

Proof. The energy decay formula (2.9) implies that

ˆ ∞0

‖∆g,hf(·, t)‖2L2(M) <∞.

Therefore there exists a sequence tn →∞ such that ‖∆g,hf(·, tn)‖L2(M) → 0.Since all higher derivatives are bounded by Proposition 2.6, this also implies‖∆g,hf(·, tn)‖ → 0 uniformly. Also using the higher derivative bounds andthe Arzela-Ascoli theorem, by passing to a subsequence we can ensure thatf(·, tn) converges in C∞ to a limit f which therefore satisfies ∆g,hf = 0. Thefunction f is in the same homotopy class as f0, since for large n we havedN (f(x, tn), f(x)) < inj(N) and there is a unique minimising geodesic fromf(x, tn) to f(x), which depends smoothly on x. Following these geodesicsdefines a homotopy from f(·, tn) to f .


Stronger convergence follows from Corollary 2.8 (essentially the argumentas [Har67, Sect. 4]): Since d(f(·, t), f) is non-increasing, and d(f(·, tn), f)→ 0,we have d(f(·, t), f) → 0, which implies f(·, t) converges to f uniformly.Convergence in C∞ follows since all higher derivatives are bounded.

2.2.7 Further Results.

2.2.7.1 Uniqueness. One can also show that the harmonic map is essen-tially unique in its homotopy class. This can be done using the followingresult by Hartman [Har67, p. 675].

Theorem 2.10. Let f1(x) and f2(x) be two homotopic harmonic maps fromM into the non-positively curved manifold N . For fixed x, let F (x, s) be theunique geodesic from F (x, 0) = f1(x) to F (x, 1) = f2(x) in the homotopyclass determined by the homotopy between f1 and f2, and let the parameters ∈ [0, 1] be proportional to arc length.

Then F (·, s) is harmonic for each s ∈ [0, 1], and E(F (·, s)) = E(f1) =E(f2). Furthermore, the length of the geodesic F (x, ·) is independent of x.

Remark 2.11. This cannot be improved since a torus S1×S1 has non-positive(zero!) curvature, and has an infinite family of homotopic harmonic mapsfrom S1 given by z 7→ (z, z0) for fixed z0 ∈ S1.

If the sectional curvatures of N are strictly negative then this cannot occur:

Theorem 2.12. If N has negative sectional curvature, then a harmonic mapf : M → N is unique in its homotopy class, unless it is constant or maps Monto a closed geodesic. In the latter case, non-uniqueness can only occur byrotations of this geodesic.

2.2.7.2 Dirichlet and Neumann Problems. It is interesting to note thatRichard S. Hamilton, with some advice and encouragement from James EellsJr., looked at solving the Dirichlet problem for harmonic mappings into non-positively curved manifolds. In [Ham74] he was able to prove the following.

Theorem 2.13. Suppose N is a compact manifold with nonempty boundary∂M , and N is complete with non-positive sectional curvature and convex (orempty) boundary. If f0 : M → N is continuous, then the parabolic system

∂f

∂t(x, t) = ∆g,hf(x, t) (x, t) ∈M × (0,∞)

f(x, 0) = f0(x) x ∈Mf(y, t) = f0(y) y ∈ ∂M

has a smooth solution f(x, t) for all t. As f(x, t) converges to the uniqueharmonic map homotopic to f0 with the same boundary values as f0 on ∂M .

Hamilton also treated natural Neumann and mixed boundary problems withsimilar results.

Chapter 3

Evolution of the Curvature

The Ricci flow is introduced in this chapter as a geometric heat-type equationfor the metric. In Section 3.4 we derive evolution equations for the curvature,and its various contractions, whenever the metric evolves by Ricci flow. Theseequations, particularly that of Theorem 3.14, are pivotal to our analysisthroughout the coming chapters. In Section 3.5.3 we discuss a historical res-ult concerning the convergence theory for the Ricci flow in n-dimensions.This will motivational much of the Bohm and Wilking analysis discussed inChapter 11.

3.1 Introducing the Ricci Flow

The Ricci flow was first introduced by Richard Hamilton in the early 1980s.Inspired by Eells and Sampson’s work on Harmonic map heat flow [ES64](where they take maps between manifolds and try to ‘make them better’),Hamilton speculated that it should be possible to take other geometric ob-jects, for instance the metric gij , and try to ‘improve it’ by means of a heat-type equation.

In looking for a suitable parabolic equation, one would like∂gij∂t to equal

a Laplacian-type expression involving second-order derivatives of the metric.Computing derivatives of the metric with the Levi-Civita connection is ofno help, as they vanish in normal coordinates. However, computing deriv-atives with respect to a fixed background connection does give somethingnon-trivial. In particular, the Ricci tensor in Lemma 1.47 has an expressionwhich involves second derivatives the components of the metric, thus it is anatural candidate for such a (2, 0)-tensor (specifically, the last term in thebracket in Lemma 1.47 is − 1

2 times a ‘Laplacian’ of the metric computedusing coordinate second derivatives). Taking this factor into account, we areled to the evolution equation

∂

∂tgij = −2 Ricij (3.1)

49

50 3 Evolution of the Curvature

known simply as the Ricci flow. Further motivation comes from the expres-sion for the metric in exponential coordinates given in Theorem 1.65, as itimplies that the Laplacian of the metric computed at the origin in exponentialcoordinates is equal to − 2

3Ricij .In contrast to other natural geometric equations (such as the minimal

surface equation and its parabolic analogue the mean curvature flow; theharmonic map equation and the associated flow; and many other examples),the Ricci flow was not initially derived as the gradient flow of a geometricfunctional. However despite this, Perelman’s work [Per02] shows there is anatural energy functional lying behind the Ricci flow. We discuss this inChapter 9.

In light of (3.1), it is clear that we are no longer interested in just a singlemanifold (M, g), but rather — assuming a suitable local existence theory —a manifold with a one-parameter family of metrics t 7→ g(t) parametrisedby a ‘time’ t. We adopt this point of view here, taking the metrics g(t) assections of the fixed bundle Sym2T ∗M . The rate of change in time of themetric makes sense since at each point p ∈ M we are simply differentiatingg(t) in the vector space given by the fibre of this bundle at p. Later (inChapter 5) we will see that there are advantages in instead working withbundles defined over M × R, where a more geometrically meaningful notionof the time derivative can be used.

3.1.1 Exact Solutions. In order to get a feel for the evolution equation,we present some simple solutions of the Ricci flow.

3.1.1.1 Einstein Metrics. If the initial metric is Ricci flat, that is Ric =0, then clearly the the metric remains stationary for all subsequent times.Concrete examples of this are the Euclidean space Rn and the flat torusTn = S1 × · · · × S1.

If the initial metric is Einstein, that is Ric(g0) = λg0 for some λ ∈ R,then a solution g(t) with g(0) = g0 is given by g(t) = (1− 2λt)g0. The casesλ > 0, λ = 0 and λ < 0 correspond to shrinking, steady and expandingsolutions. The simplest shrinking solution is that of the unit sphere (Sn, g0).Here Ric(g0) = (n − 1)g0, so g(t) = (1 − 2(n − 1)t)g0. Thus the spherewill collapse to a point in finite time T = 1/2(n − 1). By contrast, if theinitial metric g0 were hyperbolic, Ric(g0) = −(n − 1)g0 so the evolutiong(t) = (1 + 2(n−1)t)g0 will expand the manifold homothetically for all time.In this case, the solution only goes back to T = −1/2(n− 1) upon which themetric explodes out of a single point.

3.1.1.2 Quotient Metrics. If the initial Riemannian manifold N = M/Gis a quotient of a Riemannian manifold M by a discrete group of isometriesG, it will remain so under the Ricci flow — as the flow on M preserves theisometry group. For example, the projective space RPn = Sn/Z2 of constantcurvature shrinks to a point, as does its cover Sn.

3.1 Introducing the Ricci Flow 51

3.1.1.3 Product Metrics. If we take the product metric (cf. Section 1.4.2)on a product manifold M × N initially, the metric will remain a productmetric under the Ricci flow. Hence the metric on each factor evolves by theRicci flow independently of the other.

For example, on S2 × S1 the S2 shrinks to a point while S1 stays fixed sothat the manifold collapses to a circle. Moreover, if we take any Riemannianmanifold (M, g) and evolve the metric g by the Ricci flow, then it will alsoevolve on the product manifold M × Rn, n ≥ 1, with product metric g ⊕(dx2

1 + · · ·+ dx2n) — since the flow is stationary on the Rn part.

3.1.2 Diffeomorphism Invariance. The curvature of a manifold Mcan be thought of as the obstruction to being locally isometric to Euclideanspace. Indeed, a Riemannian manifold is flat if and only if its curvature tensorvanishes identically. Moreover, recall that:

Theorem 3.1. The Riemann curvature tensor R is invariant under localisometries: If ϕ : (M, g)→ (M, g) is a local isometry, then ϕ∗R = R.

So if ϕ : M → M is a time-independent diffeomorphism such that g(t) =ϕ∗g(t) and g is a solution of the Ricci flow, we see that

∂

∂tg = ϕ∗

( ∂∂tg)

= ϕ∗(− 2 Ric(g)

)= −2 Ric(ϕ∗g)

= −2 Ric(g),

where the second last equality is due to Theorem 3.1. Hence g is also a solutionto the Ricci flow. Therefore we conclude that (3.1) is invariant under the fulldiffeomorphism group.1

3.1.2.1 Preservation of Symmetries. The Ricci flow preserves any sym-metries that are present in the initial metric. To see this, note that eachsymmetry is an isometric diffeomorphism of the initial metric; so the pull-back of a solution of the Ricci flow by this diffeomorphism gives a solution ofRicci flow. Since the symmetry is an isometry of the initial metric, these aresolutions of Ricci flow with the same initial data, and so are identical (as-suming the uniqueness result proved in Chapter 4). Therefore the symmetryis an isometry of the metric at any positive time.

3.1.3 Parabolic Rescaling of the Ricci Flow. Aside from the geo-metric symmetries of diffeomorphism invariance, the Ricci flow has additionalscaling properties that are essential for blow-up analysis of singularities. The

1 Note that if ϕ is time-dependent, an extra Lie derivative term is introduced intothe equation (cf. Section 4.4).


time-independent diffeomorphism invariance allows for changes in the spatialcoordinates. We can also translate in the time coordinates: If g(x, t) satisfiesRicci flow, then so does g(x, t − t0) for any t0 ∈ R. Further, the Ricci flowhas a scale invariance: If g is a solution of Ricci flow, and λ > 0, then gλ isalso a solution, where

gλ(x, t) = λ2g(x,

t

λ2

).

The main use of this rescaling will be to analyse singularities that developunder the Ricci flow. In such a case the curvature tends to infinity, so weperform a rescaling to produce metrics with bounded curvature and try toproduce a smooth limit of these. We will discuss the machinery required forthis in Chapter 8, and return to the blow-up procedure in Section 8.5.

3.2 The Laplacian of Curvature

We devote this section to the derivation of an expression for the Laplacian ofRijk`. This will be used to derive an evolution equation for the curvature inSection 3.4, where the Ricci flow (3.1) implies a heat-type equation for theRiemannian curvature R. In order to do this, we need to introduce quadraticBijk` terms which will help simplify our computation.

3.2.1 Quadratic Curvature Tensor. Various second order derivativesof the curvature tensor are likely to differ by terms quadratic in the curvaturetensor. To this end we introduce the (4, 0)-tensor B ∈ T 4

0 (M) defined by

B(X,Y,W,Z) = 〈R(X, ·, Y, ?), R(W, ·, Z, ?)〉 ,

where the inner product 〈, 〉 is given by (1.3). In components this becomes

Bijk` = gprgqsRpiqjRrks` = Rp qi jRpkq`. (3.2)

This tensor has some of the symmetries of the curvature tensor, namely

Bijk` = Bji`k = Bk`ij .

However other symmetries of the curvature tensor may fail to hold for Bijk`.

3.2.2 Calculating the Connection Laplacian ∆Rijk`. Using thedefinition of the Laplacian explicated in Section 1.6, we compute the con-nection Laplacian acting on the T 4

0 (M) tensor bundle.

Proposition 3.2. On a Riemannian manifold (M, g), the Laplacian of thecurvature tensor R satisfies

∆Rijk` = ∇i∇kRj` −∇j∇kRi` +∇j∇`Rik −∇i∇`Rjk− 2(Bijk` −Bij`k −Bi`jk +Bikj`

)+ gpq

(Rqjk`Rpi +Riqk`Rpj

).

3.2 The Laplacian of Curvature 53

Proof. Using the second Bianchi identity — together with the linearity of ∇over the space of tensor fields — we find that

0 = (∇p∇qR)(∂i, ∂j , ∂k, ∂`)

+ (∇p∇iR)(∂j , ∂q, ∂k, ∂`) + (∇p∇jR)(∂q, ∂i, ∂k, ∂`).

Combining this identity with (1.10) implies that the Laplacian of R takes theform

∆Rijk` = (trg∇2R)ijk`

= gpq(∇∂p∇∂qR)(∂i, ∂j , ∂k, ∂`)

= gpq(− (∇p∇iR)(∂j , ∂q, ∂k, ∂`)− (∇p∇jR)(∂q, ∂i, ∂k, ∂`)

)= gpq

(∇p∇iRqjk` −∇p∇jRqik`

).

From this it suffices to express gpq∇p∇iRqjk` in terms of lower order termsby commuting derivatives and contracting with the metric.

Firstly, Proposition 1.44 implies that

∇p∇iRqjk` = ∇i∇pRqjk` +(R(∂i, ∂p)R

)(∂q, ∂j , ∂k, ∂`). (3.3)

From the second Bianchi identity, the first term on the right-hand side of(3.3) becomes

∇i∇pRqjk` = ∇i∇kRjq`p −∇i∇`Rjqkp.

Contracting with the metric and invoking Proposition 1.48, equation (1.15),gives

gpq∇i∇pRqjk` = gpq∇i∇kRjq`p − gpq∇i∇`Rjqkp= ∇i∇kRj` −∇i∇`Rjk. (3.4)

Turning our attention to the second term on the right-hand side of (3.3), wefind by Proposition 1.43 that(

R(∂i, ∂p)R)(∂q, ∂j , ∂k, ∂`)

=((((((((R(∂i, ∂p)(Rqjk`)

−R(R(∂i, ∂p)∂q, ∂j , ∂k, ∂`)− · · · −R(∂q, ∂j , ∂k, R(∂i, ∂p)∂`)

= −R nipq Rnjk` −R

nipq Rqnk` −R

nipq Rqjn` −R n

ipq Rqjkn

= gmn(RpiqmRnjk` +RpijmRqnk` +RpikmRqjn` +Rpi`mRqjkn

).

Therefore

gpq(R(∂i, ∂p)R

)(∂q, ∂j , ∂k, ∂`)

= gpqgmn(RpiqmRnjk` +RpijmRqnk` +RpikmRqjn` +Rpi`mRqjkn

).


We now look to simplify this expression by observing that the first termcontracts to

gpqgmnRpiqmRnjk` = gmnR pipm Rnjk` = gmnRnjk`Rim = gpqRpjk`Riq,

the second term contracts to

gpqgmnRpijmRqnk` = gpqgmnRpijm(−Rnkq` −Rkqn`)= gpqgmn(RpimjRq`nk −RpimjRqkn`)= Rq ni jRq`nk −R

q ni jRqkn`

= Bij`k −Bijk`,

and the third and forth terms contract to

gpqgmn(RpikmRqjn` +Rpi`mRqjkn) = gpqgmn(−RpimkRqjn` +Rpim`Rqjnk)

= −Rq ni kRqjn` +Rq ni `Rqjnk

= −Bikj` +Bi`jk.

Combining these results gives

gpq(R(∂i, ∂p)R

)(∂q, ∂j , ∂k, ∂`)

= gpqRpjk`Riq − (Bijk` −Bij`k −Bi`jk +Bikj`). (3.5)

Finally, by putting (3.4) and (3.5) together we get

gpq∇p∇iRqjk` = ∇i∇kRj` −∇i∇`Rjk− (Bijk` −Bij`k −Bi`jk +Bikj`) + gpqRpjk`Riq.

Since ∆Rijk` = gpq∇p∇iRqjk`− (i↔ j) the desired formula now follows.

3.3 Metric Variation Formulas

In this section we establish how one can formally take the time derivativeof a metric and the associated Levi-Civita connection. Thereafter we derivevarious variational equations for the Levi-Civita connection, curvature tensorand various traces thereof.

3.3.1 Interpreting the Time Derivative. Consider a one-parameterfamily of smooth metrics g = g(t) ∈ Γ (Sym2 T ∗M) parametrised by ‘time’ t.We define the time derivative ∂

∂tg : X (M)×X (M)→ C∞(M) of the metricg by letting ( ∂

∂tg)

(X,Y ) :=∂

∂tg(X,Y ) (3.6)

3.3 Metric Variation Formulas 55

for any time independent vector fields X,Y ∈X (M) (where ∂∂tg(X,Y ) is the

time derivative of the smooth function g(X,Y ) ∈ C∞(M) given by the stand-ard difference quotient). The metric in local coordinates can be expressed asg(t) = gij(t)dx

i ⊗ dxj , in which case (3.6) implies that

∂

∂tg = gij(t) dx

i ⊗ dxj .

Therefore we regard the time derivative of the metric as the derivative of itsthe component functions with respect to a fixed basis.

Since the Levi-Civita connection ∇ can be written locally in terms of themetric (1.9), it too will be time dependent. So in a similar fashion, we definethe time derivative of ∇ = ∇(t) by letting( ∂

∂t∇)

(X,Y ) :=∂

∂t∇XY (3.7)

for time-independent vector fields X and Y . As ∇ satisfies the product ruleby definition, it is not tensorial. However, we observe the following specialproperties of its time derivative:

Lemma 3.3. The time derivative ∂∂t∇ of the Levi-Civita connection ∇ is

tensorial.

Proof. For f ∈ C∞(M) and any time independent vector fields X,Y ∈X (M), we see that( ∂

∂t∇)

(X, fY ) =∂

∂t

((Xf)Y + f∇XY

)= f

∂

∂t∇XY = f

( ∂∂t∇)

(X,Y ).

Thus ∂∂t∇ is a tensor by Proposition 1.19.

Lemma 3.4. If X is a time independent vector field and V = V (t) is a timedependent vector field, then

∂

∂t∇XV =

( ∂∂t∇)

(X,V ) +∇X∂V

∂t.

Proof. Fix a time independent vector field X. As ∇X = ∇(t)X : X (M) →

X (M) we see that

∂

∂t∇XV lim

δ→0

∇(t+δ)(X,V (t+ δ))−∇(t)(X,V (t))

δ.

Since


∇(t+δ)(X,V (t+ δ))−∇(t)(X,V (t))

= ∇(t+δ)(X,V (t+ δ))−∇(t+δ)(X,V (t))

+(∇(t+δ) −∇(t)

)(X,V (t))

= ∇(t+δ)(X,V (t+ δ)− V (t)) +(∇(t+δ) −∇(t)

)(X,V (t)),

we conclude that

∂

∂t∇XV = lim

δ→0

1

δ∇(t+δ)(X,V (t+ δ)− V (t))

+ limδ→0

1

δ

(∇(t+δ) −∇(t)

)(X,V (t))

= ∇(t)X

(limδ→0

V (t+ δ)− V (t)

δ

)+( ∂∂t∇)

(X,V ).

Furthermore, we would like to differentiate the Christoffel symbols Γ kij =

dxk(∇∂i∂j) of the Levi-Civita connection ∇. To do this, we consider them asa map Γ : X (M) ×X (M) × T 1

0 (M) → C∞(M) defined by Γ (X,Y, ω) :=ω(∇XY ). With this we proceed in a similar fashion by defining ∂

∂tΓ to be( ∂∂tΓ)

(X,Y, ω) :=∂

∂tΓ (X,Y, ω). (3.8)

It is easy so see that ∂∂tΓ is tensorial since ( ∂∂tΓ )(X,Y, ω) = ∂

∂tω(∇XY ) =

ω( ∂∂t∇XY ) and ∂∂t∇ is a tensor by Lemma 3.3.

3.3.2 Variation Formulæ of the Curvature. We now derive evolutionequations for geometric quantities under arbitrary metric variations. Notethat ∂

∂tT = (δgT )(g) = (δhT )(g) for a (k, `)-tensor T , whenever the direction

h = ∂∂tg (where the first variation δh is taken in the sense of Appendix A).

Lemma 3.5. Suppose g(t) is a smooth one-parameter family of metrics on amanifold M such that ∂

∂tg = h. Then

∂

∂tgij = −gikgj`hk`. (3.9)

Proof. As gikgk` = δi` we find that 0 = ( ∂∂tgik) gk` + gikhk`. In which case

∂∂tg

ij = −gikgj`hk` = −hij .

Proposition 3.6. Suppose g(t) is a smooth one-parameter family of metricson a manifold M such that ∂

∂tg = h. Then the Levi-Civita connection Γ kij ofg evolves by

∂

∂tΓ kij =

1

2gk`(

(∇jh)(∂i, ∂`) + (∇ih)(∂j , ∂`)− (∇`h)(∂i, ∂j)).


Proof. As ∂∂tΓ and ∇h are tensorial, we are free to work in any local coordin-

ate chart (xi). In particular, by choosing normal coordinates about a point p(and evaluating the computation at that point), equation (1.9) implies that

∂

∂tΓ kij =

1

2

( ∂∂tgk`)((((

((((((

(∂jgi` + ∂igj` − ∂`gij) +1

2gk`

∂

∂t

(∂jgi` + ∂igj` − ∂`gij

).

Using Proposition 1.29 we observe that

∂

∂t∂jgi` = ∂j

(h(∂i, ∂`)

)= (∇jh)(∂i, ∂`) + h(

∇j∂i, ∂`) + h(∂i,∇j∂`),

from which the result now follows.


∂tg = h. Then the Riemannian curvature tensorevolves by

∂

∂tR ìjk =

1

2g`p(∇i∇phjk +∇j∇khip −∇i∇khjp −∇j∇phik

−R qijk hqp −R

qijp hqk

).

Proof. As the expression is tensorial, we are free to work in normal coordin-ates in a neighbourhood about a point p (and evaluating the expression atthat point). Using Lemma 1.46 in these coordinates, we find that

∂

∂tR ìjk =

∂

∂t

(∂jΓ

ìk − ∂iΓ `jk

)+(( ∂

∂tΓmik

)Γ `jm +

Γmik

( ∂∂tΓ `jm

))−XXXX(i↔ j

)= ∂j

( ∂∂tΓ ìk

)− ∂i

( ∂∂tΓ `jk

).

From this, together with Proposition 3.6, we have that

∂j

( ∂∂tΓ ìk

)=

1

2∂jg`p(∇khip +∇ihkp −∇phik

)+

1

2g`p(∂j(∇khip +∇ihkp −∇phik

))=

1

2g`p(∇j∇khip +∇j∇ihkp −∇j∇phik

),

where the last equality — using Proposition 1.29 — is due to

∂j((∇h)(∂i, ∂k, ∂`)

)=(∇j(∇h)

)(∂i, ∂k, ∂`)

+ (∇h)(∇j∂i, . . .) + · · ·+ (∇h)(. . . ,

∇j∂`)= (∇j∇ih)(∂k, ∂`).


Therefore,

∂

∂tR `ijk =

1

2g`p(∇j∇khip +∇j∇ihkp −∇j∇phik

)− 1

2g`p(∇i∇khjp +∇i∇jhkp −∇i∇phjk

)(3.10)

=1

2g`p(∇i∇phjk +∇j∇khip −∇i∇khjp −∇j∇phik

+(R(∂i, ∂j)h

)(∂k, ∂p)

)(3.11)

since ∇j∇ihkp − ∇i∇jhkp = (R(∂i, ∂j)h)(∂k, ∂p) by Proposition 1.44. Thedesired equation now follows since we find, by Proposition 1.43, that(R(∂i, ∂j)h

)(∂k, ∂p) =((((

((R(∂i, ∂j)hkp − h(R(∂i, ∂j)∂k, ∂p)− h(∂k, R(∂i, ∂j)∂p)

= −R qijk hqp −R

qijp hqk

Proposition 3.8. Suppose g(t) is a smooth family of metrics on a manifoldM with ∂

∂tg = h. Then the (4, 0)-Riemann curvature tensor R evolves by

∂

∂tRijk` =

1

2

(∇2i,`hjk +∇2

j,khi` −∇2i,khj` −∇2

j,`hik)

+1

2gpq(Rijkphq` +Rijp`hqk

)Proof. As Rijk` = R m

ijk gm`, we have

∂

∂tRijk` = R q

ijk

∂

∂tgq` + ga`

∂

∂tR aijk .

Now since

R qijk

∂

∂tgq` = gpqRijkphq`,

and (by Proposition 3.7)

ga`∂

∂tR aijk =

1

2

(∇2i,`hjk +∇2

j,khi` −∇2i,khj` −∇2

j,`hik)

− 1

2gpq(Rijkphq` +Rij`phqk

)the result naturally follows.


∂tg = h. Then the Ricci tensor Ric evolves by

∂

∂tRik =

1

2gpq(∇2q,khip −∇2

i,khqp +∇2q,ihkp −∇2

q,phik).


Proof. Recall from (3.10) that

∂

∂tR `ijk =

1

2g`p(∇2j,khip +∇2

j,ihkp −∇2j,phik

−∇2i,khjp −∇2

i,jhkp +∇2i,phjk

).

From which we find that

∂

∂tRik =

∂

∂tR jijk =

1

2gjp(∇2j,khip −∇2

i,khjp +∇2j,ihkp −∇2

j,phik)

+1

2((((((((

((((gqp(−∇2

i,qhkp +∇2i,phqk

)

Proposition 3.10. Suppose g(t) is a smooth one-parameter family of metricson a manifold M with ∂

∂tg = h. Then the scalar curvature Scal evolves by

∂

∂tScal = −∆trgh+ δ2h− 〈h,Ric〉 ,

where δ2h = gijgpq∇2q,jhpi is the ‘divergence term’.

Proof. Using equation (3.9) and Proposition 3.9 we find that

∂

∂tScal = Rik

∂

∂tgik + gik

∂

∂tRik

= −gijgk`hj`Rik +1

2gikgpq

(∇2q,khip −∇2

i,khqp +∇2q,ihkp −∇2

q,phik

)= −gijgpqhjqRip + gijgpq∇2

q,jhip − gijgpq∇2i,jhpq

= −hjqRjq + gijgpq∇2q,jhpi −∆trgh

Proposition 3.11. Suppose g(t) is a smooth one-parameter family of metricson a manifold M with ∂

∂tg = h. Then the volume form dµ(g(t)) evolves by

∂

∂tdµ =

1

2trh dµ.

To prove this, recall that:

Definition 3.12. The adjunct of a square matrix A is defined as the trans-pose of the cofactor matrix of A, that is

adjA =

detA11 −detA21 . . .−detA12 detA22 . . .

......

. . .

where Aij is obtained from A by striking out the i-th row and the j-th column.

Also recall that:


Lemma 3.13. If A is a square matrix then A adjA = adjAA = detA In×n.Moreover, if detA 6= 0 then A−1 = 1

detAadjA.

So if A is a square matrix with (i, j)-th entry aij , we can expand detA alongthe i-th row to get: detA =

∑nk=1(−1)i+k detAik. Therefore by Lemma 3.13,

the partial derivative of detA with respect to the (i, j)-th entry are

∂

∂aijdetA = (−1)i+j detAij

= (adjA)ji = detA (A−1)ji. (3.12)

Proof (Proposition 3.11). In local coordinates (xi) the volume form can bewritten as dµ =

√det g dx1 ∧ . . . ∧ xn. So by (3.12) and the chain rule,

∂

∂t

√det g =

1

2

1√det g

∂

∂tdet g

=1

2

1√det g

∂ det g

∂gij

∂gij∂t

=1

2

√det g (g−1)jihij

=1

2gijhij

√det g =

1

2trh

√det g.

Thus,∂

∂tdµ =

∂√

det g

∂tdx1 ∧ . . . ∧ xn =

1

2trh dµ.

3.4 Evolution of the Curvature Under the Ricci Flow

Using the results of the previous sections, it is now a relatively easy task toderive the evolution equations of the curvature, and its various traces, underthe Ricci flow.

Theorem 3.14. Suppose g(t) is a solution of the Ricci flow, the the (4, 0)-Riemannian tensor R evolves by

∂

∂tRijk` = ∆Rijk` + 2(Bijk` −Bij`k −Bi`jk +Bikj`)

− gpq(Rpjk`Rqi +Ripk`Rqj +RijkpRq` +Rijp`Rqk).

Proof. By Proposition 3.8, with ∂∂tgij = −2Rij , the time derivative of Rijk`

satisfies

∇2i,`Rjk +∇2

j,kRi` −∇2i,kRj` −∇2

j,`Rik

= − ∂

∂tRijk` − gpq

(RijkpRq` +Rijp`Rqk

).

3.4 Evolution of the Curvature Under the Ricci Flow 61

By Proposition 3.2, with indices k and ` switched, the Laplacian of R satisfies

∇2i,`Rjk −∇2

j,`Rik +∇2j,kRi` −∇2

i,kRj`

= ∆Rij`k + 2(Bij`k −Bijk` −Bikj` +Bi`jk)

− gpq(Rqj`kRpi +Riq`kRpj).

Combining these equations gives

−∆Rijk` = ∆Rij`k = − ∂

∂tRijk` − 2(Bij`k −Bijk` −Bikj` +Bi`jk)

+ gpq(Rqj`kRpi +Riq`kRpj)− gpq(RijkpRq` +Rijp`Rqk).

We can also derive, without too much effort, the evolution of the followingquantities under the Ricci flow.

Corollary 3.15. Under the Ricci flow, the connection coefficients evolve by

∂

∂tΓ kij = −gk`

((∇jRic)(∂i, ∂`) + (∇iRic)(∂j , ∂`)− (∇`Ric)(∂i, ∂j)

).

Corollary 3.16. Under the Ricci flow, the volume form of g evolves by

∂

∂tdµ = −Scal dµ.

Corollary 3.17. Under the Ricci flow,

∂

∂tRik = ∆Rik +∇2

ikScal− gpq(∇2q,iRkp +∇2

q,kRip),

∂

∂tScal = 2∆Scal− 2gijgpq∇2

q,jRpi + 2|Ric|2.

The proof of these corollaries follow easily by substituting ∂∂tgij = −2Rij into

Propositions 3.6, 3.11, 3.9 and 3.10 respectively.We note that the fomulæ in the last corollary can be simplified as follows.


∂

∂tRik = ∆Rik + 2gpqgrsRpikrRqs − 2gpqRipRqk.

Proof. By (3.9) the time derivative of Rik = gj`Rijk` is ∂∂tRik = gj` ∂∂tRijk`−

2gjpg`qRpqRijk`. Substituting the expression for ∂∂tRijk` in Theorem 3.14

(with gj`∆Rijk` = ∆Rik from equation (1.15)) results in

∂

∂tRik = ∆Rik + 2gj`(Bijk` −Bij`k −Bi`jk +Bikj`)

− gj`gpq(Rpjk`Rqi +Ripk`Rqj +Rijp`Rqk +RijkpRq`).


As we find that

2gj`(Bijk` −Bij`k −Bi`jk +Bikj`)

= 2gj`Bijk` − 2gj`(Bi`jk +Bij`k) + 2gprgqsRpiqkRrs

= 2gj`Bijk` − 4gj`Bij`k + 2gprgqsRpiqkRrs

= 2gj`(Bijk` − 2Bij`k) + 2gprgqsRpiqkRrs

and

gj`gpq(Rpjk`Rqi +Ripk`Rqj +Rijp`Rqk +RijkpRq`)

= 2gpqRpiRqk + gj`gpqRipk`Rqj + gj`gpqRijkpRq`

= 2gpqRpiRqk + 2gprgqsRpiqkRrs,

it follows that

∂

∂tRik = ∆Rik + 2gj`(Bijk` − 2Bij`k) + 2gprgqsRpiqkRrs − 2gpqRpiRqk.

The desired result now follows from the following claim.

Claim. For any metric gij , the tensor Bijk` satisfies the identity

gj`(Bijk` − 2Bij`k) = 0.

Proof of Claim. Using the Bianchi identities,

gj`Bijk` = gj`gprgqsRpiqjRrks`

= gj`gprgqsRpqijRrsk`

= gj`gprgqs(Rpiqj −Rpjqi)(Rrks` −Rr`sk)

= 2gj`(Bijk` −Bij`k).


∂

∂tScal = ∆Scal + 2|Ric|2.

Proof. From Corollary 3.17 it suffices to show that

2gijgpq∇2q,jRpi = gpq∇q∇pScal = ∆Scal.

To do this we claim:

Claim. The identity

1

2∇q∇pScal = gij∇q∇jRpi

holds true.

3.5 A Closer Look at the Curvature Tensor 63

Proof of Claim. By the contracted second Bianchi identity (1.16),

1

2(∇Scal)(∂p) =

1

2∇pScal = gij∇jRpi = (g−1 ⊗∇Ric)(∂i, ∂j , ∂j , ∂p, ∂i)

= (tr14 tr23 g−1 ⊗∇Ric)(∂p)

= (tr g−1 ⊗∇Ric)(∂p).

As ∇q(tr g−1 ⊗∇Ric) = tr g−1 ⊗∇q(∇Ric) we find that

1

2∇q∇pScal =

1

2(∇q(∇Scal))(∂p)

= (tr g−1 ⊗∇q(∇Ric))(∂p)

= gij∇q∇jRpi.

Remark 3.20. The evolution of the scalar curvature by Corollary 3.19 providesa simple illustration of the fact that the Ricci flow ‘prefers’ positive curvature.In this case the two components ∆Scal and 2|Ric|2 can be interpreted in thefollowing way: The dissipative term ∆Scal reflects the fact that a point inM with a higher average curvature than its neighbours will tend to revertto the mean. The nonlinear term 2|Ric|2 reflects the fact that if one is ina positive curvature region (e.g. a region behaving like a sphere), then themetric will contract, thus increasing the curvature to be even more positive.Conversely, if one is in a negative curvature region (such as a region behavinglike a saddle), then the metric will expand, thus weakening the negativity ofcurvature. In both cases the curvature is trending upwards, consistent withthe non-negativity of 2|Ric|2.

3.5 A Closer Look at the Curvature Tensor

So far we have managed to derive, in Theorem 3.14, a heat-type evolutionequation for the curvature tensor under the Ricci flow. As we seek to deformthe metric so it has constant sectional curvature, we need to look at therelationship between the algebraic properties of the curvature tensor and theglobal topological and geometry of the manifold.

In three dimensions the curvature is relatively simple algebraically, andin four dimensions still somewhat tractable, but in higher dimensions thecurvature becomes very complicated and hard to study. To get a glimpse ofthis we will look closely at the algebraic structure, in particular that of theWeyl curvature tensor. To do this though, we first need to define the followingproduct.

3.5.1 Kulkarni-Nomizu Product. Given two (2, 0)-tensors we want tobuild a (4, 0)-tensor that has the same symmetries as that of the algebraicRiemannian curvature tensor (that is, a (4, 0)-tensor that satisfies symmetryproperties (i)–(iv) in Section 1.7.4.1). To do this we need a map, called the


Kulkarni-Nomizu product, of the form

? : Sym2(M)× Sym2(M)→ Curv(M),

where Sym2(M) is the bundle of symmetric (2, 0)-tensors and Curv(M) isthe bundle of curvature tensors. Thus given α, β ∈ Sym2(M) we require theproduct α? β to satisfy the following symmetries:

(a) Antisymmetric in the first two arguments: (α? β)ijk` = −(α β)jik`(b) Antisymmetric in the last two arguments: (α? β)ijk` = −(α? β)ij`k(c) Symmetric paired arguments: (α? β)ijk` = (α? β)k`ij(d) Satisfy the Bianchi identity: (α? β)ijk` + (α? β)jki` + (α? β)kij` = 0

A natural way to build α ? β is to use the tensor product. By linearitywe expect α? β to be a sum of α⊗ β with components permuted in such away that the symmetries matches that of the algebraic Riemann curvaturetensor. Hence all that is needed is to find the correct permutations.

Now as α and β are symmetric, terms of the form αijβk` are disallowedsince this would contradict (a) and (b). So we need to mix i, j with k, ` acrossthe tensors α and β. If we naıvely suppose one of the terms is αikβj`, then(c) implies we will also need αj`βik and by (a), (b) we will also need −αjkβi`and −αi`βjk. Therefore by defining

(α? β)ijk` := αikβj` + αj`βik − αjkβi` − αi`βjk (3.13)

or alternatively

(α? β)(v1, v2, v3, v4) = (α⊗ β)(v1, v3, v2, v4) + (α⊗ β)(v2, v4, v1, v3)

− (α⊗ β)(v1, v4, v2, v3)− (α⊗ β)(v2, v3, v1, v4),

properties (a)–(c) are immediately satisfied; inspection also shows that theBianchi identity holds as well. Thus we have constructed a (4, 0)-tensor thathas all of the symmetries of that of the algebraic curvature tensor.

3.5.2 Weyl Curvature Tensor. The Weyl curvature tensor is definedto be the traceless component of the Riemann curvature tensor. It can beobtained from the full curvature tensor R by subtracting out various traces.To find an exact expression, we seek to subtract a tensor C which is the sumof the scalar and traceless Ricci2 parts of R, with the additional conditionsthat C must have the same traces and algebraic structures as that of R. Byusing the Kulkarni-Nomizu product, we consider the tensor C as taking theform

C = c1Scal g ? g + c2Ric

? g,

2 Where the traceless Ricci tensor Ric

:= Ric − Scalng, since gikRic

ik = Scal −

ScalngikRicik = Scal− Scal

nδii = 0.


where c1 and c2 are scalars. As trR = trC, where the trace is taken over anypair of indices, we must have

Ricik = gj`Rijk` = gj`Cijk`.

Now since

Cijk` = 2c1Scal(gikgj` − gi`gjk)

+ c2(Ricikgj` + Ric

j`gik − Ric

i`gjk − Ric

jkgi`),

it follows that

1

nScal gik + Ric

ik = Ricik = gj`Cijk`

= 2c1(n− 1)Scal gik + c2(n− 2) Ricik.

In which case the scalars c1 = 12n(n−1) and c2 = 1

n−2 .

Therefore the Weyl tensor W must be of the form

Weyl = R− 1

n− 2Ric

? g − Scal

2n(n− 1)g ? g. (3.14)

Alternatively, by using Ric

= Ric− Scaln g, we could also write

Weyl = R− 1

n− 2Ric ? g +

Scal

2(n− 1)(n− 2)g ? g. (3.15)

Since W is defined using the Kulkarni-Nomizu product it has the same sym-metries as the curvature tensor, but by definition all of its traces vanish.Indeed, one can check by hand that W j

ijk = 0, or indeed any other trace, asper definition. Also note that∣∣R∣∣2 =

∣∣ Scal

2n(n− 1)g ? g

∣∣2 +∣∣ 1

n− 2Ric

? g∣∣2 +

∣∣Weyl∣∣2

as the decomposition is orthogonal and that the metric g has constant sec-

tional curvature if and only if Ric

= 0 and Weyl = 0 (cf. Section 11.3.1).

3.5.3 Sphere Theorem of Huisken-Margerin-Nishikawa. In threedimensions the Weyl tensor vanishes, and the curvature can be understoodsolely in terms of the Ricci tensor (see Section 6.5.3 for one way of doingthis). One of the first insights into the differentiable pinching problem, fordimensions n ≥ 4, was made independently by Huisken [Hui85], Nishikawa[Nis86] and Margerin [Mar86]. By using the Ricci flow, they were able showthat a manifold is diffeomorphic to a spherical space form, provided the norm


of the Weyl curvature tensor and the norm of the traceless Ricci tensor arenot too large compared to the scalar curvature at each point.

Theorem 3.21. Let n ≥ 4. If the curvature tensor of a smooth compactn-dimensional Riemannian manifold M of positive scalar curvature satisfies∣∣Weyl

∣∣2 +∣∣ 1

n− 2Ric

? g∣∣2 ≤ δn 2

n(n− 1)Scal2 (3.16)

with δ4 = 15 , δ5 = 1

10 , and δn = 2(n−2)(n+1) for n ≥ 6, then the Ricci flow

has a solution g(t) on a maximal finite time interval [0, T ), and g(t)2(n−1)(T−t)

converges in C∞ to a metric of constant curvature 1 as t→∞. In particular,M is diffeomorphic to a spherical space form.

Remark 3.22. The Riemann curvature tensor can be considered as an elementof the vector space of symmetric bilinear forms acting on the space

∧2TM

(see Section 11.2). The inequality (3.16) defines a cone in this vector spacearound the line of constant curvature tensors, in which the Riemann tensormust lie. In Chapters 11 and 12 we will return to this idea of constructingcones in the space of curvature tensors.

The technique of the proof follows Hamilton’s original paper [Ham82b].The approach is to show that if the scalar-free part of the curvature is smallcompared to the scalar curvature initially, then it must remain so for alltime. In which case we need to show, under the hypothesis (3.16) with anappropriate δn, there exists ε > 0 such that∣∣Weyl

∣∣2 +∣∣ 1

n− 2Ric

? g∣∣2 ≤ δn (1− ε)2 2

n(n− 1)Scal2 (3.17)

remains valid as long as the solution to the Ricci flow exists for times t ∈[0, T ). This is achieved by working with

R

:= R− 2 Scal

n(n− 1)g ? g,

which measures the failure of g to have (pointwise) constant positive sec-tional curvature (cf. Section 11.3.1). It is proved using the maximum principle(which we discuss in Chapter 6) that a bound can be obtained on a functionof the form

Fσ =|R|2

Scal2−σ,

for σ ≥ 0, yielding the following result.3

3 The quantity |R|2 is a higher dimensional analogue of the quantity |Ric

|2 = |Ric|2−

13

Scal2 used by Hamilton [Ham82b] in the n = 3 case, which measures how far theeigenvalues of the Ricci tensor diverge from each other.


Proposition 3.23 ([Hui85, Sect. 3]). If the inequality (3.17), that is

|R|2 ≤ δn (1− ε)2 2

n(n− 1)Scal2,

holds at time t = 0, then it remains so on 0 ≤ t < T . Moreover, there areconstants C <∞ and σ > 0, depending on n and the initial metric, such that

|R|2 ≤ C Scal2−σ

holds for 0 ≤ t < T .

Remark 3.24. From this analysis we see that the main obstacle to obtainingsimilar sphere-type theorem, under weaker pinching conditions than that ofTheorem 3.21, is the Weyl curvature tensor W . Controlling the behaviourof W under the Ricci flow, for dimensions n ≥ 4, has proved to be a majortechnical hurdle, which was finally overcome by the efforts of Bohm andWilking [BW08]. We discuss their method in Chapter 11.

Chapter 4

Short-Time Existence

An important foundational step in the study of any system of evolution-ary partial differential equations is to show short-time existence and unique-ness. For the Ricci flow, unfortunately, short-time existence does not followfrom standard parabolic theory, since the flow is only weakly parabolic. Toovercome this, Hamilton’s seminal paper [Ham82b] employed the deep Nash-Moser implicit function theorem to prove short-time existence and unique-ness. A detailed exposition of this result and its applications can be foundin Hamilton’s survey [Ham82a]. DeTurck [DeT83] later found a more directproof by modifying the flow by a time-dependent change of variables to makeit parabolic. It is this method that we will follow.

4.1 The Symbol

To investigate short-time existence and uniqueness for the Ricci flow, onenaturally looks to the theory of non-linear pde’s on vector bundles. Here weestablish the symbol which will be used to determine a pde’s type.

4.1.1 Linear Differential Operators. Let E and F be bundles overa manifold M . We say L : Γ (E) → Γ (F ) is a linear differential operator oforder k if it is of the form

L(u) =∑|α|≤k

Lα∂αu,

where Lα ∈ Hom(E,F ) is a bundle homomorphism and α is a multi-index.

Example 4.1. Let (ek) be local frame for E over a neighbourhood of p ∈ Mwith local coordinates (xi). Then a second order linear differential operatorP : Γ (E)→ Γ (E) has the form

P(u) =

((λij)

k`

∂2u`

∂xi∂xj+ (µi)

k`

∂u`

∂xi+ νk` u

`

)ek.

69

70 4 Short-Time Existence

Here λ ∈ Γ (Sym2(T ∗M) ⊗ Hom(E,E)), while µ ∈ Γ (T ∗M ⊗ Hom(E,E))and ν ∈ Γ (Hom(E,E)).

The total symbol σ of L in direction ζ ∈X (M) is the bundle homomorph-ism

σ[L](ζ) =∑|α|≤k

Lαζα.

Thus in Example 4.1, (σ[P](ζ))(u) = ((λij)k` ζiζju` + (µi)`

kζiu` + λk`u`)ek.

In the familiar case of scalar equations, the bundle is simply M × R sothat Hom(E,E) is one-dimensional, and we can think of λ as a section ofSym2(T ∗M), µ as a vector field and ν as a scalar function.

The principal symbol σ of L in direction ζ is defined to be the bundlehomomorphism of only the highest order terms, that is

σ[L](ζ) =∑|α|=k

Lαζα.

The principal symbol captures algebraically the analytic properties of L thatdepend only on its highest derivatives. In Example 4.1, the principal symbol indirection ζ is σ[P](ζ) = (λij)

k` ζiζj(e∗)`⊗ek. As Hamilton noted in [Ham82b,

Sect. 4], computing the symbol is easily obtained (at least heuristically) byreplacing the derivative ∂

∂xi by the Fourier transformation variable ζi.The principal symbol determines whether an evolution equation is of para-

bolic type: We say a differential operator L of order 2m is elliptic if forevery direction ζ ∈ TxM the eigenvalues of the principal symbol σ[L](ζ) havestrictly positive real part, or equivalently if there exists c > 0 such that forall ζ and u we have 〈σ[L](ζ)u, u〉 ≥ c|ζ|2m|u|2. This implies in particular thatthe principal symbol in any direction is a linear isomorphism of the fibre. Alinear equation of the form ∂tu = Lu is parabolic if L is elliptic.

An important property of the principal symbol is the following: If G inanother vector bundle over M with a operator O : Γ (F )→ Γ (G) of order `,then the symbol of O L in direction ζ is the bundle homomorphism

σ[O L](ζ) = σ[O](ζ) σ[L](ζ) : E → G (4.1)

It is of degree at most k + ` in direction ζ.

4.1.2 Nonlinear Differential Operators. When faced with a nonlinearpartial differential equation, one attempts to linearise the equation in such away that linear theory can be applied. That is, if one has a solution u0 to agiven nonlinear pde, it is possible to linearise the equation by considering asmooth family u = u(s) of solutions with a variation v = δu = ∂

∂su∣∣s=0

. Bydifferentiating the pde with respect to s, the result is a linear pde in termsof v.

For example, if the nonlinear pde is of the form

4.2 The Linearisation of the Ricci Tensor 71

∂u

∂t= F (D2u,Du, u, x, t),

where F = F (p, q, r, x, t) : Sym2(Rn)×Rn×R×Rn×R→ R. The linearisationabout a solution u0 is given by

∂v

∂t=

∂F

∂pij

∣∣∣u0

DiDjv +∂F

∂qk

∣∣∣u0

Dkv +∂F

∂r

∣∣∣u0

v.

The result is a linear pde with coefficients depending on u0.Moreover for a (nonlinear) differential operator P : Γ (E) → Γ (F ) with a

given solution u0, the linearisation DP of P at u0 (if it exists) is defined tobe the linear map DP : Γ (E)→ Γ (F ) given by

DP∣∣u(v) =

∂

∂tP (u(t))

∣∣∣t=0

,

where u(0) = u0 and u′(0) = v. From this, we say a nonlinear equation isparabolic if and only if its linearisation about any u0 ∈ Γ (E) is parabolic.

Of particular interest is the evolution equation of the form

∂f

∂t= E(f),

where E(f) is a nonlinear differential operator of degree 2 in f over a bundleπ : E →M . If f is a variation of f then

∂f

∂t= DE

∣∣f(f)

where DE∣∣f

is the linear operator of degree 2. If DE∣∣f

is elliptic, then the

evolution equation ∂f∂t = E(f) has a unique smooth solution for the initial

value problem f = f0 at t = 0 for at least a short time interval 0 ≤ t < ε(where ε may depend on the initial data f0).

4.2 The Linearisation of the Ricci Tensor

When considering the Ricci flow ∂∂tg = −2Ric(g), one would like to regard

the Ricci tensor as a nonlinear partial differential operator on the space ofmetrics g. That is, as an operator Ric : Γ (Sym2

+T∗M) → Γ (Sym2T ∗M).

By using the variation formula from Proposition 3.9 with hij = ∂∂tgij , the

linearisation of Ric is given by

(DRicg)(h)ik =∂

∂tRicg(g(t))

∣∣∣t=0

=1

2gpq(∇q∇khip +∇q∇ihkp −∇i∇khqp −∇q∇phik

).


So the principal symbol

σ[DRicg]ζ : Sym2+T∗M → Sym2T ∗M

in direction ζ can be obtained by replacing the covariant derivative ∇i bythe covector ζi. Hence

(σ[DRicg]ζ)(h)ik =1

2gpq(ζqζkhip + ζqζihkp − ζiζkhqp − ζqζphik

)(4.2)

4.3 Ellipticity and the Bianchi Identities

In investigating the principal symbol σ[DRic(g)], Hamilton [Ham82b, Sect.4] observed that the failure of ellipticity is principally due to the Bianchiidentities. We present this result with a discussion on the link between theBianchi identities and the diffeomorphism invariance of the curvature tensor.The result of this investigation shows the failure of Ricci flow to be parabolicis a consequence of its geometric nature.

To begin, recall that the divergence operator δg : Γ (Sym2T ∗M) →Γ (T ∗M) is defined by

(δgh)k = −gij∇ihjk. (4.3)

The adjoint of δg, denoted by δ∗g : Γ (T ∗M) → Γ (Sym2T ∗M), with respectto the L2-inner product is given by1

(δ∗gω)jk =1

2(∇jωk +∇kωj) =

1

2(Lω]g)jk. (4.4)

Now consider the composition

DRic(g) δ∗g : Γ (T ∗M)→ Γ (Sym2T ∗M).

By (4.1) this is a priori a (2+1)-order differential operator, so its principalsymbol σ[DRicg δ∗g ](ζ) is the degree 3 part of its total symbol. However, bycommuting derivatives and the contracted second Bianchi identity one canshow that

(DRic(g) δ∗g)(ω) =1

2(ωp∇pRik +Rpk∇iωp +Rpi∇jωp) =

1

2Lω](Ric(g)).

As the right-hand side involves only one derivative of ω, its total symbol is ofdegree at most 1. In other words, the principal (degree 3) symbol σ[DRicg δ∗g ](ζ) is in fact the zero map. Thus

0 = σ[DRicg δ∗g ](ζ) = σ[DRicg](ζ) σ[δ∗g ](ζ),

1 Since´M

(δgh)kωkdµ =´Mhjk(δ∗gω)jkdµ, for compact M .

4.3 Ellipticity and the Bianchi Identities 73

and so Im σ[δ∗g ](ζ) ⊂ ker σ[DRicg](ζ). Therefore σ[DRicg](ζ) has at least ann-dimensional kernel in each fibre.

In fact one can go further by showing dim ker σ[DRicg](ζ) = n. Here webriefly sketch the idea as follows: First consider the first order linear operatorBg : Γ (Sym2T ∗M)→ Γ (T ∗M) defined by

Bg(h)k = gij(∇ihjk −1

2∇khij). (4.5)

As any metric satisfies the contracted second Bianchi identity,

Bg(Ric(g)) = 0.

By linearising this pde we obtain

Bg((DRicg)(h)) + (DBg)(Ric(g + h)) = 0.

Now Bg DRicg is a priori a degree 3 differential operator. However DBgis of order 1, so its degree 3 symbol is zero. Thus the principal (degree 3)symbol σ[Bg D(Ricg)](ζ) must be the zero map, and so

Im σ[DRicg](ζ) ⊂ ker σ[Bg](ζ) ⊂ Sym2T ∗M.

From here one can combine maps Bg DRicg and DRicg δ∗g to show

0 −→ T ∗Mσζ [δ∗g ]−−−−→ Sym2T ∗M

σζ [DRicg]−−−−−−→ Sym2T ∗Mσζ [Bg]−−−−→ T ∗M −→ 0

constitutes a short exact sequence. A discussion on this can be found in[CK04, pp. 77-8]. The desired result follows, however the emphasis lies in thefact there are no degeneracies other than those implied by the contractedsecond Bianchi identity.

4.3.1 Diffeomorphism Invariance of Curvature and the BianchiIdentities. So far we have seen the degeneracy of the Ricci tensor resultsfrom the contracted second Bianchi identity. In this section we show theBianchi identities are a consequence of the invariance of the curvature tensorunder the full diffeomorphism group (which is of course infinite dimensional).An upshot of this — as mentioned in [Kaz81] — is a natural and conceptuallytransparent proof of the Bianchi identities. As a consequence, the failure ofRic(g) to be (strongly) elliptic is due entirely to this geometric invariance.

The simplest illustration of this involves the scalar curvature Scal. To start,let φt be the one-parameter group of diffeomorphisms generated by the vectorfield X with φ0 = idM . By the diffeomorphism invariance of the curvature,

φ∗t (Scal(g)) = Scal(φ∗t g).

Now the linearisation of Scal is given by


DScalg(LXg) =d

dtScal(φ∗t g)

∣∣∣t=0

=d

dtφ∗t (Scal(g))

∣∣∣t=0

= LXScal = ∇XScal.

On the other hand, Proposition 3.10 implies that

DScalg(h) = −∆trgh+ δ2h− 〈h,Ric〉= −gijgk`(∇i∇jhk` −∇i∇khj` +Rikhj`),

where h is the arbitrary variation of g. By setting this variation hij =(LXg)ij = ∇iXj +∇jXi we find (by commuting covariant derivatives) that

DScalg(LXg) = 2gjkXi∇kRij = Xi∇iScal.

As X is arbitrary, it follows that

gjk∇kRij =1

2∇iScal.

So the diffeomorphism invariance of Scal implies the contracted second Bi-anchi identity.

The same method works for the full curvature tensor (see [CLN06, Ex.1.26]). Here we sketch the main result. The diffeomorphism invarianceR(φ∗t g) =φ∗t (R(g)) implies

DRg(LXg) =d

dtR(φ∗t g)

∣∣∣t=0

=d

dtφ∗t (R(g))

∣∣∣t=0

= LXR,

where

(LXR) ìjk = Xp∇pR `

ijk +R `pjk ∇iX

p +R ìpk ∇jX

p

+R ìjp ∇kX

p −R pijk ∇pX

`.

However by Proposition 3.7 and (3.10) we also have

2[DRg(h)] ìjk = g`p

(∇j∇khip +∇j∇ihkp +∇j∇phik

− ∇i∇khjp −∇i∇jhkp −∇i∇phjk).

By substituting h = LXg into the previous equation and rewriting LXR as

(LXR) ìjk = g`p

(−∇i(R q

kpj Xq)−∇j(R qpki Xq)−∇kXqR

qijp

−∇qXqRq

ijk +Xq

(gqr∇rRijkp +∇iR q

kpj +∇jR qpki

)),

we find that

4.4 DeTurck’s Trick 75

0 = [DRg(LXg)] ìjk − (LXR) `

ijk

=1

2g`p(−∇i

((R q

jpk −Rq

kpj −Rq

jkp )Xq)

+ ∇j((R q

ipk −Rq

kpi −Rq

ikp )Xq)

− 2Xq(∇qRijkp +∇iRkpjq +∇jRpkiq)).

To get the desired Bianchi identities from this, we evaluate this expressionpointwise with an appropriate choice of X. Firstly, by prescribing X(p) = 0and ∇iXj(p) = gij(p) it follows that

0 = −(Rj`ki −Rk`ji −Rjkì) + (Ri`k −Rkìj −Rik`j).

In which case symmetries (i)–(iii) in Section 1.7.4.1 imply the first Bianchiidentity: 0 = Rijk` +Rik`j +Ri`jk. Similarly, if X is chosen to be an elementof a local orthonormal frame, we can obtain the second Bianchi identity0 = ∇qRijk` +∇iRjqk` +∇jRqik`.

4.4 DeTurck’s Trick

Despite the failing of Ric(g) (as a nonlinear differential operator) to be el-liptic, the Ricci flow still enjoys short-time existence and uniqueness:

Theorem 4.2. If (M, g0) is a compact Riemannian manifold, there exists aunique solution g(t), defined for time t ∈ [0, ε), to the Ricci flow such thatg(0) = g0 for some ε > 0.

In proving this theorem, DeTurck [DeT83] showed it is possible to modify theRicci flow in such a way that the nonlinear pde in fact becomes parabolic.As we shall see, this is done by adding an extra term which is a Lie derivativeof the metric with respect to a certain time dependent vector field.

4.4.1 Motivation. Here we closely examination the Ricci tensor. Tomotivate DeTurck’s idea, rewrite the linearisation of the Ricci tensor as

−2[DRicg(h)]ik = ∆hik + gpq(∇i∇khqp −∇q∇ihkp −∇q∇khip).

Define the 1-form V = Bg(h) ∈ Γ (T ∗M), where Bg is defined by (4.5).Observe that

Vk = gpq(∇qhpk −1

2∇khpq)

∇iVk = gpq(∇i∇qhpk −1

2∇i∇khpq).

Thus− 2[DRicg(h)]ik = ∆hik −∇iVk −∇kVi + Sik, (4.6)


where the lower order tensor

Sik = gpq(1

2(∇i∇khqp −∇k∇ihqp)

+ (∇i∇qhkp −∇q∇ihkp) + (∇k∇qhip −∇q∇khip))

= gpq(1

2(R r

ikq hrp +R rikp hrq)

+ (R riqp hrk +R r

iqk hrp) + (R rkqi hrp +R r

kqp hri))

= gpq(2R rqik hrp −Riphkq −Rkphiq)

since ∇i∇jhpq − ∇j∇ihpq = R rijp hrq + R r

ijq hrp by Proposition 1.43. It isclear Sik is symmetric and involves no derivatives of h. Therefore (4.6) impliesthat the linearisation of the Ricci tensor is equal to a Laplacian term minusa Lie derivative term ∇iVk +∇kVi with a lower order symmetric term Sik.

Moreover, by Proposition 3.6 we can write V (at least locally) as

Vk =1

2gpq(∇phqk +∇qhpk −∇khpq) = gpqgkr(DΓg(h))rpq,

where h is the variation of g and

DΓg : Γ (Sym2T ∗M)→ Γ (Sym2T ∗M ⊗ TM)

is the linearisation of the Levi-Civita connection Γ (g). We now wish to addan appropriate correction term to the Ricci tensor to make it elliptic.

To do this, fix a background metric g on M with Levi-Civita connectionΓ . By our above investigation, define a vector field W by

W k = gpq(Γ kpq − Γ kpq). (4.7)

As it is the difference of two connections, it is a globally well defined vectorfield. Since W only involves one derivative of the metric g, the operator

P = P (Γ ) : Γ (Sym2T ∗M)→ Γ (Sym2T ∗M),

define byP (g) := LW g,

is a second order differential operator in g. The linearisation of P is given by(DP (h)

)ik

= ∇iVk +∇kVi + Tik, (4.8)

where Tik is a linear first order expression in h. Comparing this with (4.6)leads one to consider the operator

Q := −2Ric + P : Γ (Sym2T ∗M)→ Γ (Sym2T ∗M).


Therefore by (4.6) and (4.8) we find that

DQ(h) = ∆h+A,

where Aik = Tik − 2Sik is a first order linear term in h. Hence the principalsymbol of DQ is

σ[DQ](ζ)h = |ζ|2h. (4.9)

Therefore Q is elliptic, and so by the standard theory of partial differentialequations the modified Ricci flow

∂

∂tg = −2Ric(g) + P (g) = −2Ric(g) + LW g,

also referred to as the Ricci-DeTurck flow, enjoys short-time existence anduniqueness.

4.4.2 Relating Ricci-DeTurck Flow to Ricci Flow. We now followDeTurck’s strategy of proving short time existence and uniqueness for theRicci flow.

Proof (Theorem 4.2). We proceed in stages, first starting with existence.Step 1: Fix a background metric g on M , define the vector field W by (4.7)

and let the Ricci-DeTurck flow be given by

∂

∂tgij = −2Rij +∇iWj +∇jWi (4.10)

g(0) = g0

where Wj = gjkWk = gjkg

pq(Γ kpq − Γ kpq). From (4.9), the Ricci-DeTurk flowis strictly parabolic. So for any smooth initial metric g0 there exists ε > 0such that a unique smooth solution g(t) to (4.10) flow exists for 0 ≤ t < ε.

Step 2: As there exists a solution to the Ricci-DeTurck flow, the one-parameter family of vector fields W (t) defined by (4.7) exists for 0 ≤ t < ε.In which case there is a 1-parameter family of maps ϕt : M → M (i.e. theflow along the vector field −W ) defined by the ode

∂

∂tϕt(p) = −W (ϕt(p), t)

ϕ0 = idM

As M is compact, one can combine the escape lemma (for instance see[Lee02, p. 446]) with the existence and uniqueness of time-dependent flows(for instance see [Lee02, p. 451]) to conclude there exists a unique familyof diffeomorphisms ϕt(p) which is defined for all times t in the interval ofexistence 0 ≤ t < ε.

Step 3: We now show:


Claim 4.3. The family of metrics g(t) := ϕ∗t g(t) defined for 0 ≤ t < ε is aunique solution to the Ricci flow

∂

∂tg = −2Ric(g) (4.11)

g(0) = g0

To show that g satisfies (4.11), note that g(0) = g(0) = g0 (since ϕ0 = idM )and that

∂

∂tg =

∂

∂t(ϕ∗t g(t))

=∂

∂s

∣∣∣s=0

ϕ∗s+tg(t+ s)

= ϕ∗t( ∂∂tg(t)

)+

∂

∂s

∣∣∣s=0

(ϕ∗s+tg(t))

= ϕ∗t (−2Ric(g(t)) + LW (t)g(t)) +∂

∂s

∣∣∣s=0

ϕ∗s+tg(t)

= −2Ric(ϕ∗t g(t)) + ϕ∗t (LW (t)g(t)) +∂

∂s

∣∣∣s=0

ϕ∗s+tg(t).

As ϕ∗t+s = (ϕ−1t ϕt+s ϕt)∗ = (ϕ−1

t ϕt+s)∗ ϕ∗t and

∂

∂s

∣∣∣s=0

(ϕ−1t ϕt+s) = (ϕ−1

t )∗(∂

∂s

∣∣∣s=0

ϕt+s) = (ϕ−1t )∗W (t),

we conclude that

∂

∂tg = −2Ric(ϕ∗t g(t)) + ϕ∗t (LW (t)g(t))− L(ϕ−1

t )∗W (t)(ϕ∗t g(t))

= −2Ric(g(t)).

Hence g(t) = ϕ∗t g(t) is indeed a solution of (4.11) for t ∈ [0, ε). This completesthe existence part of the the claim.

Step 4: All we need to show now is the uniqueness for the Ricci flow. Itsuffices to prove that a solution of the Ricci-DeTurck flow is produced froma solution of Ricci flow after a reparametrisation defined by harmonic mapheat flow.

Precisely, let (M, g(t)) satisfy Ricci flow. Fix N with a background metric

h and an associated Levi-Civita connection ∇. Let ϕ0 : M → N be adiffeomorphism. Define ϕ : M × [0, T )→ N by the harmonic map heat flow

from the (time-dependent) metric g to h so that

ϕ∗∂t = ∆g(t),hϕ = gij∇iϕ∗(∂j),

where we take ∇ to be defined by the Levi-Civita connection ∇g (at time t)

on T ∗M and the pullback connection ϕ(t)∇ on ϕ(t)∗TN . In which case


∆g,hϕ = gij(ϕ(t)∇i (ϕ(t)∗∂j)− ϕ(t)∗

(∇gi ∂j

))= gij

(∂i∂jϕ

γ + Γ γαβ ∂iϕ

α∂jϕβ − Γ k

ij ∂iϕγ)∂γ . (4.12)

Step 5: Now define g(t) = (ϕ(t)−1)∗g(t), a time-dependent metric on N .We claim this metric g is a solution of the Ricci-DeTurck flow. To show this,note that a direct computation — similar to that of Step 3 — gives thefollowing:

Lemma 4.4.∂g

∂t= (ϕ−1)∗

(∂g

∂t

)+ LV g,

where Vp = −ϕ∗∂t∣∣ϕ−1(p)

.

The geometric invariance of the curvature implies (ϕ−1)∗(∂g∂t ) = −2Ric(g),since g is the pullback of g. So all we need to do is relate the Lie derivativeterm to that in the Ricci-DeTurck equation, i.e. we want to show that V = W .

The key observation here is the geometric invariance of the ‘map Laplacian’reflected in the following proposition.

Proposition 4.5. Suppose K, M and N are smooth manifolds with a diffeo-morphism ψ : K →M and a smooth map ϕ : M → N . Let g be a metric onM , h be a metric on N and g = ψ∗g. Then

∆g,h (ϕ ψ) =(∆g,hϕ

)ψ.

Note that ∆g,hϕ ∈ Γ (ϕ∗TN), so by restriction (∆g,hϕ)ψ ∈ Γ (ψ∗ϕ∗TN) =

Γ ((ϕ ψ)∗TN).

The result is not surprising: It states that the harmonic map Laplacian ofa map from M to K is unchanged if we apply an isometry to M .

Proof. For any p ∈ K, we need to show that

∆g,h (ϕ ψ) (p) =(∆g,hϕ

)(ψ(p)).

To do this for any fixed p, choose local coordinates (xi) for M near ψ(p) andinduce local coordinates on K near p by yi = xi ψ. Fix local coordinates(zα) for N near ϕ ψ(x). In these coordinates we have that z ϕ x−1 =z (ϕ ψ) y−1, gij = gij and hence Γij

k = Γijk everywhere on the chart.

Therefore in these coordinates ϕα = (ϕ ψ)α and so

80 4 Short-Time Existence(∆g,hϕ

)(ψ(p)) = gij

(∂i∂jϕ

α − Γijk∂kϕα + Γβγα∂iϕ

β ∂jϕγ) ∂

∂zα

= gij(∂i∂j(ϕ ψ)α − Γijk∂k(ϕ ψ)α

+ Γβγα∂i(ϕ ψ)β ∂j(ϕ ψ)γ

) ∂

∂zα

=(∆g,h (ϕ ψ)

)(p).

We can now complete the argument. As V (p) = − (ϕ∗∂t) (ϕ−1(p)) =−(∆g,hϕ)ϕ−1(p), Proposition 4.5 (with ψ = ϕ−1, N = M) and equation

(4.12) gives

V = −∆g,h

(ϕ ϕ−1

)= −gij

(Γij

k − Γijk)∂k = W.

Step 6: Finally we now prove the uniqueness result for the Ricci flow.Suppose there are solutions gi(t) of the Ricci flow, for i = 1, 2, with initialcondition g1(0) = g2(0). Taking N = M and ϕ0(x) = x, we produce solutionsgi(t) of the Ricci-DeTurck flow, with g2(0) = g2(0) = g1(0) = g1(0). Byuniqueness of solutions of the Ricci-DeTurck flow, g2(t) = g1(t) for all t in

their common interval of existence. Hence W = gij(Γijk − Γijk) is the same

for the two solutions. The diffeomorphisms ϕi(t) are given by the harmonicmap heat flow, and as before Proposition 4.5 gives

∂tϕi(x, t) = (∆gi,hϕi)(x, t) = −W (ϕi(x, t));

ϕi(x, 0) = x.

Thus ϕ1 and ϕ2 are solutions of the same initial value problem for a systemof ordinary differential equations, and hence ϕ1(x, t) = ϕ2(x, t) and g2 =ϕ∗2g2 = ϕ∗1g1 = g1, proving uniqueness for the Ricci flow.

Chapter 5

Uhlenbeck’s Trick

In Theorem 3.14 we derived an evolution equation for the curvature R underthe Ricci flow, which took the form

∂

∂tRijk` = ∆Rijk` + 2(Bijk` −Bij`k −Bi`jk +Bikj`)

− (RpiRpjk` +RpjRipk` +RpkRijp` +Rp`Rijkp)

where the time derivative is interpreted as in Section 3.3.1. In this chapter weexamine a trick attributed to Karen Uhlenbeck [Ham86, p. 155] that allowsone to simplify the above equation by removing the last collection of termswith a ‘change of variables’. We pursue this idea via three different methodswhich correspond to different bundle constructions.

5.1 Abstract Bundle Approach

Let V → M be an abstract vector bundle isomorphic to the tangent bundleTM over M with fixed (time-independent) metric h = g(0). Let ι0 : V → TMbe the identity map, so that h = ι∗0(g(0)). We aim to extend ι0 to a familyof bundle isometries ιt, so that h = ι∗t (g(t)) for every t. To do this, we evolveι according to the ode

∂

∂tι = Ric] ι (5.1a)

ι(·, 0) = ι0( · ) (5.1b)

where Ric] ∈ End(TM) is the endomorphism defined by raising the secondindex (i.e. Ric](∂i) = R j

i ∂j). This defines a 1-parameter family of bundleisomorphisms ι : V × [0, T )→ TM .

Lemma 5.1. If ι evolves by (5.1), then the bundle map ιt : (V, h) →(TM, g(t)) is an isometry for every t.

To see this, observe that

81

82 5 Uhlenbeck’s Trick

∂

∂tg(ι(v), ι(v)) =

( ∂∂tg)

(ι(v), ι(v)) + g( ∂∂tι(v), ι(v)

)+ g(ι(v),

∂

∂tι(v)

)= −2 Ric(ι(v), ι(v)) + g(Ric](ι(v)), ι(v)) + g(ι(v),Ric](ι(v)))

= 0.

for any v ∈ V . Therefore ι∗g(t) is independent of time t, and so continues toequal the fixed metric h.

Now consider the pullback of R by ι: Since ι∗∂a = ιia∂i we have

Rabcd = (ι∗Rg(t))(∂a, ∂b, ∂c, ∂d) = ιiaιjbιkc ι`dRijk`

andBabcd = (ι∗B)(∂a, ∂b, ∂c, ∂d) = ιiaι

jbιkc ι`dBijk`.

Thus as ∂∂t ι

ia = ιpaR

ip , we find that

∂

∂tRabcd = ιiaι

jbιkc ι`d

∂

∂tRijk` +

(( ∂∂tιia

)ιjbι

kc ι`dRijk` + · · ·+ ιiaι

jbιkc

( ∂∂tι`d

)Rijk`

)= ιiaι

jbιkc ι`d

∂

∂tRijk` +

((ιpaR

ip)ι

jbιkc ι`dRijk` + · · ·+ ιiaι

jbιkc ιqd(R

`qRijk`)

)= ιiaι

jbιkc ι`d

∂

∂tRijk` +

(ιqaι

jbιkc ι`dR

pqRpjk` + · · ·+ ιiaι

jbιkc ιqdR

pqRijkp

).

So by letting ∆D := trg(ι∗∇ ι∗∇), where ι∗∇ is the pullback connection,

Theorem 3.14 implies that

∂

∂tRabcd = ιiaι

jbιkc ι`d∆Rijk` + 2(Babcd −Babdc −Badbc +Bacbd)

= ∆DRabcd + 2(Babcd −Babdc −Badbc +Bacbd) (5.2)

where Babcd = −heghfiRaebfRcgdi. That is, the last collection of terms fromthe evolution equation for R in Theorem 3.14 have been eliminated.

5.2 Orthonormal Frame Approach

There is an alternative approach to Uhlenbeck’s trick put forth by Hamilton[Ham93, Sect. 2] (see also [CCG+08, App. F]). In this treatment the ideais to work on the bundle of frames (see Section 1.2.2) where we define anatural direction in which to take time derivatives. This is a much moreinvolved construction, but we will see later that this can be derived by arather standard procedure from the more elementary structure introduced inSection 5.3.

First, for completeness, we give an overview of the frame bundle machineryused in this approach.

5.2 Orthonormal Frame Approach 83

5.2.1 The Frame Bundle. Let M be a differentiable manifold andπ : FM → M the general linear frame bundle (thus the elements of FMx

are nonsingular linear maps from Rn to TxM). The group GL(n) acts bycomposition on each fibre, so that M = (Ma

b ) ∈ GL(n) acts on a frameY = (Ya) ∈ FMx to give the frame YM := Y M = (Ma

b Ya). Suppose (xi)is a local coordinate system defined on an open set U ⊂ M , so that

(∂∂xi

)is a local frame on U — which is a local section of the frame bundle FMrestricted to U . A frame Y = (Ya) can be written in local coordinates (xi) as

Ya = yia(Y )∂

∂xi,

where yia : FM |U → R assigns to a frame Y the i-th component of the a-thvector. The vector valued function y = (yia) : FM → GL(n,R) describes thetransition of the frame

(∂∂xi

)to the frame Y = (Ya).1

Now let xi = xi π : FM |U → R. The collection (xj , yia) is a coordinatesystem defined on the open set FM |U ⊂ FM . In particular ( ∂

∂xj ,∂∂yia

) is a

basis for the tangent space of FM at points in FM |U . Note that ( ∂∂yia

) is a

basis for the tangent space of the fibres FxM , for x ∈ U : The vectors ( ∂∂yia

)

are vertical whereas ( ∂∂xj ) are transverse to the fibres.

Remarkably, the tangent space of FM is trivial: We can define globalindependent vertical vector fields (Λab ) and horizontal vector fields (∇a). Inlocal coordinates (yia(Y ), xj(Y )) for FM , these can be expressed as

Λab = yib∂

∂yia

∇a = yja

(∂

∂xj− ykbΓ ikj

∂

∂yib

).

Here moving in the direction (∇a) in FM at Y corresponds to parallel trans-lating a frame along a path in M with initial velocity Ya; the vectors ∇aspan a natural ‘horizontal subspace’ in T (FM) defined by the connection.The vectors (Λab ) are generators of the action of GL(n) on the fibres of FM .

Each frame Y ∈ FMx has a dual co-frame (Y a∗ ) for T ∗xM , defined byY a∗ (Yb) = δba. Given a (p, q)-tensor on M we can define a function from FMto⊗p

(Rn)∗ ⊗⊗q Rn with components given by

Ta1...aq

b1...bp(Y ) := T

(Yb1 , . . . , Ybp , Y

a1∗ , . . . , Y

aq∗).

We can then differentiate this function in the direction of any vector fieldon FM , such as the vector fields defined above. The notation ∇a for the

1 We follow the convention that indices i, j, k, . . . are reserved for coordinates on Mand indices a, b, c, . . . are used for components of the frame. For instance in localcoordinates gij = g( ∂

∂xi, ∂∂xj

), whereas gab(Y ) = g(Ya, Yb) for a given frame Y . If

Ya = yia∂∂xi

then we also have gab = gijyiayjb .


horizontal vector fields is justified by the observation that differentiating theabove function in direction ∇a gives the components of the covariant deriv-ative ∇aT . We also have an expression for derivatives in vertical directions:

Lemma 5.2. If T is a (p, q)-tensor, then

(ΛcdTa1···aq

b1···bp )(Y ) =

p∑`=1

δcb` Ta1···aq

b1···b`−1 d b`+1···bp (Y )

−q∑

k=1

δakd Ta1···bk−1 c ak+1···aq

b1···bp (Y ).

In particular, if Vab = yiayjbVij is a covariant (2, 0)-tensor then

(ΛabVcd)(Y ) = δacVbd(Y ) + δadVcb(Y ).

The splitting of T (FM) into horizontal and vertical components alsomakes it possible to define a metric gF on FM to make π : (FM, gF ) →(M, g) a Riemannian submersion, so that (π∗g)(V,W ) = gF (V,W ) for V andW horizontal. We have the vertical and horizontal subspaces orthogonal, andon the vertical parts we take the natural metric defined by

⟨Λba, Λ

dc

⟩= δacδ

bd.The orthonormal frame bundle OM is the subbundle of FM defined by

OM := Y ∈ FM : gab(Y ) = δab.

As above, the metric g = (gab)na,b=1 can be considered as a function

g : FM → Sym+(n),

where Sym+(n) is the space of symmetric positive definite n×n matrices; inwhich case OM = g−1(In×n), where In×n is the identity matrix. Also, to betangent to OM a vector field V on FM must satisfy V (gab) = 0 on OM .

Lemma 5.3. The globally defined vector fields (∇a) and (Λbc) form a basison FM which — when restricted to the subbundle OM — is orthonormalwith respect to the Riemannian metric gF .

Proof. By construction gF (∇a, Λbc) = 0, and the vertical vectors (Λbc) are or-thonormal. Also, gF (∇a,∇c) = g(π∗∇a, π∗∇b) = g(Ya, Yc) and as g(Ya, Yc) =δac whenever Y ∈ OM , we also have gF (∇a,∇c) = δac.

Remark 5.4. The global framing of FM by the n2 +n vector fields Λba and ∇awas noted by Ambrose-Singer [AS53] and Nomizu [Nom56, p. 49], who calledthem ‘fundamental vector fields’ and ‘basic vector fields’ repectively. Thecorrespondence between tensors on M and functions on the frame bundle wasobserved by Wong [Won61, Theorem 2.4], who also noted that the basic vectorfields ∇a act to give the components of the covariant derivative [Won61,Theorem 2.15].

5.2 Orthonormal Frame Approach 85

5.2.2 Time-dependent Frame Bundles and the Ricci Flow. Whenconsidering a solution to the Ricci flow (M, g(t)), for t ∈ [0, T ), it is naturalto work on the space-time manifold M × [0, T ). Likewise the product of theframe bundle with the time axis gives a bundle over M×[0, T ) with projectionπ : FM × [0, T )→M × [0, T ) defined by π = π× id[0,T ). The time-dependentmetric g(t) can be considered as a function

g : FM × [0, T )→ Sym(n),

where Sym(n) is the space of symmetric n× n matrices.The orthonormal frame bundles at each time combine to give a bundle

over space-time also:

OM :=⋃

t∈[0,T )

OMg(t) × t = g−1(In×n) ⊂ FM × [0, T ).

Note that the time-like vector field ∂∂t is not always tangent to OM unless

g(t) is independent of time! Since the metric varies according to

∂

∂tgab = −2Rab,

the orthonormal frame bundle, with gab = δab, will now vary with time.Therefore all that is needed is to modify the time-like vector field to make ittangent to the orthonormal frame bundle OM .

To achieve this we consider the time derivative as the directional derivativewith respect to the following vector field.

Definition 5.5. The vector field ∇t on FM × [0, T ) is defined by

∇t :=∂

∂t+Rabg

bcΛac . (5.3)

This is characterised as follows:

Lemma 5.6. The vector field ∇t, restricted to OM ⊂ FM × [0, T ), is the

unique vector field tangent to the subbundle OM for which ∇t− ∂∂t is vertical

and perpendicular to each OMg(t) ⊂ FM .

Proof. On OM we have gab = δab. Now if∇t− ∂∂t is vertical and perpendicular

to each OMg(t) ⊂ FM , it follows that:

1. ∇t − ∂∂t = αdcΛ

cd as it is vertical.

2. 0 =⟨Λab − Λba, αdcΛcd

⟩= αba − αab , as ∇t − ∂

∂t is perpendicular to thesubbundle OMg(t), so that α is symmetric.

3. As ∇tgab = 0, we have that


0 =∂

∂tgab + αdagbd + αcbgac

= −2Rab + αba + αab

since ∇t is tangent to OM .

It follows that ∇t− ∂∂t = RabΛ

ab . Conversely, if ∇t is defined in this way then

it certainly satisfies the above properties.

Lemma 5.7. If T is a time-dependent (p, q)-tensor then

∇tTa1···aq

b1···bp =∂

∂tT

a1···aqb1···bp

+

n∑c,d=1

p∑j=1

RdbjTa1···aq

b1···bj−1 d bj+1···bp

+

n∑c,d=1

q∑i=1

Raid Ta1···ai−1 d ai+1···aq

b1···bp . (5.4)

In particular, if T is a (2, 0)-tensor then

∇tTab =∂

∂tTab +Racg

cdTdb +RbcgcdTad.

Proof. When T is a (2, 0)-tensor, Lemma 5.2 implies that ΛdcTab = δdaTcb +δdbTac so we find that

∇tTab =∂

∂tTab +Rde g

ec ΛdcVab

=∂

∂tTab +Rae g

ec Tcb +Rbe gec Tac.

The equation for general tensors follows likewise.

Finally, by defining the Laplacian ∆ :=∑ne=1∇e∇e — which acts on

vector valued functions on OM — we see by (5.4) that

∇tRabcd =∂

∂tRabcd +RpaRpbcd +RpbRapcd +RpcRabpd +RpdRabcp.

So by Theorem 3.14 it follows that the evolution equation of Rabcd takes thedesired form

(∇t −∆)Rabcd = 2(Babcd −Babdc +Bacbd −Badbc). (5.5)

Remark 5.8. This approach is in fact equivalent to the method discussed inSection 5.1. To see this, start by evolving a frame Y according to

d

dtYa = Rabg

bc Yc. (5.6)

5.3 Time-Dependent Metrics and Vector Bundles over M × R 87

With this one finds that ∂∂t (g(Ya, Yb)) = 0 under the Ricci flow; in particular,

if a frame is initially orthonormal then it remains so. We see that the pathγ(t) := (Y (t), t) lies in OM , so d

dtγ(t) ∈ T (OM) and

d

dtγ(t) = ∇t.

To check this equation, observe that ddtγ(t) = Ric(y) + ∂

∂t which in localcoordinates takes the form

d

dtγ(t) = Rabg

bcykc (Y )∂

∂yka

∣∣∣γ(t)

+∂

∂t

∣∣∣γ(t)

= RabgbcΛac

∣∣γ(t)

+∂

∂t

∣∣∣γ(t)

= ∇t.

In which case ∇t, defined by (5.3), corresponds to taking the time-derivative∂∂t

(T (Ya1 , . . . , Yap)

)of a tensor T where Yai satisfy (5.6). Thus the evolution

of the curvature Rabcd = Rijk`FiaF

jb F

kc F

`d in a frame moving according to

(5.6) is equivalent to (5.2) seen in Section 5.1.

5.3 Time-Dependent Metrics and Vector Bundles overM × R

Now we consider an alternative approach: Instead of using an abstract bundleand constructing an identification with the tangent bundle at each time, weput the tangent bundles at different time together to form a vector bundleover the ‘space-time’ M × R and place a natural connection on this bundle.This reproduces very simply the first method described above, and we willsee that the frame bundle machinery also relates closely to this method.

5.3.1 Spatial Tangent Bundle and Time-Dependent Metrics. Webegin with a rather general setting: Let g(t) be an arbitrary smooth familyof Riemannian metrics on a manifold M , parametrised by ‘time’ t. That is,we have for each (p, t) ∈M ×R an inner product g(p,t) on TpM . We interpretthis as a metric acting on the spatial tangent bundle S, defined by

S := v ∈ T (M × R) : dt(v) = 0,

as a vector bundle over M×R. With this, the metric g is naturally a metric onS, since S(p,t) is naturally isomorphic to TpM via the projection π : (p, t)→p. In which case a local frame for S consists of the coordinate tangent vectorfields (∂i) for TM , and from Example 1.62 we have the decomposition ofT (M × R) as a direct sum

T (M × R) = S⊕ R∂t.


Since S is a subbundle of T (M × R), any section of S is also a section ofT (M × R); we call these spatial vector fields.

5.3.1.1 The Canonical Connection on the Spatial Tangent Bundle. Thenext step is establish a result analogous to the Levi-Civita theorem — whichsays that for any Riemannian metric there is a unique compatible connectionwhich is symmetric. We want to construct a canonical connection on S forany given metric g on S. This is provided by the following theorem:

Theorem 5.9. Let g be a metric on the spatial tangent bundle S→M ×R.Then there exists a unique connection ∇ on S satisfying the following threeconditions:

1. ∇ is compatible with g: For any X ∈ Γ (T (M×R)) and any spatial vectorfields Y,W ∈ Γ (S),

Xg(Y,W ) = g(∇XY,W ) + g(Y,∇XW ).

2. ∇ is spatially symmetric: If X,Y ∈ Γ (S) are any two spatial vector fields,so they are in particular vector fields on M × R,

∇XY −∇YX = [X,Y ].

3. ∇ is irrotational: The tensor S ∈ Γ (S∗ ⊗S), defined by

S(V ) = ∇∂tV − [∂t, V ] (5.7)

for any V ∈ Γ (S), is symmetric with respect to g:

g(S(V ),W ) = g(V,S(W )),

for any V,W ∈ S(p,t).

Remark 5.10. Observe that S is S-valued and tensorial, since

S(fV ) = ∇∂t(fV )− [∂t, fV ]

= f∇∂tV + (∂tf)V − (∂tf)V − f [∂t, V ]

= fS(V )

and [∂t, V ] = [∂t, Vi∂i] = (∂tV

i)∂i ∈ S.

Proof. We show uniqueness. Working in local coordinates, we can write ∇ interms of its coefficients:

∇∂i∂j = Γijk∂k 1 ≤ i, j, k ≤ n

∇∂t∂j = Γ0jk∂k 1 ≤ j, k ≤ n

Now the first conditions implies that ∂tgij = g(∇t∂i, ∂j) + g(∂i,∇t∂j) =Γ0ij + Γ0ji. Combined with the second condition, we find that the spatial


components Γijk are given by the Christoffel symbols of the metric at fixed

time t, that is Γijk = 1

2gkl (∂igjl + ∂jgil − ∂lgij). The third is used as follows:

As S(∂i) = ∇∂t∂i−[∂t, ∂i] = Γ0ik∂k, the symmetry with respect to g amounts

toΓ0ij = g(S(∂i), ∂j) = g(∂i,S(∂j)) = Γ0ji.

So with compatibility with g, we have 2Γ0ij = ∂tgij . Therefore

Γ k0i =

1

2gkj∂tgij . (5.8)

It is now a simple matter to check that these formulas define a connectionwith the required properties.

Remark 5.11. The connection restricted to each fixed t is simply the Levi-Civita connection of the metric g(t). The importance of the compatibilitycondition will become apparent when we discuss the maximum principle forvector bundles in Section 6.4. The irrotational condition says that paralleltransport in the time direction does not have any rotation component (recallthat antisymmetric matrices are the generators of rotations).

5.3.1.2 The Time Derivative of the Curvature Tensor. In the special casewhen the metric evolves by Ricci flow, the above proof — in particular (5.8)— implies in local coordinates that

∇∂t∂i = Γ0ij∂j =

1

2gjp∂tgip∂j = −Ricji∂j .

From this we compute ∇∂tR:

(∇∂tR)ijkl = ∂t (R(∂i, ∂j , ∂k, ∂l))

−R(∇∂t∂i, ∂j , ∂k, ∂l)− · · · −R(∂i, ∂j , ∂k,∇∂t∂l)= ∂tRijkl −R(−Rpi ∂p, ∂j , ∂k, ∂l)− · · · −R(∂i, ∂j , ∂k,−Rpl ∂p)= ∂tRijkl +RpiRpjkl + · · ·+RplRijkp

= ∆Rijkl + 2 (Bijkl −Bijlk −Biljk +Bikjl) .

Thus again, the last collection of terms has been eliminated by computingwith respect to the connection on S rather than the time derivative definedin Section 3.3.1.

5.3.1.3 Relation to the Frame Bundle Method. Now we clarify the rela-tionship between our method and that involving computation on the framebundle: The construction given in Section 5.2.1 can be straightforwardly gen-eralised to the following situation: Let E be an arbitrary vector bundle of rankk over M , with a connection ∇. Then TE has a canonical splitting into ho-rizontal and vertical subspaces, where the vertical subspace is tangent to the


fibre, and the horizontal consists of the directions corresponding to paralleltranslation in E.

As in Section 1.2.2, the frame bundle FE of E is the bundle with fibreat x ∈ M given by the space of nonsingular linear maps from Rk to Ex. Asbefore this bundle has a left action of GL(k), and each frame (ξα) has a dualframe (φα) for E∗. An arbitrary tensor field T constructed on E (that is, amultilinear function at each x ∈ M acting on copies of Ex and E∗x) can bewritten as a vector-valued function on FE: If T ∈ Γ (

⊗pE∗ ⊗

⊗qE), then

we associate to T the function from FE to T pq (Rk) ' Rk(p+q) given by

(T (ξ))β1...βq

α1...αp= T

(ξα1

, . . . , ξαp , φβ1 , . . . , φβq

).

We are not quite in the situation we had previously where we could definea global frame for T (FM). In general this can be achieved by taking a directproduct of FE with the frame bundle FM . Assuming we have a connection onTM , there is again a canonical choice of horizontal subspace, and a canonicalframing of T (FM ⊕ FE).

However in the present situation a simpler construction suffices: Here wereplace M by M × [0, T ), and E is the spatial tangent bundle S. Then F (S)is isomorphic to FM × [0, T ). Since T (M × [0, T )) = S ⊕ R∂t, we have aglobal framing for TF (S), given by the fibre directions Λba, 1 ≤ a, b ≤ n,and the horizontal directions, given by ∇a for 1 ≤ a ≤ n and ∇t given bythe direction corresponding to parallel translation in the ∂t direction. This isexactly the framing constructed in Section 5.2.2.

In most of the computations we will undertake in this book we will findit much more convenient to work directly with the connection on S ratherthan working on the frame bundle.

5.3.2 Alternative Derivation of the Evolution of Curvature Equa-tion. We will now discuss an alternative derivation of the evolution equationfor the curvature tensor, making use of the canonical connection on the spa-tial tangent bundle defined above. We will first carry this out in the settingof a general time-dependent metric, then specialise to the Ricci flow.

5.3.2.1 A Canonical Connection on the Space-Time Tangent Bundle. Wehave constructed a connection on the spatial subbundle S of T (M × R). Itwill be convenient to extend this to a connection on all of T (M ×R) (mostlyfor the application of the Bianchi identity). There are several ways to do this,but we will choose the following obvious construction:

Theorem 5.12. There exists a unique connection ∇ on T (M ×R) for which∇∂t = 0 and ∇X is as constructed in Theorem 5.9 for all X ∈ Γ (S).

Remark 5.13. This choice of connection is not symmetric: We have in generalthat∇∂t∂i 6= 0, while∇i∂t = 0 always. However, this choice of connection hasseveral good points: Each of the submanifolds M×t is totally geodesic (and


so importantly for us computing derivatives of spatial tangent vector fieldsgives the same answer whether computed as section of S or of T (M × R).Also, choosing ∂t to be parallel has some benefits: In particular it ensuresthat the projection from T (M × R) to S is a parallel tensor.

We compute the torsion tensor: This clearly vanishes if both argumentsare spatial, so the only non-zero components (up to symmetry) are

τ(∂t, ∂i) = ∇∂t∂i −∇∂i∂t = S(∂i), (5.9)

where S is defined by (5.7).

5.3.2.2 The Spatial and Temporal Curvature Tensors. We wish to relatethe curvatures of the connections we have constructed to the curvature of theLevi-Civita connections of each of the metrics g(t).

To do this we define the spatial curvature R ∈ Γ (⊗4S∗) by

R(X,Y,W,Z) = R∇(X,Y,W,Z) = g(R∇(X,Y )W,Z)

for any X,Y,W,Z ∈ Γ (S). Moreover, the temporal curvature tensor P ∈Γ (⊗3S∗) is defined by

P(X,Y, Z) = g(R∇(∂t, X)Y, Z) (5.10)

for any X,Y, Z ∈ Γ (S). Note that our choice of connections ensures thatthese are the same whether we interpret the connection as acting on S or onT (M × R).

Furthermore, if X,Y,W,Z ∈ T(p,t)M , then R(X,Y,W,Z) is the Rieman-nian curvature of the metric g(t) at the point p, acting on π∗X, π∗Y , π∗Wand π∗Z where π : M × R → M ; (p, t) 7→ p is the projection map. Theother components (up to symmetry) of the curvature of the connection ∇ onT (M × R) all vanish.

Our next step will be to compute the temporal curvature tensor P interms of the tensor S, but to do this we must first make a digression todiscuss Bianchi identities.

5.3.2.3 Generalised Bianchi Identities. Let ∇ be a connection over a vec-tor bundle E over a manifold M , and suppose there is also a symmetricconnection ∇ defined on TM . As R∇ ∈ Γ (⊗2T ∗M ⊗ E∗ ⊗ E) we can —by the discussion in Section 1.5.3 — make sense of the covariant derivative∇R∇ ∈ Γ (⊗3T ∗M ⊗ E∗ ⊗ E).

Theorem 5.14. For any vector fields X,Y, Z ∈X (M) and ξ ∈ Γ (E),

(∇XR∇)(Y, Z, ξ) + (∇YR∇)(Z,X, ξ) + (∇ZR∇)(X,Y, ξ) = 0.

Proof. Since the equation is tensorial, it is enough to check the identity withX = ∂i, Y = ∂j and Z = ∂k about a point p ∈ M . We work in local


coordinates defined by exponential coordinates about p, so that ∇i∂j |p = 0for all i and j. Extend ξ ∈ Ep to a section of E such that ∇ξ|p = 0 (forexample, construct ξ on a neighbourhood of p by parallel transport alonggeodesics from p). Then as [∂i, ∂j ] = 0 everywhere, we find at the point pthat

(∇iR∇)(∂j , ∂k, ξ) = ∇i (R∇(∂j , ∂k, ξ))

−R∇(∇i∂j , ∂k, ξ)−R∇(∂j , ∇i∂k, ξ)−R∇(∂j , ∂j ,∇iξ)= ∇i (∇j (∇kξ)−∇k (∇jξ)) . (5.11)

Using this, we find a the point p that

(∇iR∇)(∂j , ∂k, ξ) + (∇jR∇)(∂k, ∂i, ξ) + (∇kR∇)(∂i, ∂j , ξ)

= ∇i (∇j (∇kξ))−∇i (∇k (∇jξ))+∇j (∇k (∇iξ))−∇j (∇i (∇kξ))+∇k (∇i (∇jξ))−∇k (∇j (∇iξ))

= R∇(∂j , ∂i,∇kξ) +R∇(∂k, ∂j ,∇iξ) +R∇(∂i, ∂k,∇jξ)= 0

where we grouped the first term on the first line with the second on the third,the first term on the second line with the second on the first, and the firstterm on the third line with the second on the second.

Unfortunately we will not always be in this convenient situation: It willturn out to be natural to work with a non-symmetric connection when work-ing with time-dependent metrics, so we will need the following variation onthe Bianchi identity in the non-symmetric case. In this case we define thetorsion tensor τ ∈ Γ (⊗2T ∗M ⊗ TM) by

τ(X,Y ) = ∇XY − ∇YX − [X,Y ]. (5.12)

Theorem 5.15. For any vector fields X,Y, Z ∈X (M) and ξ ∈ Γ (E),

(∇XR∇)(Y,Z, ξ) + (∇YR∇)(Z,X, ξ) + (∇ZR∇)(X,Y, ξ)

+R∇(τ(X,Y ), Z, ξ) +R∇(τ(Y,Z), X, ξ) +R∇(τ(Z,X), Y, ξ) = 0.

Proof. As before, choose X = ∂i, Y = ∂j , and Z = ∂k about a point p ∈M .Extend ξ ∈ Ep to a smooth section with ∇ξ|p = 0. The difference arisesbecause the connection in geodesic coordinates from p is no longer symmetric:Instead we have ∇vi∂i(vj∂j)|p = 0 for every vector v ∈ TpM (this is thegeodesic equation along the geodesic with direction vi∂i at p). It follows that

Γijk is antisymmetric in the first two arguments at p, so we have

2Γijk = Γij

k − Γjik = τijk.


So rather than (5.11), we find that

(∇iR∇)(∂j , ∂k, ξ) = ∇i (R∇(∂j , ∂k, ξ))

−R∇(∇i∂j , ∂k, ξ)−R∇(∂j , ∇i∂k, ξ)−R∇(∂j , ∂j ,∇iξ)= ∇i (∇j (∇kξ)−∇k (∇jξ))

− 1

2R∇(τij , ∂k, ξ)−

1

2R∇(∂j , τik, ξ).

The result now follows as before by observing the antisymmetry of τ and ofthe first two argument of R∇.

In the special case where we are working with a connection on TM , wealso have a version of the first Bianchi identity, which we state in the case ofconnections which may be non-symmetric:

Theorem 5.16. For any vector fields X,Y, Z ∈X (M),

R(X,Y )Z +R(Y, Z)X +R(Z,X)Y

= ∇Y τ(X,Z) +∇Zτ(Y,X) +∇Xτ(Z, Y )

+ τ(τ(Y,X), Z) + τ(τ(Z, Y ), X) + τ(τ(X,Z), Y ).

Proof. As before we work in exponential coordinates, so that at a point p wehave ∇i∂j = 1

2τ(∂i, ∂j). Then we compute

R(∂i, ∂j)∂k +R(∂j , ∂k)∂i +R(∂k, ∂i)∂j

= ∇j(∇i∂k)−∇i(∇j∂k) +∇k(∇j∂i)−∇j(∇k∂i) +∇i(∇k∂j)−∇k(∇i∂j)

= ∇j (τ(∂i, ∂k)) +∇k (τ(∂j , ∂i)) +∇i (τ(∂k, ∂j))

= ∇jτik +∇kτji +∇iτkj + τ(τji, ∂k) + τ(τkj , ∂i) + τ(τik, ∂j)

where we used the antisymmetry of τ in the last line.

Finally, a result which applies when the connection is compatible with ametric g (or, more generally, a parallel symmetric bilinear form).

Theorem 5.17. Let E be a vector bundle over M , ∇ a connection on Ecompatible with the form g. For all X,Y ∈X (M) and ξ, η ∈ Γ (E),

R(X,Y, ξ, η) +R(X,Y, η, ξ) = 0

where we define R(X,Y, ξ, η) = g(R(X,Y )ξ, η).

Proof. By compatibility and the definition of the Lie bracket,


0 = Y (X g(ξ, η))−X(Y g(ξ, η))− [Y,X] g(ξ, η)

= Y(g(∇Xξ, η) + g(ξ,∇Xη)

)−X

(g(∇Y ξ, η) + g(ξ,∇Y η)

)− g(∇[Y,X]ξ, η)− g(ξ,∇[Y,X]η)

= g(∇Y (∇Xξ), η) + g(∇Xξ,∇Y η) + g(∇Y ξ,∇Xη) + g(ξ,∇Y (∇Xη)

− g(∇X(∇Y ξ), η)− g(∇Y ξ,∇Xη)− g(∇Xξ,∇Y η)− g(ξ,∇X(∇Y η)

− g(∇[Y,X]ξ, η)− g(ξ,∇[Y,X]η)

= R(X,Y, ξ, η) +R(X,Y, η, ξ).

5.3.2.4 Expression for the Temporal Curvature. We see that the temporalcurvature tensor P can be computed in terms of the tensor S:

Theorem 5.18. For X,Y, Z ∈ Γ (S),

P(X,Y, Z) = ∇ZS(Y,X)−∇Y S(Z,X).

Proof. Observing that R(·, ·, ∂t, ·) = 0 and the expression for the torsion givenby (5.9) with the the Bianchi identity proved in Theorem 5.16 gives

R(∂t, X, Y, Z)−R(∂t, Y,X,Z) = g(∇XS(Y ), Z)− g(∇Y S(X), Z)

= ∇XS(Y,Z)−∇Y S(X,Z)

since the connection is spatially symmetric. From this we obtain, using The-orem 5.17, the following expression:

2R(∂t, X, Y, Z) = R(∂t, X, Y, Z)−R(∂t, X, Z, Y )

= R(∂t, Y,X,Z) +∇XS(Y,Z)−∇Y S(X,Z)

−R(∂t, Z,X, Y )−∇XS(Z, Y ) +∇ZS(X,Y )

= R(∂t, Z, Y,X)−R(∂t, Y, Z,X)−∇Y S(X,Z) +∇ZS(X,Y )

= −2∇Y S(Z,X) + 2∇ZS(Y,X)

where we used the symmetry of S from Theorem 5.9.

5.3.2.5 The Temporal Bianchi Identity and Evolution of Curvature. TheBianchi identity from Theorem 5.15 gives the usual second Bianchi identity inspatial directions. The remaining identities are those where one of the vectorfields is ∂t, which provide the following evolution equation for the spatialcurvature tensor:

Theorem 5.19. For X,Y, Z ∈ Γ (S),

∇∂tR(X,Y,W,Z) = ∇X∇ZS(Y,W )−∇X∇WS(Y, Z)

−∇Y∇ZS(X,W ) +∇Y∇WS(X,Z)

−R(S(X), Y,W,Z)−R(X,S(Y ),W,Z).


Proof. From the Bianchi identity in Theorem 5.15 and the torsion identity(5.9) we have:

∇∂tR(X,Y,W,Z) = −∇XR(Y, ∂t,W,Z)−∇YR(∂t, X,W,Z)

−R(τ(∂t, X), Y,W,Z)−R(τ(X,Y ), ∂t,W,Z)

−R(τ(Y, ∂t), X,W,Z)

= ∇XP(Y,W,Z)−∇Y P(X,W,Z)

−R(S(X), Y,W,Z)−R(X,S(Y ),W,Z) (5.13)

= ∇X∇ZS(Y,W )−∇X∇WS(Y, Z)

−∇Y∇ZS(X,W ) +∇Y∇WS(X,Z)

−R(S(X), Y,W,Z)−R(X,S(Y ),W,Z)

where we used the result of Theorem 5.18. Note the difference in the last twoterms compared to the expression in Proposition 3.8.

5.3.2.6 The Evolution of Curvature in the Ricci Flow Case. Suppose nowthat that the metric g evolves by Ricci flow. With the vector bundle ma-chinery presented so far, we can now derive the evolution equation for thespatial curvature. Working with the connection ∇ on M×R constructed fromTheorems 5.9 and 5.12, we first observe directly from Proposition 3.2 — as∇ is spatially symmetric — that:

Theorem 5.20. For X,Y,W,Z ∈ Γ (S), the Laplacian of the spatial curvaturetensor R satisfies

(∆R)(X,Y,W,Z) = ∇X∇WRic(Y,Z)−∇Y∇WRic(X,Z)

+∇Y∇ZRic(X,W )−∇X∇ZRic(Y,W )

− 2(B(X,Y,W,Z)−B(X,Y, Z,W )

+B(X,W, Y, Z)−B(X,Z, Y,W ))

+ Ric(R(W,Z)X,Y )− Ric(R(W,Z)Y,X).

Now since the time-dependent metric g = g(t) evolves according to ∂tgij =−2Ricij , equation (5.8) implies that

S(∂i) = ∇∂t∂i −[∂t, ∂i]

= Γ k0i ∂k

=1

2(gkj∂tgij)∂k

= −Ric ki ∂k.

So we conclude for X,Y ∈ Γ (S) that

S(X) = −Ric](X) and S(X,Y ) = −Ric(X,Y ).


As ∇ is compatible with g, we also see that

∇X∇Y S(Z,W ) = −∇X∇Y Ric(Z,W ) = −g(∇X∇Y Ric](Z),W ).

Thus, by combining Theorems 5.19 and 5.20, we obtain the following desiredreaction-diffusion type equation for the curvature.

Theorem 5.21. If the metric g evolves by Ricci flow, then the curvature Revolves according to

∇∂tR(X,Y, Z,W ) = (∆R)(X,Y, Z,W )

+ 2(B(X,Y, Z,W )−B(X,Y,W,Z)

+B(X,W, Y, Z)−B(X,Z, Y,W )),

where X,Y, Z and W are spatial vector fields.

Chapter 6

The Weak Maximum Principle

The maximum principle is the main tool we will use to understand the beha-viour of solutions to the Ricci flow. While other problems arising in geometricanalysis and calculus of variations make strong use of techniques from func-tional analysis, here — due to the fact that the metric is changing — mostof these techniques are not available; although methods in this direction aredeveloped in the work of Perelman [Per02]. The maximum principle, thoughvery simple, is also a very powerful tool which can be used to show thatpointwise inequalities on the initial data of parabolic pde are preserved bythe evolution. As we have already seen, when the metric evolves by Ricci flowthe various curvature tensors R, Ric, and Scal do indeed satisfy systems ofparabolic pde. Our main applications of the maximum principle will be toprove that certain inequalities on these tensors are preserved by the Ricciflow, so that the geometry of the evolving metrics is controlled.

6.1 Elementary Analysis

Suppose U ⊂ Rn is open and let f : U ⊂ Rn → R be a smooth function. Iff has a local minimum at some p ∈ U , then it follows that ∇f(p) = 0 and∆f(p) ≥ 0. This follows from the second derivative test for functions of onevariable: Given any direction v ∈ Rn, the function t 7→ fv(t) := f(p+tv) has alocal minimum at t = 0, so f ′v(0) = ∇vf(p) = 0 and f ′′v (0) = ∇v∇vf(p) ≥ 0.The Laplacian is proportional to the average of f ′′v (0) over all unit vectors v,and so is also non-negative.

On a Riemannian manifold the same argument applies if we define fv(t) :=f(expp(tv)

), so that we have the following result:

Lemma 6.1 (Second Derivative Test). Let M be a n-dimensional Rieman-nian manifold, and u : M → R a C2-function. If u has a local minimum ata point p ∈M , then

∇u(p) = 0 and ∆u(p) ≥ 0.

97

98 6 The Weak Maximum Principle

As we shall see, Lemma 6.1 is the main ingredient in the proof of themaximum principle.

6.2 Scalar Maximum Principle

The simplest manifestation of the weak maximum principle is the followingscalar maximum principle for time dependent metrics.

Proposition 6.2. Suppose g(t), t ∈ [0, T ), is a smooth family of metrics ona compact manifold M such that u : M × [0, T )→ R satisfies

∂

∂tu−∆g(t)u ≥ 0.

If u ≥ c at t = 0, for some c ∈ R, then u ≥ c for all t ≥ 0.1

Proof. Fix ε > 0 and define uε = u + ε(1 + t), so by hypothesis uε > c att = 0. We claim uε > c for all t > 0.

To prove this, suppose the result is false. That is, there exists ε > 0such that uε ≤ c somewhere in M × [0, T ). As M is compact, there exists(x1, t1) ∈ M × (0, T ) such that uε(x1, t1) = c and uε(x, t) ≥ c for all x ∈ Mand t ∈ [0, t1]. From this it follows that at (x1, t1) we have ∂uε

∂t ≤ 0 and∆uε ≥ 0, so that

0 ≥ ∂uε∂t≥ ∆g(t)uε + ε > 0,

which is a contradiction. Hence uε > c on M × [0, T ); and since ε > 0 isarbitrary, u ≥ c on M × [0, T ).

This proposition can be generalised by considering the semi-linear second-order parabolic operator:

Lu :=∂u

∂t−∆g(t)u− 〈X(t),∇u〉 − F (u, t),

where X(t) is a time-dependent vector field and F = F (x, t) : R× [0, T )→ Ris continuous in t and locally Lipschitz in x. We say u is a supersolution ifLu ≥ 0, and a subsolution if Lu ≤ 0.

Proposition 6.3 (Comparison Principle). Suppose that u and v are C2

and satisfy Lv ≤ Lu on M× [0, T ), and v(x, 0) ≤ u(x, 0) for all x ∈M . Then

v(x, t) ≤ u(x, t)

holds on M × [0, T ).

1 Note that the Laplacian is defined by ∆g(t) := g(t)ij∇(t)i ∇

(t)j , where ∇(t) is the

covariant derivative associated to g(t).

6.2 Scalar Maximum Principle 99

Proof. We apply an argument to w = u − v similar to that of the previousproposition. Firstly, compute

0 ≤ Lu− Lv =∂w

∂t−∆w − 〈X,∇w〉 − F (u, t) + F (v, t),

and note that the main difficulty is in controlling the last two terms. To dothis, let τ ∈ (0, T ), so that u and v are C2 on M × [0, τ ]. In particular,since M × [0, τ ] is compact and F is locally Lipschitz in the first argument,there exists a constant C such that |F (u(x, t), t)− F (v(x, t), t)| ≤ C|u(x, t)−v(x, t)|, for all (x, t) ∈ M × [0, τ ]. Now let ε > 0, and define wε(x, t) =w(x, t) + εe2Ct. Then wε(x, 0) ≥ ε > 0 for all x ∈M , while

∂wε∂t≥ ∆w + 〈X,∇w〉 − C|w|+ 2Cεe2Ct.

At a first point and time (x0, t0) where wε(x0, t0) = 0, we have w = −εe2Ct0 ,∇w = 0, and ∆w ≥ 0, while ∂wε

∂t ≤ 0. So at this point

0 ≥ ∂wε∂t≥ ∆w + 〈X,∇w〉 − Cεe2Ct0 + 2Cεe2Ct0 ≥ Cεe2Ct0 > 0,

which is a contradiction. Therefore wε > 0 for all ε > 0, and hence w ≥ 0 onM × [0, τ ]. Since τ ∈ (0, T ) is arbitrary, w ≥ 0 on M × [0, T ).

From the above result we can conclude that super/sub-solutions of heattype equations can be bounded by solutions to associated ode:

Theorem 6.4 (The Scalar Maximum Principle). Suppose u is C2 andsatisfies Lu ≥ 0 on M × [0, T ), and u(x, 0) ≥ c for all x ∈M . Let φ(t) be thesolution to the associated ode:

dφ

dt= F (φ, t),

φ(0) = c.

Thenu(x, t) ≥ φ(t)

for all x ∈M and all t ∈ [0, T ) in the interval of existence of φ.

Proof. Apply Proposition 6.3 with v(x, t) = φ(t): This can be done sinceLu ≥ 0 = Lφ, and u(x, 0) ≥ c = φ(0).

Remark 6.5. Of course a similar result holds if both inequalities are reversed,and in particular if u satisfies Lu = 0 then both upper and lower bounds foru can be obtained from solutions of the associated ode.


6.2.1 Lower Bounds on the Scalar Curvature. As a simple illustra-tion of how the scalar maximum principle can be applied to the Ricci flow,consider the evolution equation for the scalar curvature given in Corollary3.19. Since the reaction term is always non-negative, we have

∂

∂tScal ≥ ∆Scal.

So any initial lower bound for scalar curvature is preserved (while upperbounds are in general not preserved). A stronger result can be obtained byestimating the reaction term more carefully: Since

|Ric|2 ≥ 1

nScal2,

we have∂

∂tScal ≥ ∆Scal +

2

nScal2.

Writing S0 = infScal(p, 0) : p ∈ M, we can apply Theorem 6.4 to proveScal(p, t) ≥ φ(t), where

φ(t) =

− n|S0|n+2|S0|t if S0 < 0,

0 if S0 = 0,nS0

n−2S0tif S0 > 0.

In particular, if S0 > 0 then the lower bound φ(t) approaches infinity ast → n

2S0, thus giving an upper bound on the maximal interval existence for

the solution of Ricci flow.

6.2.2 Doubling-Time Estimates. Another useful application of thescalar maximum principle is the so-called doubling time estimate, which givesa lower bound on the time taken for the curvature to become large. From theevolution equation for curvature (as in the proof of Lemma 7.3) we find that

∂

∂t|R|2 ≤ ∆|R|2 + C(n)|R|3.

So if |R| ≤ K at t = 0, then

|R|2(x, t) ≤(

1

K− C(n)

2t

)−2

=: ρ(t) (6.1)

for all x ∈M and t ≥ 0 where ρ(t) is the solution of the ode: dρdt = C(n)ρ3/2

with ρ(0) = K2. In particular |R|(t) ≤ 2K, whenever t ∈ [0, 1/C(n)K]. Thisproves the following result which states that the maximum of the curvaturecannot grow too fast:

6.3 Maximum Principle for Symmetric 2-Tensors 101

Lemma 6.6 (Doubling Time Estimate). Let g(t) be a solution of theRicci flow on a compact manifold M , with |R(g(0))| ≤ K. Then |R(g(t))| ≤2K, for all t ∈ [0, 1/C(n)K].

6.3 Maximum Principle for Symmetric 2-Tensors

To go beyond controlling the scalar curvature to controlling the Ricci curvature,it is natural to consider a generalisation of the maximum principle which ap-plies to symmetric 2-tensors. This was done by Hamilton in his paper onthree-manifolds [Ham82b]. In order to state his result, we need the followingdefinition:

Definition 6.7 (Null-eigenvector Assumption). We say β : Sym2T ∗M×[0, T )→ Sym2T ∗M satisfies the null-eigenvector assumption if whenever ωijis a nonnegative symmetric 2-tensor at a point x, and if V ∈ TxM is suchthat ωijV

j = 0, thenβij(ω, t)V

iV j ≥ 0

for any t ∈ [0, T ).

Note that a symmetric tensor ωij is defined to be non-negative if and onlyif ωijv

ivj ≥ 0 for all vectors vi (i.e. if the quadratic form induced by ωij ispositive semi-definite). In this situation we write ωij ≥ 0.

Theorem 6.8. Suppose that g(t), t ∈ [0, T ), is a smooth family of metricson a compact manifold M . Let α(t) ∈ Γ (Sym2T ∗M) be a symmetric 2-tensorsatisfying

∂α

∂t≥ ∆g(t)α+ 〈X,∇α〉+ β,

where X is a (time-dependent) vector field and β = β(α, t) is a symmetric2-tensor locally Lipschitz in α and continuous in t.

If α(p, 0) ≥ 0 for all p ∈M and β satisfies the null-eigenvector assumption,then α(p, t) ≥ 0 for all p ∈M and t ∈ [0, T ).

We naturally want to apply this theorem to the evolution of Ricij underthe Ricci flow, which by Corollary 3.18 takes the form

∂

∂tRicij = ∆Ricij + 2(RicpqRpijq − RicipRicpj).

The problem here is trying to control the curvature term Rpijq which po-tentially could be quite complicated. Fortunately Hamilton [Ham82b] notedthat the Weyl curvature tensor, discussed in Section 3.5.2, vanishes when thedimension n = 3.2 The significance of this is that it allows one to write the

2 When n = 3 there are only two possible types of nonzero components of W . Eitherthere are three distinct indices, such as W1231, or there are two distinct indicessuch as W1221. Using the trace-free property, W1231 = −W2232 −W3233 = 0. Also,W1221 = −W2222−W3223 = −W3223 = W3113 = −W2112 = −W1221 so W1221 = 0.


full curvature tensor:

Rijk` = gikRicj` + gj`Ricik − gi`Ricjk − gjkRici` −1

2Scal(gikgj` − gi`gjk).

It follows that the evolution of the Ricci tensor, when n = 3, is given by

∂

∂tRicij = ∆Ricij −Qij ,

where Qij = 6RicipRicpj − 3Scal Ricij − (Scal2 − 2|Ric|2)gij is completelyexpressed in terms of the Ricci tensor Ricij and its contractions with themetric gij . With the equation now in this form, one can simply check thenull-eigenvector condition in Theorem 6.8, with X = 0, αij = Ricij andβij = −Qij , to prove the following:

Theorem 6.9 ([Ham82b, Sect. 9]). Suppose g(t), t ∈ [0, T ), is a solution ofthe Ricci flow on a closed 3-manifold M . If Ricij ≥ 0 at t = 0, then Ricij ≥ 0on 0 ≤ t < T .

Remark 6.10. However in general for n ≥ 3 neither the condition of nonneg-ative Ricci curvature nor the condition of nonnegative sectional curvature ispreserved under the Ricci flow on closed manifolds (see [CCG+08, Chap. 13]).It is precisely the problem of controlling the Weyl part in the evolutionaryequation that has remained a major obstruction to pinching results in higherdimensions. We will return to this problem in Chapters 11 and 12.

6.4 Vector Bundle Maximum Principle

Consider a compact Riemannian manifold (M, g) with a (possibly time-

varying) metric g with Levi-Civita connection ∇, and a rank k-vector bundleπ : E → M × R with a connection ∇. Let h be a metric on the bundle Ewhich is compatible with the connection ∇.

Now consider a heat-type pde in which sections u ∈ Γ (E) evolve by

∇∂tu = ∆u+∇V u+ F (u), (6.2)

where the Laplacian ∆ acting on sections of E is defined by

∆u = gij(∇i(∇ju)−∇∇i∂ju);

V is a smooth section of the spatial tangent bundle S, i.e. a smoothtime-dependent vector field on M ; and F is a time-dependent vertical vec-tor field on each fibre of E, i.e. F ∈ Γ (π∗E → E) where π∗E is thepullback bundle. Thus we have F (x, t, u) ∈ (π∗E)(x,t,u) = E(x,t) for any(x, t, u) = (x, t, u(x, t)) ∈ E. In particular F(x,t) := F |π−1(x,t) : u → v =F (x, t, u) is a map from E(x,t) to itself; in which case (6.2) takes the form∇∂tu(x, t) = ∆u(x, t) + F (x, t, u(x, t)).

6.4 Vector Bundle Maximum Principle 103

6.4.1 Statement of Maximum Principle. In order to state an ana-logous maximum principle for vector bundles, we need the following threeconditions.

Definition 6.11 (Convex in the Fibre). A subset Ω ⊂ E is said to beconvex in the fibre if for each (x, t) ∈M × R, the set Ω(x,t) = Ω ∩E(x,t) is aconvex subset of the vector space E(x,t).

With reference to Appendix B, we define the support function s : E∗ → Rof Ω by

s(x, t, `) = sup`(v) : v ∈ Ω(x,t) ⊂ E(x,t).

The normal cone NvΩ(x,t) of Ω(x,t) at a point v ∈ ∂Ω(x,t) is then defined by

NvΩ(x,t) = ` ∈ E∗(x,t) : `(v) = s(x, t, `),

and the tangent cone is defined by

TvΩ(x,t) =⋂

`∈NvΩ(x,t)

z ∈ E(x,t) : `(z) ≤ 0

.

Definition 6.12 (Vector Field Points into the Set). Let Ω be a subsetof E which is convex in the fibre. The vector field F ∈ Γ (π∗E) is said topoint into Ω if F (x, t, v) ∈ TvΩ(x,t) for every (x, t, v) ∈ E with v ∈ ∂Ω(x,t).Furthermore, F is said to strictly point into Ω if F (x, t, v) is in the interiorof TvΩ(x,t) for every (x, t, v) ∈ E with v ∈ ∂Ω(x,t).

Remark 6.13. We will prove in Corollary 6.16 that F points into Ω preciselywhen the flow of the vector field F on each fibre E(x,t) takes Ω(x,t) into itself.

Definition 6.14 (Invariance Under Parallel Transport). Let Ω ⊂ E bea subset. We say Ω is invariant under parallel transport by the connection∇ if for every curve γ in M × R and vector V0 ∈ Ωγ(0), the unique parallelsection V along γ with V (0) = V0 is contained in Ω.

We now state the maximum principle for vector bundles as follows.

Theorem 6.15 (Maximum Principle for Vector Bundle). Let Ω ⊂ Ebe closed, convex in the fibre, and invariant under parallel transport withrespect to ∇. Let F be a vector field which points into Ω. Then any solutionu to the pde (6.2) which starts in Ω will remain in Ω: If u(x, t0) ∈ Ω for allx ∈M , then u(x, t) ∈ Ω for all x ∈M and t ≥ t0 in the interval of existence.

Proof. We apply a maximum principle argument to a function f on the‘sphere bundle’ S∗ over M × R (with fibre at (x, t) given by S∗(x,t) = ` ∈E∗(x,t) : ‖`‖ = 1) defined by

f(x, t, `) = `(u(x, t))− s(x, t, `)


where s is the support function defined above. By Theorem B.2 in Appendix Bwe note that for each (x, t), the supremum of f(x, t, `) over ` ∈ S∗(x,t) gives

the distance of u(x, t) from Ω(x,t). Thus the condition: u(x, t) ∈ Ω, for all(x, t), is equivalent to the inequality f ≤ 0 on S∗. In particular the initialcondition u(x, t0) ∈ Ω(x,t0), for all x ∈M is equivalent to the condition:

f(x, t0, `) ≤ 0, ∀ (x, t0, `) ∈ S∗.

We seek to prove the same is true for positive times t.Our strategy will be to show that f − εeCt remains negative for all small

ε > 0, for a suitable constant C. The choice of C will depend on Lipschitzbounds for the vector field F , so we must first restrict to a suitable compactregion: Let t1 > t0 be any time less than the maximal time of existence of thesolution, so that u is a smooth section on E over M×[t0, t1]. Since M×[t0, t1]is compact, there exists K > 0 such that ‖u(x, t)‖ ≤ K. Since F is smooth,there exists a constant L such that ‖F (x, t, v2) − F (x, t, v1)‖ ≤ L‖v2 − v1‖for all (x, t) ∈ M × [t0, t1] and all v2, v1 ∈ E(x,t) with ‖vi‖ ≤ 2K, i = 1, 2.

Now choose ε0 > 0 such that ε0e(L+1)(t1−t0) ≤ K. We will prove that:

f − εe(L+1)(t−t0) ≤ 0 on M × [t0, t1]

for any ε ≤ ε0.First note that the inequality f − εe(L+1)(t−t0) holds strictly for t = t0. If

the inequality does not hold on the entire domain M×[t0, t1], then there exists

(x, t, ˜) with t0 < t ≤ t1 and ` ∈ S∗(x,t)

such that f(x, t, `0) = εe(L+1)(t−t0)

and f(x, t, `) ≤ εe(L+1)(t−t0) for all (x, t, `) ∈ S∗ with t0 ≤ t ≤ t.Let v be the closest point in Ω(x,t) to u(x, t). By Theorem B.2 we have

‖v − u(x, t)‖ = εe(L+1)(t−t0) ≤ K since ε ≤ ε0, and ‖u(x, t)‖ ≤ K, so‖v‖ ≤ 2K. Thus by the definition of L we have∥∥F (x, t, v)− F (x, t, u(x, t))

∥∥ ≤ L‖v − u(x, t)‖

= εLe(L+1)(t−t0).

We claim that ` ∈ NvΩ(x,t): From the proof of Theorem B.2, for ` to at-

tain the maximum of f we must have `( · ) =⟨u(x, t)− v, ·

⟩/‖u(x, t)− v‖,

and then the proof of Theorem B.1 shows that `(v) = s(x, t, `), so ` ∈NvΩ(x,t) as required. The assumption that F points into Ω then implies

that `(F (x, t, v)) ≤ 0.In a neighbourhood U of (x, t), extend ˜ to a smooth section ` of E∗ by

parallel translation along spatial geodesics from x, then along lines of fixed xin the t direction. Since the metric is compatible with the connection on E,we have `(x, t) ∈ S∗ for (x, t) ∈ U .

6.4 Vector Bundle Maximum Principle 105

Claim. For this `, the function (x, t) 7→ s(x, t, `(x,t)) is constant on U .

Proof of Claim. Parallel transporting along the curves indicated above definesthe parallel transport maps

P : U × E(x,t) → π−1E U ⊂ E and P ∗ : U × E∗(x,t) → π−1

E∗(U) ⊂ E∗.

Thus `(x, t) = P ∗(x, t, ˜) by definition. The invariance of Ω under paralleltransport implies that P (x, t,Ω(x,t)) = Ω(x,t) for every (x, t) ∈ U . The defin-

ition of the dual connection implies that (P ∗(x, t, ˜))(P (x, t, v)) is parallel,hence constant, along each of the curves. In particular this implies

s(x, t, P ∗(x, t, ˜)) = sup

(P ∗(x, t, ˜))(v) : v ∈ Ω(x,t)

= sup

(P ∗(x, t, ˜))(P (x, t, v)) : v ∈ Ω(x,t)

= sup

˜(v) : v ∈ Ω(x,t)

= s(x, t, ˜)

which proves the claim.

Define a function f on U by setting f(x, t) = f(x, t, `(x, t)). By the

construction of `, f is smooth on U . Furthermore, f(x, t) = εe(L+1)(t−t0)

and f(x, t) ≤ εe(L+1)(t−t0) for all (x, t) with t0 ≤ t ≤ t. It followsfrom this that spatial derivatives of f vanish at (x, t), ∆f(x, t) ≤ 0, and∂∂t f(x, t)− ε(L+ 1)e(L+1)(t−t0) ≥ 0.

Since ` is parallel along spatial geodesics through (x, t), we have that(∇2`(x, t))(v, v) = 0 for every v, and hence ∇2`(x, t) = 0 since ∇ is symmet-ric. In particular, ∆`(x, t) = 0.

Now we compute

∂

∂tf∣∣∣(x,t)

=∂

∂t

(`(u)− s(`)

)=

∇∂t`(u) + `(∇∂tu)−∂

∂ts(`)

= `(∆u+∇V u+ F (u)

)= ∆f(x, t) +∇V f(x, t)

+ `(F (x, t, u(x, t))− F (x, t, v)

)+ `(F (x, t, v)),

since ∇f = `(∇u) + (∇`)(u) − ∇(s(`)) = `(∇u) and ∆f = (∆`)(u) +2∇i`(∇iu) + `(∆u)−∆(s(`)) = `(∆u).

Combining the inequalities ∆f ≤ 0, ∇f = 0, ∂∂t (f − εe

(L+1)(t−t0)) ≥ 0,‖F (u)− F (v)‖ ≤ L‖u− v‖ and `(F (v)) ≤ 0 noted previously, we see that


0 ≤ ∂

∂tf(x, t)− ε(L+ 1)e(L+1)(t−t0)

= ∆f(x, t) +∇V f(x, t) + `(F (x, t, v))

+ `(F (x, t, u(x, t))− F (x, t, v))− ε(L+ 1)e(L+1)(t−t0)

≤ L‖u(x, t)− v‖ − ε(L+ 1)e(L+1)(t−t0)

= −εe(L+1)(t−t0) < 0

which is a contradiction, since ‖u(x, t)− v‖ = εe(L+1)(t−t0).Therefore the inequality

f − εe(L+1)(t−t0) ≤ 0

holds true on M × [t0, t1] for any 0 < ε < ε0, and so f ≤ 0 on M × [t0, t1].Since t1 is an arbitrary time before the maximal time, we have f ≤ 0 for allt ≥ t0 in the interval of existence, and hence u(x, t) ∈ Ω for all x ∈ M andt ≥ t0 as desired.

As a first application of the maximum principle for vector bundles, weapply Theorem 6.15 in the simplest possible case to prove the following char-acterisation of the flow when a vector field preserves a convex set:

Corollary 6.16. Let Ω be a closed convex subset of a finite-dimensional vec-tor space V , and let F : [0, T ] × V → V be a smooth time-dependent vectorfield which points into Ω, so that F (t, v) ∈ TvΩ for all v ∈ ∂Ω and t ∈ [0, T ].Then the flow of F preserves Ω, in the sense that for any u0 ∈ Ω the solutionof the ode

d

dtu(t) = F (t, u(t))

u(0) = u0

has u(t) ∈ Ω for all t ≥ 0 in its interval of existence.

Proof. Let M to be the zero-dimensional manifold consisting of a single pointx. Let E the trivial bundle x × V with the trivial connection given bydifferentiation in the t direction, and metric given by a time-independentinner product on V . Also take Ω constant. Then ∆u = 0 since there are nospatial directions. Equation (6.2) reduces to the ordinary differential equationabove, and the maximum principle applies.

Remark 6.17. The converse also holds: If F does not point into Ω, then thereexists some v ∈ ∂Ω and ` ∈ NvΩ such that `(F (v)) > 0. But then thesolution of the ode with initial value v has d

dt`(u(t)) > 0 at t = 0. Therefore`(u(t)) > `(v) = s(`) = sup`(x) : x ∈ Ω for small t > 0, so u(t) /∈ Ω.

6.5 Applications of the Vector Bundle Maximum Principle 107

6.5 Applications of the Vector Bundle MaximumPrinciple

6.5.1 Maximum Principle for Symmetric 2-Tensors Revisited. Toillustrate the application of the vector bundle maximum principle, we showhow Theorem 6.15 implies the maximum principle for symmetric 2-tensorsgiven in Theorem 6.8. Here we take S to be the spatial tangent bundle definedin Section 5.3.1, with a time-dependent metric g and connection given byTheorem 5.9. We take E to be the bundle of symmetric 2-tensors over S, i.e.E = Sym2S∗, with the metric and connection induced from S, and considerevolution equation for A ∈ Γ (E) given in local coordinates by

∂

∂tAij = ∆Aij +∇VAij + Fij(A),

where V is a smooth section of S and F : E → E is a smooth section ofπ∗EE, i.e. for each α ∈ E(x,t), F (α) ∈ E(x,t). We note that since ∇t∂i = S(∂i),the equation can be rewritten as follows:

∇∂tAij =∂

∂tAij −A(S(∂i), ∂j)−A(∂i,S(∂j))

= ∆Aij +∇VAij + Fij(A)−A(S(∂i), ∂j)−A(∂i,S(∂j))

= ∆Aij +∇VAij + Fij(A).

Now let Ω ⊂ E be the set of positive definite symmetric 2-tensors acting onS. The set Ω is certainly convex in the fibres of E, since it is given as anintersection of half-spaces by

Ω(x,t) =⋂

v∈S(x,t): ‖v‖=1

α ∈ E(x,t) : `v(α) ≤ 0 (6.3)

where `v ∈ E∗(x,t) is defined by `v(α) = −α(v, v). This has the form of equa-

tion (B.1), where B = `v : v ∈ S, ‖v‖ = 1 and φ(`v) = 0 for each v.Note that Ω is invariant under parallel transport: Let γ be a smooth

curve in M , and let Ps be the parallel transport operator along γ fromγ(0) to γ(s). Suppose α ∈ Ωγ(0). Then Psα ∈ Eγ(s) is also in Ω, sincedds (Ps(α))(Psv, Psv) = (∇sPsα)(Psv, Psv) + 2(Psα)(∇sPsv, Psv) = 0, and so

(Ps(α))(v, v) = α(P−1s v, P−1

s v) ≥ 0

for any v ∈ Sγ(s).For the maximum principle to apply, we need the vector field F to point

into Ω. That is, we require that for any α ∈ ∂Ω, F (α) ∈ TαΩ. Since Ω isgiven in the form (6.3), Theorem B.7 implies that F (α) ∈ TαΩ if and onlyif `v(F (α)) ≤ 0 for any v ∈ S with ‖v‖ = 1 and `v(α) = 0. That is, if


α(v, v) = 0 and α is non-negative definite (so that v is a null eigenvector ofα) then we need F (v, v) ≥ 0. This reduces to the same condition for F , since

F (v, v) = F (v, v)− 2α(S(v), v) = F (v, v)

because v is a null-eigenvector of α. Thus the null-eigenvector condition ofDefinition 6.7 is precisely the condition that F points into the set Ω of weaklypositive definite symmetric bilinear forms. The tensor maximum principlenow follows.

6.5.2 Reaction-Diffusion Equation Applications. Reaction-diffusionsystems describe interaction and diffusion of a density or concentration —often typified by a chemical process. As we have see, the systems is charac-terised by the pde:

∂tq = κ∆q + F (q)

where q = q(x, t) represents the concentration of a substance, κ is the diffu-sion coefficient and F is the reaction term.

Example 6.18. One of the simplest examples is the chemical reaction:

X + Y 2Y

This reaction is one in which a molecule of species X interacts with a moleculeof species Y . The X molecule is converted into a Y molecule. The finalproduct consists of the original Y molecule plus the Y molecule created inthe reaction. It is know as an autocatalytic reaction as at least one of theproducts (in this case species Y ) is also a reactant.

Let x denote the concentration of X and y denote the concentration of Y .In a well stirred reaction the concentration can be modeled as satisfying therate reaction ode:

dx

dt= −k1xy + k2y

2

dy

dt= k1xy − k2y

2

where k1 and k2 are the respective forward and backwards reaction rates.The system is clearly in equilibrium when k1xy = k2y

2.If both chemicals are able to diffuse within the underlying medium the

(unstirred) reaction satisfies the reaction-diffusion pde:

∂x

∂t= κ1∆x− k1xy + k2y

2

∂y

∂t= κ2∆y + k1xy − k2y

2.


Example 6.19. The Lotka-Volterra equations (also known as the predator-prey equations) are used to describe the dynamics of biological systems inwhich two species interact, one a predator and one its prey. They were pro-posed independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926.The ode describing the interaction between x number of prey (i.e. rabbits)and y number of some predator (i.e. foxes) is given by:

dx

dt= x(α− βy)

dy

dt= −y(γ − δx)

where α represents the exponential growth of the prey, γ represents the decayof the predator and β, δ quantifies the interaction of the species (all arepositive). The fixed points of this system are (0, 0) and (γ/δ, α/β); the firstof which is a saddle point.

Our interest here is to show the corresponding reaction-diffusion equationsobeys the weak maximum principle discussed in the above. Crucially, thissystem has an integral of motion of the form

K = yαe−βyxγe−δx.

We observe K is log-concave, so the super-level sets are convex by the fol-lowing lemma:

Lemma 6.20. If f : C → R is a convex function on a convex domain C ⊂Rn, then the super-level sets Lλ = x : f(x) ≤ λ are convex.

Proof. For any points x1, x2 ∈ Lλ, one has f(x1) ≤ λ, f(x2) ≤ λ and x1, x2 ∈C. Consider the point x = θx1 + (1 − θ)x2, for any 0 < θ < 1. Clearlyx ∈ C by the convexity of C and since f is a convex function, f(x) ≤θf(x1)+(1−θ)f(x2) ≤ θλ+(1−θ)λ = λ. Thus x ∈ Lλ and so Lλ is a convexset.

This is easily seen Figure 6.1. The orbits are cyclic with four phases in thepositive quadrant. Since K is a constant of the motion, the flow of the vectorfield

F (x, y) = (αx− βxy,−γy + δxy). (6.4)

preserves the super-level sets of K. By Remark 6.17 the vector field F pointsinto Ω. Thus by the maximum principle, solutions u : M × [0, T ) → R2 ofthe reaction-diffusion equation ∂

∂tu = ∆u+F (u) which start in a super-levelset of K will remain so.

Remark 6.21. In the last example one could deduce the same result by ap-plying the scalar maximum principle to ψ u, where ψ is the convex functionψ = − logK: A direct computation gives


Fig. 6.1 Contourplots for Lotka-Volterramodel with fixed point(γ/δ, α/β). Note the levelsets are convex in theplane and the vector field(6.4) is tangent to them.

prey

predator

0 1 2 3

01

23

∂

∂t(ψ u) = ∆(ψ u)−D2ψ(∇u,∇u) +DF (u)ψ(u).

The fact that K is a constant of the motion implies that the last term van-ishes. The convexity of ψ makes the second term non-positive, so we canapply the maximum principle to show that the maximum of ψ u does notincrease.

This gives some intuition for why the convexity condition is important.The power of the vector bundle maximum principle, Theorem 6.15, is thatit can be applied to cases where the set Ω is not smooth or is not naturallydescribed as the sub-level set of a smooth convex function.

6.5.3 Applications to the Ricci Flow when n = 3. In three di-mensions we can define an isomorphism from S to Λ2(S) (up to sign in thenon-oriented case) by ι : ∂i 7→ 1

2εijkgjpgkq∂p ∧ ∂q, where εijk is the Levi-

Civita symbol, or elementary alternating 3-tensor (i.e. the volume form), sothat in an oriented orthonormal basis e1, e2, e3 we have εijk = 1 if (i j k)is a positive permutation of (1 2 3), −1 for a negative permutation, and zerootherwise. Using this we can rewrite the Riemann tensor as a symmetricbilinear form Λ ∈ Γ (S∗ ⊗S∗) by Λ(u, v) = R(ιu, ιv), yielding

Λij =1

4εiabεjcdRabcd. (6.5)

Thus we find that


Λ =

R2323 R1332 R1223

R2331 R1313 R2113

R3221 R3112 R1212

.

The geometric interpretation of Λ is that Λ(v, v) is ‖v‖2 times the sectionalcurvature of the 2-plane normal to v. We will now consider applications ofthe maximum principle to prove that the Ricci flow keeps Λ inside suitablesubsets of the bundle Sym2(S) of symmetric 2-tensors acting on the spatialtangent bundle S, if this is true initially.

6.5.3.1 Subsets of the Bundle of Symmetric Tensors. Let E be a vec-tor bundle of rank k over a manifold M , equipped with a metric g and acompatible connection ∇. Then the constructions in Sections 1.4 and 1.5provide metrics and compatible connections on each of the tensor bundlesT qp (E) =

⊗pE ⊗

⊗qE∗ constructed from E, so in particular on the bundle

of (2, 0)-tensors. By Example 1.64 the bundle Sym2(E) of symmetric 2-tensorsacting on E is a parallel subbundle of T 2

0 (E), and so inherits a metric andcompatible connection. We will discuss a natural construction which producessubsets of Sym2(E) which are automatically convex and invariant under par-allel transport.

The idea is as follows: Let K be a closed convex subset in the spaceSym2(Rk) of symmetric k × k matrices, which is invariant under the ac-tion of O(k) (or, if E is oriented, of SO(k)): That is, for O ∈ O(k) andT ∈ Sym2(Rk), define TO by

TO(u, v) = T (Ou,Ov)

for all u, v ∈ Rk. Then our requirement is that T ∈ K implies that TO ∈ K.Now define Ω ⊂ Sym2(E) as follows: T ∈ Sym2(Ex) ∈ Ω if and only if for

any orthonormal frame Y for Ex (that is, Y : (Rk, δ) → (Ex, gx) is a linearisometry), the element TY of Sym2(Rk) defined by TY (u, v) = T (Y (u), Y (v))lies in K.3 We note that this is independent of the choice of Y , since anyother frame is of the form Y O for some O ∈ O(k), and

TY O(u, v) = TY (Ou,Ov) = TOY (u, v),

and by assumption TOY ∈ K if TY ∈ K.

We can check that Ω is convex in the fibre: If T and S are in Ωx anda ∈ (0, 1), then for any frame Y , TY and SY are in K, so (aT + (1−a)S)Y =aTY + (1− a)SY ∈ K since K is convex, and so aT + (1− a)S ∈ Ωx.

Finally, we can check that Ω is invariant under parallel transport: Supposeσ is a smooth curve in M , and T is a parallel section of σ∗(Sym2(E)) with

3 This can be conveniently expressed in terms of the frame bundle machinery discussedin Section 5.2: As noted there, a symmetric 2-tensor gives rise to a function from theorthonormal frame bundle to Sym2(Rk). Our condition is simply that this functiontakes values in K.


T (0) ∈ Ωσ(0). Let Y be a parallel frame along σ, defined by σ∇∂s(Y (u)) = 0

for all u ∈ Rk (see Section 1.8.5). Then Y (s) is an orthonormal frame forEσ(s) for each s, since

∂

∂sg(Y (u), Y (v)) = g(σ∇∂s(Y (u)), Y (v)) + g(Y (u), σ∇∂sY (v)) = 0

for every u, v ∈ Rk, where we used the result from Proposition 1.58 thatshows the compatibility of the pullback connection and metric. But since T ,Y (u) and Y (v) are parallel, we have

∂

∂s(TY (s)(u, v)) =

∂

∂s(T (Y (u), Y (v))) = 0,

and so (T (s))Y (s) = (T (0))Y (0) ∈ K, so T (s) ∈ Ωσ(s) for every s, as required.Now let us look a little closer at the convex set K: A special property of

the space of symmetric k × k matrices is that they can always be diagonal-ised by an orthogonal transformation. That is, for each T ∈ Sym2(Rk) thereexists O ∈ O(k) such that TO is diagonal. Then by the invariance of K, T isin K if and only if the diagonal matrix TO is in K. Thus it is sufficient forus to consider the set KD given by the intersection of K with the space ofdiagonal matrices, which we identify with Rk. Since the diagonal matrices area vector subspace of Sym2(Rk), KD is convex. Furthermore, KD is invariantunder those orthogonal matrices which send diagonal matrices to diagonalmatrices, which are exactly the permutation matrices which act by inter-changing basis elements of Rk. That is, K determines a convex set KD ⊂ Rkwhich in symmetric, in the sense that it is invariant under permutation of thecoordinates. In fact the converse is also true: If KD is any symmetric convexset in Rk, then the space K of symmetric matrices having eigenvalues in KD

is convex and O(k)-invariant, and so our construction above gives a subset ofSym2(E) which is convex in the fibre and invariant under parallel transport.

6.5.3.2 Checking that the Vector Field Points into the Set. We now applythe above construction taking E to be the spatial tangent bundle S over themanifold M × I, for an interval I ⊂ R. This is a vector bundle of rank 3, sofor any symmetric convex set KD in R3 we produce a suitable parallel convexset Ω ⊂ Sym2(S). We want to use the maximum principle to show that ifΛ(x, 0) ∈ Ω(x,0) for all x ∈M , then Λ(x,t) ∈ Ω(x,t) for all x ∈M and all t ≥ 0in I. There is one more condition we need to check: The vector field arisingin the reaction-diffusion equation for Λ must point into Ω.

By Theorem 5.21 the pde system for R is of the form

∇∂tRijk` = ∆Rijk` + 2Q(R)ijkl,

where Q(R)ijkl = Bijk` − Bij`k + Bikj` − Bi`jk. The fact that the metric isparallel implies also that the alternating tensor ε is parallel, so Λ satisfies thereaction-diffusion equation


∇∂tΛij = ∆Λij + Q(Λ)ij .

A direct computation shows that in a basis Yi for which ΛY is diagonal,the reaction term (Q(Λ))Y is also diagonal, and is given as follows:4

Q

x 0 00 y 00 0 z

= 2

x2 + yz 0 00 y2 + xz 00 0 z2 + xy

.In particular, the flow of this vector field keeps stays inside the subspace ofdiagonal matrices (with respect to the given basis), and so the flow staysinside Ω provided the flow of ΛY by the vector field QY stays inside K. Thishappens precisely when the vector field

V (x, y, z) = (x2 + yz, y2 + xz, z2 + xy)

on R3 points into the symmetric convex set KD ⊂ R3.We now check this condition for several concrete examples:

1. The cone of positive sectional curvature operators: Here

KD = (x, y, z) ∈ R3 : x, y, z ≥ 0.

This is convex since it is an intersection of three half-spaces: We can writeit in the form

KD =⋂

i=1,2,3

(x, y, z) : ì(x, y, z) ≤ 0 , (6.6)

where `1 = (−1, 0, 0), `2 = (0,−1, 0) and `3 = (0, 0,−1). We must checkthat V is in the tangent cone at any boundary point, which by TheoremB.7 in Appendix B amounts to showing that if ì(x, y, z) = 0 for some(x, y, z) ∈ KD, then ì(V (x, y, z)) ≤ 0. By symmetry we need only provethis for i = 1, in which case we have x = 0, y, z ≥ 0, and

`1(V (x, y, z)) = −x2 − yz = −yz ≤ 0

as required.2. The cone of positive Ricci curvature operators: Here

KD =⋂P∈S3

(x, y, z) : `(P (x, y, z)) ≤ 0 ,

where S3 is the set of permutations of three objects, and `(x, y, z) =−X · (1, 1, 0). By symmetry it suffices to check `(V ) ≤ 0 at a point with`(x, y, z) = x+ y = 0 and `(P (x, y, z)) ≤ 0 for all P : This is true, since

4 See Example 11.14 in Chapter 11 for the derivation of this.


Fig. 6.2 Radial pro-jection of the vectorfield V onto the planex + y + z = 1. The linesshown are lines of vanish-ing sectional curvatures,and the shaded regioncorresponds to the cone ofpositive Ricci curvature.The inner triangle corres-ponds to the cone of pos-itive sectional curvature.

`(V ) = −x2 − yz − y2 − xz = −x2 − y2 − z(x+ y) = −x2 − y2 ≤ 0.

3. Cones of pinched Ricci curvatures: Here we fix ε ∈ (0, 2) and define

KD =⋂P∈S3

(x, y, z) : `(P (x, y, z)) ≤ 0

where `(x, y, z) = εz − x − y. Again by symmetry we need only prove`(V (x, y, z)) ≤ 0 at any point with εz − x − y = `(x, y, z) = 0 and`(P (x, y, z)) ≤ 0 for all P ∈ S3:

`(V ) = −x2 − yz − y2 − xz + εz2 + εxy

= −x2 − y2 + εxy − z(x+ y − εz)= −x2 − y2 + εxy

≤ 0

since ε ≤ 2. Notice that the inequality is strict for ε < 2, unless (x, y, z) =(0, 0, 0).

4. Pinching sets: The fact that the vector field points strictly into the conesof pinched Ricci curvature means that we can find a preserved set whichgets near the constant curvature line when the curvature is large. Let usconsider sets defined by

KD =⋂P∈S3

(x, y, z) : ϕ(P (x, y, z)) ≤ 0 ,

where ϕ(x, y, z) = z − f(x+y

2

)and f is an increasing concave differenti-

able function on [0,∞) with f(0) = 0. Since f is concave, ϕ is convex, sothe sub-level set ϕ ≤ 0 is convex, and hence KD is an intersection of


Fig. 6.3 Radial pro-jection of the vectorfield V onto the planex+ y+ z = 1, showing theregions corresponding topinched Ricci curvaturecones. Note that the vec-tor field points into eachof these sets.

convex sets, hence convex. We can write this as an intersection of half-spaces as follows: Since a concave function always lies below its tangentline, we can write

f(a) = inff(b) + f ′(b)(a− b) : b ≥ 0,

and hence

KD =⋂

P∈S3,b≥0

(x, y, z) : `b(P (x, y, z)) ≤ f(b) ,

where `b(x, y, z) = z − f ′(b)(x+y2 − b). This gives KD as an intersection

of half-spaces in the form used in Section B.5 of Appendix B. We need toprove that if `b(x, y, z) = f(b) for some b ≥ 0, and `b′(P (x, y, z)) ≤ f(b′)for all b′ and P , then `b(V (x, y, z)) ≤ 0. The conditions imply that `b =` x+y

2and z = f

(x+y

2

), so we have

`b(V (x, y, z)) = z2 + xy − f ′

2(x2 + yz + y2 + xz)

= f2 +(x+ y

2

)2

−(x− y

2

)2

− f ′((x+ y

2

)2

+(x− y

2

)2

+(x+ y

2

)f).

Thus we require that the right-hand side be non-positive for all suchpoints. Since we are assuming that f ′ ≥ 0, we observe that for a fixedvalue of ξ = x+y

2 , the right-hand side is maximised by choosing x = y, sofor the inequality to hold it is necessary and sufficient that

f(ξ)2 + ξ2 − f ′(ξ)(ξ2 + ξf(ξ)

)≤ 0.


This differential equation has the solutions (f − ξ)2 = Cξe−f/ξ for arbit-rary C > 0. One can verify directly that the solutions are concave andincreasing and have f(0) = 0, and satisfy f(ξ) ∼ ξ − C

√ξ as ξ →∞.

Fig. 6.4 Graphs of f fora family of pinching setsdefined by z ≤ f

(x+y

2

).

The diagonal correspondsto the constant curvatureline z = (x+y)/2, and theentire family of pinchingsets fills out the cone ofpositive Ricci curvature.

Any metric with strictly positive Ricci curvature on a compact 3-manifoldhas principal sectional curvatures lying in one of the pinching sets of the lastexample for some value of C. In particular this implies that the sectionalcurvatures have ratios approaching 1 wherever the curvature becomes large.A manifold which has sectional curvatures equal at each point is necessar-ily a constant curvature space (see Schur’s theorem in Section 11.3.1). Wewould like to conclude that the metric on the manifold approaches a constantcurvature metric. However to conclude this we must first establish two things:We need to know that the solution of Ricci flow continues to exist until thecurvature ‘blows up’ somewhere, and we need to develop some machinery ofcompactness for Riemannian metrics in order to produce a limiting metricto which Schur’s theorem can be applied. We will address these points in thenext two chapters.

Chapter 7

Regularity and Long-Time Existence

In Chapters 3 and 5 we saw that the curvature under Ricci flow obeys aparabolic equation with quadratic nonlinearity. By appealing to this view,we would expect the same kind of regularity that is seen in parabolic equa-tions to apply to the curvature. In particular we want to show that boundson curvature automatically induce a priori bounds on all derivatives of thecurvature for positive times. In the literature these are known as Bernstein-Bando-Shi derivative estimates as they follow the strategy and techniquesintroduced by Bernstein (done in the early 20th century) for proving gradi-ent bounds via the maximum principle and were derived for the Ricci flowin [Ban87] and comprehensively by Shi in [Shi89]. Here we will only needthe global derivative of curvature estimates (for various local estimates see[CCG+08, Chap. 14]). In the second section we use these bounds to provelong-time existence.

7.1 Regularity: The Global Shi Estimates

We seek to prove the following global derivative estimates — which says thatfor short times there is an estimate which depends on the initial bound, andfor large times there is a bound independent of initial bounds. As we shall see,the short-time estimate follows by applying the maximum principle directlyto equation (7.3).

Theorem 7.1 (Bernstein-Bando-Shi Estimate). Let g(t), t ∈ [0, T ), bea smooth solution of the Ricci flow on a compact manifold Mn. Then foreach p ∈ N there exists a constant Cp depending only on n such that if|R(x, t)|g(t) ≤ K for all x ∈M and t ∈ (0, T ) then

|∇pR(x, t)|g(t) ≤ min

supM×0

|∇pR(x, 0)|g(0) eCpKt, Cp max

K

p+22 ,

K

tp2

for all x ∈M and t ∈ (0, T ).

117

118 7 Regularity and Long-Time Existence

Remark 7.2. To avoid a notational quagmire in the proof, we adopt the fol-lowing convention: If A and B are two tensorial quantities on a Riemannianmanifold, we let A ∗ B be any linear combination of tensors obtained fromthe tensor product A⊗B by one or more of these operations:

1. summation over pairs of matching upper and lower indices;2. contraction on upper indices with respect to the metric;3. contraction on lower indices with respect to the dual metric.

Lemma 7.3. Suppose, under the Ricci flow, that a tensor A satisfies

∇∂tA = ∆A+B,

where B is a tensor of the same type as A, and ∇ is the connection on thespatial tangent bundle S defined in Theorem 5.9. Then the square of its normsatisfies the heat-type equation

∂

∂t|A|2 = ∆|A|2 − 2|∇A|2 + 2 〈B,A〉 .

Proof. Using the identity ∆|A|2 = 2 〈∆A,A〉+ 2|∇A|2, and noting that ∇ iscompatible with the metric, we find that

∂

∂t〈A,A〉 = 2

⟨ ∂∂tA,A

⟩= ∆|A|2 − 2|∇A|2 + 2〈B,A〉.

Moreover, using the formulae

[∇t,∇i]A = R(∇i,∇t)A = ∇R ∗A

from Theorem 5.18, and

[∇, ∆]A = ∇∆A−∆∇A = R ∗ ∇A+∇R ∗A,

we see that the derivative ∇A also satisfies a heat-type equation, of the form

∇t∇A = ∆(∇A) +∇B +R ∗ ∇A+∇R ∗A. (7.1)

Proof (Theorem 7.1, sketch only). The proof follows by induction on p, firststarting with the case p = 1. Using our ∗-convention, the evolution equationfor R, derived in Theorem 3.14, takes the form

∂

∂tR = ∆R+R ∗R. (7.2)

So by Lemma 7.3, with A = R and B = R ∗R, we find that

∂

∂t|R|2 = ∆|R|2 − 2|∇R|2 +R ∗R ∗R.

7.1 Regularity: The Global Shi Estimates 119

Moreover, by using Lemma 7.3 again, now with A = ∇R and B = R ∗ ∇R,it is also possible to show (in conjunction with (7.1)) that

∂

∂t|∇R|2 = ∆|∇R|2 − 2|∇2R|2 +R ∗ ∇R ∗ ∇R. (7.3)

We would now like to estimate |∇R|2 from these equations. However, thereare one or two difficulties. The first is the potentially bad term R∗(∇R)∗2 onthe right-hand side of (7.3), the second is that we have no control on |∇R|2at t = 0. To clear these obstacles, we define

F := t|∇R|2 + |R|2. (7.4)

With this choice we find that the uncontrollable |∇R|2 vanishes at t = 0, soan initial upper bound F |t=0 ≤ K2 is obtained by hypothesis, and when t issmall the bad term from differentiating |∇R|2 can be controlled by the goodterm −2|∇R|2 from differentiating |R|2: We have

∂F

∂t≤ ∆F + (t C1K − 1)|∇R|2 + C2K

3.

Now by applying the maximum principle on 0 < t ≤ 1C1K

we get

t|∇R|2 ≤ K2 + t C2K3 ≤

(1 +

C2

C1

)K2.

If t > 1C1K

, the same argument on the interval [t− 1C1K

, t] gives

1

C1K|∇R|2 ≤

(1 +

C2

C1

)K2.

So by combining these two results we find that

|∇R|2 ≤ max(1 + C2

C1

)K2

t, (C1 + C2)K3

.

Thus proving the case p = 1.To get the necessary results for higher derivatives, we bootstrap the p = 1

case. Using equation (7.1) one can prove by induction that

∂

∂t∇kR = ∆∇kR+

k∑j=0

∇jR ∗ ∇k−jR.

It follows from Lemma 7.3 that

∂

∂t|∇kR|2 = ∆|∇kR|2 − 2|∇k+1R|2 +

k∑j=0

∇jR ∗ ∇k−jR ∗ ∇kR. (7.5)


Now by taking

G = tp|∇pR|2 + βp

p∑k=1

αptp−k|∇p−kR|2,

we look to apply the same pde techniques as before with appropriately chosenconstants βp. Indeed, on [0, 1

CK ] one can show that

∂

∂tG ≤ ∆G+BpK

3.

By the maximum principle we find Cp depending on p and n such that

supxG(x, t) ≤ C2

pK2.

If t > 1CK , the same argument on [t− 1

CK , t] gives

|∇pR|(t) ≤ (C(p)K)Kp/2 = C(p)K1+p/2.

7.2 Long-Time Existence

Suppose (M, g(t)), t ∈ [0, T ), is a solution of the Ricci flow. We say that [0, T )is the maximum time interval of existence if either T = ∞, or that T < ∞and there does not exist ε > 0 and a smooth solution g(t), t ∈ [0, T + ε), ofthe Ricci flow such that g(t) = g(t) for t ∈ [0, T ). In the latter case, when Tis finite, we say g(t) forms a singularity at time T or simply g(t), t ∈ [0, T ),is a singular solution. Henceforth, with this terminology, we will talk aboutthe Ricci flow with initial metric g0 on a maximal time interval [0, T ).

In this section we are interested in proving long-time existence for the Ricciflow with a particular interest in the limiting behaviour of singular solutions.

Theorem 7.4 (Long-time Existence). On a compact manifold M withsmooth initial metric g0, the unique solution g(t) of Ricci flow with g(0) = g0

exists on a maximal time interval 0 ≤ t < T ≤ ∞. Moreover, T <∞ only if

limtT

supx∈M|R(x, t)| =∞.

In order to prove this, we need bounds on the metric and its derivatives.

Lemma 7.5 ([Ham82b, Lemma 14.2]). Let g(t), t ∈ [0, τ ] be a solution tothe Ricci flow for τ ≤ ∞. If |Ric| ≤ K on M × [0, τ ] then

e−2Ktg(0) ≤ g(t) ≤ e2Ktg(0)

for all t ∈ [0, τ ].1

1 We say g2 ≥ g1 if g2 − g1 is weakly positive definite.

7.2 Long-Time Existence 121

Proof. For any nonzero v ∈ TxM , ∂∂tg(x,t)(v, v) = −2Ric(x,t)(v, v) so that∣∣∣∣ ∂∂tg(x,t)(v, v)

∣∣∣∣g(t)

≤ 2|Ric| g(x,t)(v, v).

Hence for times 0 ≤ t1 ≤ t2 ≤ τ , we find that∣∣∣∣logg(x,t2)(v, v)

g(x,t1)(v, v)

∣∣∣∣ =

∣∣∣∣∣ˆ t2

t1

∂∂tg(x,t)(v, v)

g(x,t)(v, v)dt

∣∣∣∣∣≤ˆ t2

t1

∣∣∣∣∣ ∂∂tg(x,t)(v, v)

g(x,t)(v, v)

∣∣∣∣∣g(t)

dt

≤ 2|Ric| (t2 − t1). (7.6)

Therefore by letting t1 = 0 and t2 = t we have∣∣∣∣logg(x,t)(v, v)

g(x,0)(v, v)

∣∣∣∣ ≤ 2Kt.

So the results follows by exponentiation.

Lemma 7.6. Fix a metric g and connection ∇ on M . For any smooth solu-tion g(t) of the Ricci flow on M× [0, T ), for which supM×[0,T ) |R| ≤ K, thereexist constants Cq for each q ∈ N such that

supM×[0,T )

∣∣∣∇(q)g(x, t)∣∣∣g≤ Cq.

Proof. It suffices to prove by induction that:

∂

∂t∇qg =

∑j0+j1+···+jm=q

∇j0R ∗ ∇j1g ∗ · · · ∗ ∇jmg. (7.7)

The case q = 1 is proved as follows: Since ∇ is independent of time,

∂

∂t∇igkl = ∇i∂tgkl

= ∇i (−2Rickl)

= −2∇iRickl − 2Ricpl(Γ − Γ

)ikp − 2Rickp

(Γ − Γ

)ilp

= ∇R+R ∗ ∇g,

since (Γ − Γ )ikp = 1

2gpr(∇igkr + ∇kgir − ∇rgik

).

The induction step is now straightforward, for if (7.7) holds for some q,then


∂

∂t∇q+1g = ∇

( ∂∂t∇qg

)= ∇

( ∑j0+j1+···+jm=q

∇j0R ∗ ∇j1g ∗ · · · ∗ ∇jmg)

=∑

j0+j1+···+jm=q

(∇j0+1R+ ∇j0R ∗ ∇g

)∗ ∇j1g ∗ · · · ∗ ∇jmg

+∑

j0+j1+···+jm=q

∇j0R ∗(∇j1+1g ∗ · · · ∗ ∇jmg

+ · · ·+ ∇j1g ∗ · · · ∗ ∇jm+1g)

=∑

j0+j1+···+jm=q+1

∇j0R ∗ ∇j1g ∗ · · · ∗ ∇jmg.

We now can use (7.7) to deduce the bounds needed for the Lemma by in-duction on q. That is, if we have controlled ∇-derivatives of g up to order q,then we have

∂

∂t

∣∣∇(q+1)g∣∣2g≤ C

(1 +

∣∣∇(q+1)g∣∣g

),

which implies a bound on the order q + 1 derivatives since the time intervalis finite.

Remark 7.7. The derivatives in the t direction are also controlled, since theyare related to the spatial derivatives by the Ricci flow equation.

Proof (Theorem 7.4). By taking the contrapositive, suppose there exists asequence ti T and a constant K <∞ independent of i such that

supM|R(·, ti)| ≤ K.

In particular, by the doubling time estimate (6.1) we have that |R(x, t)| ≤K/(1 − CK(t − ti)), for t ≥ ti and all x ∈ M . So for large i (i.e. whenti = T − ε for some 0 < ε 1) we have

supM|R(·, t)| ≤ K

1− CK(t− (T − ε))

for t ∈ [T − ε, T ]. Hence |R| ≤ K on [0, T ). We claim that:

Claim. The metric g(t) may be extended smoothly from [0, T ) to [0, T ].

Proof of Claim. By Lemma 7.5 and equation (7.6) we find that:∣∣∣∣logg(x,t2)(v, v)

g(x,t1)(v, v)

∣∣∣∣ ≤ 2K|t2 − t1|.

So g(x,t)(v, v) is Cauchy at t → T . By Lemma 7.6 it follows that g(x, t) is

Cauchy in Ck, for every k, as t→ T .

7.2 Long-Time Existence 123

So from this result, we take g(T ) as the ‘initial’ metric. By Theorem 4.2,the short-time existence implies that we can extend the flow for times t ∈[0, T + ε), thus contradicting the maximality of the finite final time T .

Chapter 8

The Compactness Theorem forRiemannian Manifolds

The compactness theorem for the Ricci flow tells us that any sequence ofcomplete solutions to the Ricci flow, having uniformly bounded curvatureand injectivity radii uniformly bounded from below, contains a convergentsubsequence. This result has its roots in the convergence theory developed byCheeger and Gromov. In many contexts where the latter theory is applied,the regularity is a crucial issue. By contrast, the proof of the compactnesstheorem for the Ricci flow is greatly aided by the the fact that a sequence ofsolutions to the Ricci flow enjoy excellent regularity properties (which werediscussed in the previous chapter). Indeed, it is precisely because bounds onthe curvature of a solution to the Ricci flow imply bounds on all derivativesof the curvature that the compactness theorem produces C∞-convergence oncompact sets.

The compactness result has natural applications in the analysis of sin-gularities of the Ricci flow by ‘blow-up’, discussed here in Section 8.5: Theidea is to consider shorter and shorter time intervals leading up to a sin-gularity of the Ricci flow, and to rescale the solution on each of these timeintervals to obtain solutions on long time intervals with uniformly boundedcurvature. The limiting solution obtained from these gives information aboutthe structure of the singularity.

As a remark concerning notation in this chapter, quantities dependingon the metric gk or gk(t) will have a subscript k. For instance ∇k and Rkdenote the Riemannian connection and Riemannian curvature tensor of gk.Quantities without a subscript will depend on the background metric g.

8.1 Different Notions of Convergence

In order to establish the C∞-convergence of a sequence of metrics (gk) uni-formly on compact sets, we need to recall some of the different ways a se-quence of functions can converge.

125

126 8 The Compactness Theorem for Riemannian Manifolds

8.1.1 Convergence of Continuous Functions. Consider the BanachSpace C(X) where X is a topological space (typically taken to be compactHausdorff). Recall that sequence (fn) ⊂ C(X) converges uniformly to alimiting function f if in heuristic terms ‘the rate of convergence of fn(x) tof(x) is independent of x’. Formally we say fn → f uniformly if for everyε > 0 there exists N = N(ε) > 0 such that for all x ∈ X, |fn(x)− f(x)| < εwhenever n ≥ N . A generalisation of this is the notion of uniform convergenceon compact sets.1 In this case, a sequence (fn) converges to f uniformly oncompact subsets of X (in short we just say fn → f compactly) if for everycompact K ⊂ X and for every ε > 0 there exists N = N(K, ε) > 0 suchthat for all x ∈ K, |fn(x)− f(x)| < ε whenever n > N . Intuitively, uniformconvergence on compact sets says that every point has a neighbourhood inwhich the convergence is uniform.

To illustrate the different types of convergence, consider C(0, 1) withfn(x) = xn. In this situation fn converges compactly, but not uniformly,to the zero function. However if we change the space to C(0, 1], then fn con-verges pointwise, but not uniformly on compact subsets, to the function thatis zero on (0, 1) and one at x = 1.

In general it is easy to see that if (fn) ⊂ C(X) and fn → f uniformly thenfn → f compactly. The above example shows the converse is false; howeverwe do have the following partial results:

- If X is compact then fn → f compactly implies fn → f uniformly.- If X is locally compact2 and fn → f compactly then the limit f is

continuous.

As our metrics gk are smooth, we need to take into consideration thedifferentiability aspect of our convergence.

8.1.2 The Space of C∞-Functions and the Cp-Norm. Let Ω ⊂ Rnbe an open set with compact closure, and let Cp(Ω) be the set of functionswith continuous derivatives up to order p. Moreover, let the space Cp(Ω) bethe set of functions on Ω which extend to a Cp function on some open setcontaining Ω. It is equipped with the following norm:

‖u‖Cp = sup0≤|α|≤p

supx∈Ω|Dαu(x)|,

where Dα is the derivative corresponding to the multi-index α. It can beshown that the norm is complete, thus making the space (Cp(Ω), ‖ · ‖Cp) aBanach space. However, if we consider the space of smooth functions

C∞(Ω) =

∞⋂p=0

Cp(Ω),

1 Also known as compact convergence or the topology of compact convergence.2 That is, if every point has a neighbourhood whose closure is compact.

8.1 Different Notions of Convergence 127

one notes that it is only a metric space, not a Banach space, with metric

d(f, g) =

∞∑p=0

1

2p‖f − g‖Cp

1 + ‖f − g‖Cp.

A sequence converges in C∞ if and only if it converges in Cp for every p.

8.1.3 Convergence of a Sequence of Sections of a Bundle. We willbe interested in convergence of a sequence of metrics, which are of coursesections of a certain vector bundle. To make sense of this we define whatis meant by Cp convergence or C∞ convergence for sequences of sections ofvector bundles.

Definition 8.1 (Cp-Convergence). Let E be a vector bundle over a man-ifold M , and let metrics g and connections ∇ be given on E and on TM .Let Ω ⊂ M be an open set with compact closure Ω in M , and let (ξk) be asequence of sections of E. For any p ≥ 0 we say that ξk converges in Cp toξ∞ ∈ Γ (E

∣∣Ω

) if for every ε > 0 there exists k0 = k0(ε) such that

sup0≤α≤p

supx∈Ω|∇α(ξk − ξ∞)|g < ε

whenever k > k0. We say ξk converges in C∞ to ξ∞ on Ω if ξk converges inCp to ξ∞ on Ω for every p ∈ N.

Note that since we are working on a compact set, the choice of metric andconnection on E and TM have no affect on the convergence.

Next we define smooth convergence on compact subsets for a sequence ofsections: To do this we require an exhaustion of M — that is, a sequence ofopen sets (Uk) in M such that Uk is compact and Uk ⊂ Uk+1 for all k, and⋃k≥1 Uk = M . Note that if K ⊂ M is compact, then there exists k0 such

that K ⊂ Uk for all k ≥ k0 (in particular if M is compact then Uk = M forall large k).

Definition 8.2 (C∞-Convergence on Compact Sets). Let (Uk) be anexhaustion of a smooth manifold M , and E a vector bundle over M . Fixmetrics g and connections ∇ on E and TM . Let (ξi) be a sequence of sectionsof E defined on open sets Ai ⊂ M , and let ξ∞ ∈ Γ (E). We say ξi convergessmoothly on compact sets to ξ∞ if for every k ∈ N there exists i0 such thatUk ⊂ Ai for all i ≥ i0, and the sequence (ξi

∣∣Uk

)i≥i0 converges in C∞ to ξ∞

on Uk.

Again, we remark that this notion of convergence does not depend on thechoice of the metric and connection on E and TM .


8.2 Cheeger-Gromov Convergence

We are interested in analysing the convergence of a sequence of Riemannianmanifolds arising from dilations about particular singularities. In order to doso, we need to build into our definitions which part of the manifold we areinterested in. This is done by including a base point into the definition. Wewill see later some examples where different choices for a sequence of basepoints on the same sequence of Riemannian manifolds can give quite differentlimits.3

Definition 8.3 (Pointed Manifolds). A pointed Riemannian manifold is aRiemannian manifold andO ∈M together with a choice of origin or basepointO ∈ M . If the metric g is complete, we say that the tuple is a completepointed Riemannian manifold. If (M, g(t)) is a solution to the Ricci flow, wesay (M, g(t), O), for t ∈ (a, b), is a pointed solution to the Ricci flow.

Moreover, we would like a notion of convergence for a sequence (Mk, gk, Ok)of pointed Riemannian manifolds that takes into account the action ofbasepoint-preserving diffeomorphisms on the space of metrics Met.

Definition 8.4 (Cheeger-Gromov Convergence in C∞). A sequence(Mk, gk, Ok) of complete pointed Riemannian manifolds converges to acomplete pointed Riemannian manifold (M∞, g∞, O∞) if there exists:

1. An exhaustion (Uk) of M∞ with O∞ ∈ Uk;2. A sequence of diffeomorphisms Φk : Uk → Vk ⊂Mk with Φ(O∞) = Ok

such that(Φ∗kgk

)converges in C∞ to g∞ on compact sets in M∞.

This notion of convergence is often referred to as smooth Cheeger-Gromovconvergence. The corresponding convergence for a sequence of pointed solu-tions to the Ricci flow is as follows.

Definition 8.5. A sequence (Mnk , gk(t), Ok), for t ∈ (a, b), of complete

pointed solutions to the Ricci flow converges to a complete pointed solutionto the Ricci flow (M∞, g∞(t), O∞), for t ∈ (a, b), if there exists:

1. An exhaustion (Uk) of M∞ with O∞ ∈ Uk;2. A sequence of diffeomorphisms Φk : Uk → Vk ⊂Mk with Φ(O∞) = Ok

such that(Φ∗kgk(t)

)converges in C∞ to g∞(t) on compact sets in M∞×(a, b).

8.2.1 Expanding Sphere Example. An explicit illustration of theCheeger-Gromov convergence can be seen in the following example whichshows: A sequence of pointed manifolds (Snk , gcan, N), where Snk is the stand-ard n-sphere of radius k (taken with the usual canonical metric gcan) andN is the basepoint corresponding to the sphere’s ‘north pole’, has a limitingpointed manifold (Rn, δij , 0) centred at the origin 0.

3 The definitions here are essentially due to Hamilton [Ham95a], but following[CCG+07] we do not include in the definition a choice of orthonormal frame at thebasepoint.

8.2 Cheeger-Gromov Convergence 129

Fig. 8.1 Cheeger-Gromov convergence ofexpanding spheres to aflat metric.

Rn

R

Bk(0) x

To see this, first take an exhaustion Uk = Bk(0) Rn of balls centeredat the origin with a sequence of diffeomorphisms Φk : Bk(0) → Snk ⊂ Rn+1,defined by

Φk : x 7→ (x,√k2 − |x|2),

with Vk = Φk(Bk(0)). For such Φk, we are able to explicitly compute thepullback metric:

(Φ∗kgk)ij =

⟨∂Φk∂xi

,∂Φk∂xj

⟩Rn+1

= δij +xixj

k2 − |x|2,

since ∂Φk∂xi = (ei, xi/

√k2 − |x|2). Thus, it suffices to show that(Bk(0), δij +

xixj

k2 − |x|2

)C∞−→ (Rn, δij)

uniformly on compact sets, which can be achieved simply by showing that1/(1− |x/k|2) converges to 1 in C∞ on compact subsets of Rn.

8.2.2 The Rosenau Metrics. We will illustrate various aspects ofCheeger-Gromov convergence by investigating the following specific family:4

For each α ∈ [0, 1), define a metric gα on the sphere S2 ⊂ R3 by

gα =1

1− α2x2g,

where g is the standard metric on S2 coming from its inclusion in R3, and x isone of the coordinate functions on R3. When α = 0 this is the standard metricon S2, but as α → 1 the manifold becomes longer in the x direction, andapproaches a long cylinder with capped-off ends. We will make this precisebelow, by producing the cylinder as a limit in the Cheeger-Gromov sense

4 Up to rescaling by a factor depending on k, these are exactly the metrics arising inan important explicit solution of the Ricci flow on S2 known as the Rosenau solution.


when we keep the base-points away from the ends x = ±1, and the capped-off cylinder as a limit when the base-points are at the ends.

Fig. 8.2 The Rosenaumetrics gα = 1

1−α2x2 g

with base points awayfrom the ends, convergingto a cylinder as α in-creases from 0 towards 1(shown embedded in R3).

Consider the convergence of the metrics gk = gα(k) with limk→∞ α(k) = 1,with base points chosen away from the poles, such as Ok = (0, 1, 0) for everyk. We will prove that the sequence (S2, gk, Ok) converges in the Cheeger-Gromov sense to a flat cylinder (R× (R/(2π)), du2 + dv2, (0, 0)). Choose

Φk(u, v) = Φ(u, v) =(sinhu, cos v, sin v)

coshu,

so that Φk(0, 0) = (0, 1, 0) for every k.5 Then we compute

Φ∗(∂u) =(1,− cos v sinhu,− sin v sinhu)

cosh2 u; Φ∗(∂y) =

(0,− sin v, cos v)

coshu.

So

Φ∗gk(∂u, ∂u) =1

1− α2 tanh2 ug(Φ∗∂u, Φ∗∂u)

=1

1− α2 tanh2 u

1

cosh2 u

= 1− (1− α2) tanh2 u

1− α2 tanh2 u;

Φ∗gk(∂v, ∂v) =1

1− α2 tanh2 ug(Φ∗∂v, Φ∗∂v)

= 1− (1− α2) tanh2 u

1− α2 tanh2 u;

5 This is a conformal map from the cylinder to the sphere without its poles. It canbe produced by composing the complex exponential map (which maps R× (R/2π) toC \ 0) with the stereographic projection from C to S2.

8.2 Cheeger-Gromov Convergence 131

and

Φ∗gk(∂u, ∂v) =1

1− α2 tanh2 ug(Φ∗∂u, Φ∗∂v)

= 0.

This shows that

Φ∗gk =

(1− (1− α2) tanh2 u

1− α2 tanh2 u

)(du2 + dv2),

and the quantity in the brackets converges in C∞ to 1 on any compact subsetof R × (R/(2π)). Thus Φ∗gk → du2 + dv2, so we have proved the Cheeger-Gromov convergence to a the cylinder metric.

Now let us see what happens when we take the base points to be at a pole,say Ok = (1, 0, 0) for every k. We will prove that the sequence of pointedmetrics

(S2, gk, Ok) −→(R2,

du2 + dv2

1 + u2 + v2, (0, 0)

)converge in the Cheeger-Gromov sense. To do this choose Φk to be a com-position of stereographic projection with a suitable dilation depending onk:

Φk(u, v) =1

1 + 1−α2

2 (u2 + v2)

(1− 1− α2

2(u2 + v2),

√1− α2u,

√1− α2v

).

One can then compute directly

(Φk)∗gk =du2 + dv2

1 + u2 + v2 − 1−α2

2 (u2 + v2) + (1−α2)2

16 (u2 + v2)2

→ du2 + dv2

1 + u2 + v2

as α→ 1 smoothly on compact subsets.

Remark 8.6. This example illustrates several features of Cheeger-Gromovconvergence: First, the need for specifying the base points is clear here, sincewe produce very different limits for different choices of base-point. Second,the limit of a sequence of compact spaces can be non-compact. Note that thispossibility is made possible since we use an exhaustion by open sets on thelimit manifold, and say nothing about surjectivity of the diffeomorphisms Φkonto the spaces Mk. On the other hand one cannot produce a compact limitfrom a sequence of complete non-compact spaces.

Remark 8.7. The limiting metric above is also important in Ricci flow: Itis an example of a ‘soliton’ metric, which evolves without changing shape— more precisely, the solution of Ricci flow starting from the above metric


Fig. 8.3 The Rosenaumetrics gα = 1

1−α2x2 g

with base points nearan end, converging to the

cigar metric du2+dv2

1+u2+v2 as α

increases from 0 towards 1(shown embedded in R3).

is given at any positive time by the same metric pulled back by a suitablediffeomorphism. This metric is called the ‘cigar’, and it is asymptotic to acylinder at infinity.

8.3 Statement of the Compactness Theorem

Having established the convergence criterion for a sequence of pointedRiemannian manifolds, we seek to find sufficient conditions under which agiven sequence (Mk, gk, Ok) has a convergent subsequence. As we shallsee, it turns out that there are two such conditions needed for this to oc-cur: The first is that of (globally) uniform bounds on the curvature and itshigher derivatives; the second is a lower bound on the injectivity radius atthe basepoint:

(i) |∇pR(gk)| ≤ Cp on Mk for each p ≥ 0; and(ii) injgk(Ok) ≥ κ0 for some κ0 > 0.

8.3 Statement of the Compactness Theorem 133

We say that the sequence of Riemannian manifolds has (uniformly) boundedgeometry whenever the curvature and its derivatives of all orders have uniformbounds (i.e. condition (i) holds).

The resulting compactness theorem for metrics is stated as follows. It is afundamental result in Riemannian geometry independent of the Ricci flow.

Theorem 8.8 (Compactness for Metrics). Let (Mk, gk, Ok) be a se-quence of complete pointed Riemannian manifolds satisfying (i) and (ii)above. Then there exists a subsequence (jk) such that (Mjk , gjk , Ojk) con-verges in the Cheeger-Gromov sense to a complete pointed Riemannian man-ifold (M∞, g∞, O∞) as k →∞.6

Remark 8.9. Our discussion here will take this theorem for granted as itsproof is non-trivial and does not directly relate to the Ricci flow (see [Ham95a,Sect. 3] or [CCG+07, Chap. 4] for further details).

It is clear that all of the conditions are needed:7 For example, if thereis no bound on the curvatures then one could have a sequence like the oneshown in Figure 8.4 (explicitly, take the manifolds Mk to be hypersurfacesin Rn+1 given by the graphs of y =

√1/k2 + |x|2, with the induced metric).

Similarly, one can construct examples where smooth convergence fails if thep-th derivative is not uniformly bounded, for any p > 0. As we shall see,the Ricci flow’s excellent regularity properties mean that this condition doesnot present too great a challenge in the analysis — at least, once we have abound on curvature (corresponding to p = 0) then the bounds for higher pare guaranteed).

Fig. 8.4 A sequenceof manifolds with infin-ite injectivity radius atthe central point butcurvature unbounded,and which do not con-verge to a manifold (cf.[Pet87, Example 1.1]).

A more subtle matter is that of a positive lower bound on the injectivityradius (see Figure 8.5). For an explicit example of what can go wrong, con-sider the sequence given by Mk = S1 × S1 = R2/Z2, with metrics given by

6 Due to a estimate from [CGT82] we only need injectivity radius lower bounds atthe basepoints, thereby avoiding any assumption of a uniform lower bounds for theinjectivity radius over the whole manifold.7 Condition (i) can be weakened in the following way: Rather than requiring uniformbounds on all of Mk, it suffices to have local uniform bounds, i.e. for each p ≥ 0 andeach R > 0 there exists C(p,R) such that |∇pkRk|k ≤ C(p,R) on BR(Ok).


gk(x, y) = dx2 + k−1dy2, and any choice of base points Ok. These metrics allhave zero curvature, so the first condition is fulfilled with Cp = 0 for every p.However (Mk, gk) has a closed geodesic (a circle in the y direction) of length1/√k, so the injectivity radius can be no greater than 1/(2

√k). This sequence

‘collapses’, in a certain sense converging to the lower-dimensional limit givenby the circle S1. This condition has proved to be been a major obstacle inthe analysis of Ricci flow. Establishing a suitable bound is highly non-trivial,and was one of the fundamental breakthroughs in the work of Perel’man.

Fig. 8.5 A sequenceof tori with injectivityradius approaching zero(cf. [Pet87, Example 1.2]).

8.3.1 Statement of the Compactness Theorem for Flows. Whenconsidering a sequence of complete pointed solutions to the Ricci flow, thecorresponding compactness theorem takes the following form:

Theorem 8.10 (Compactness for Flows). Let (Mk, gk(t), Ok), for t ∈(a, b), where −∞ ≤ a < 0 < b ≤ ∞, be a sequence of complete pointedsolutions to the Ricci flow such that:

1. Uniformly bounded curvature:

|Rk|k ≤ C0 on Mk × (a, b)

for some constant C0 <∞ independent of k; and2. Injectivity radius estimate at t = 0:

injgk(0)(Ok) ≥ κ0

for some constant κ0 > 0.

Then there exists a subsequence (jk) such that (Mjk , gjk(t), Ojk) convergesto a complete pointed solution to the Ricci flow (M∞, g∞(t), O∞) as k →∞,for t ∈ (a, b).

Remark 8.11. As remarked previously, there is no requirement for bounds onderivatives of curvature, since these are deduced from Theorem 7.1. We alsoremark that the curvature bound can be replaced by local uniform bounds,in the sense that we need only require bounds independent of k on each setBr(Ok) × I, where r ≥ 0, Br(Ok) is the ball of radius r with respect to themetric gk at t = 0, and I is a compact sub-interval of (a, b).

8.4 Proof of the Compactness Theorem for Flows 135

8.4 Proof of the Compactness Theorem for Flows

In this section we prove the compactness theorem for flows, Theorem 8.10,given the compactness theorem for metrics, Theorem 8.8. We will give theproof only for the case where the manifolds Mk are compact (though thereis no such assumption on the limit M∞). The proof for the more generalcase of complete solutions is not significantly more difficult, but requires astronger version of the regularity result of Theorem 7.1, which we do not wishto prove. The main tools of the proof are the theorem of Arzela-Ascoli andthe regularity results presented in the previous chapter.

8.4.1 The Arzela-Ascoli Theorem. Let X and Y be metric spaces,and fix y0 ∈ Y . We say F ⊂ C(X,Y ) is equicontinuous if for every ε > 0and x ∈ X there is a δ > 0 such that dY (f(x), f(y)) < ε for every f ∈ Fand for all y ∈ X with d(x, y) < δ. Also, we say F is pointwise bounded if forevery x ∈ X there exists C = C(x) < ∞ such that dY (f(x), y0) ≤ C(x) forall f ∈ F .

Theorem 8.12 (Arzela-Ascoli). Let X and Y be locally compact metricspaces. Then a subset F of C(X,Y ) is compact in the compact-open topologyif and only if it is equicontinuous, pointwise bounded and closed.8

Note that if F ⊂ C(X,Y ) is compact, then any sequence (fn) ⊂ F hasa subsequence that converges uniformly on every compact subset of X. Wecan apply this result to extract a convergent subsequence from a sequence ofsections of a bundle with bounded derivatives.

Corollary 8.13. Let (M, g) be a manifold, and E a vector bundle over M ,and fix metrics g and connections ∇ on E and TM . Let Ω ⊂ M an openset with compact closure Ω, and let p ∈ N ∪ 0. Let (ξk) be a sequence ofsections of E over Ω such that

sup0≤α≤p+1

supx∈Ω|∇αξk| ≤ C <∞.

Then there exists a section ξ∞ of E∣∣Ω

and a subsequence of (ξk) which con-

verges to ξ∞ in Cp on Ω.

Remark 8.14. To reduce the proof to the Arzela-Ascoli theorem, we can fixa finite collection of charts defined on compact domains covering Ω, withcorresponding trivialisations of E, and apply the Arzela-Ascoli theorem tothe components of ∇pξk in each chart. We observe that the components of∇pξk are equicontinuous, since their derivatives are controlled by |∇p+1ξk|.

8 Note that the collection of all such UK,U = f ∈ C(X,Y ) : f(K) ⊂ U, whereK ⊂ X is compact and U ⊂ Y open, defines a subbase for the compact-open topologyon C(X,Y ).


There are two situations in which Corollary 8.13 is directly applicable tothe compactness theorems we are discussing: The first is where E is the bundleof symmetric bilinear forms on TM , and the sections ξk are Riemannianmetrics on M . The second is where E is the bundle of symmetric bilinearforms on the space-like tangent bundle S over M × I, where I is a timeinterval. Then the sections ξk correspond to a sequence of time-dependentmetrics on M , such as a sequence of solutions to the Ricci flow.

8.4.2 The Proof. With the Arzela-Ascoli theorem and the regularityfor the Ricci flow discussed in Chapter 7, we are now in a position to provethe compactness theorem for flows from the general compactness theorem formetrics.

Proof (Theorem 8.10, given Theorem 8.8). We prove only the case where eachMk is compact. Consider a sequence of pointed solutions (Mk, gk(t), Ok), fort ∈ (a, b), to the Ricci flow where supMk×(a,b) |R(gk)| ≤ K. The Bernstein-Bando-Shi estimates, Theorem 7.1, give bounds of the form

|∇pR(x, t)| ≤ C(p, ε,K)

for all x ∈ M and t ∈ [a + ε, b), for each small ε > 0. The assumptionon the injectivity radius at Ok, when t = 0, fulfils the conditions of theCheeger-Gromov compactness theorem for metrics applied to the sequence(Mk, gk(0), Ok). Thus there exists a subsequence which converges in theCheeger-Gromov sense to a complete limit (M, g, O). That is, (passing to asubsequence if necessary) there exists an exhaustion Uk of M , and smoothinjective maps Φk : Uk →Mk taking O to Ok such that Φ∗k(gk(0)) convergesin C∞ on compact sets of M to g.

The idea now is to obtain uniform C∞ control on gk(t) = Φ∗k(gk(t)) oncompact subsets of M × (a, b) (note these are also solutions of Ricci flow).To do this, fix a compact set Z in M and consider only k sufficiently large sothat Z ⊂ Uk. The metrics gk(t) are uniformly comparable to g, since by theconvergence statement they are comparable to g at t = 0 and by Lemma 7.5they remain comparable for other t ∈ (a, b). To bound higher derivatives weapply, for each k, the evolution equation (7.7) derived in the proof of thelong-time existence theorem:

∂

∂t∇q gk =

∑j0+j1+···+jm=q

∇j0R(gk) ∗ ∇j1 gk ∗ · · · ∗ ∇jm gk. (8.1)

In the case q = 1 this gives∣∣∣∣ ∂∂t∇gk∣∣∣∣ ≤ C (1 +

∣∣∇gk∣∣) ,

8.5 Blowing Up of Singularities 137

where C depends on K and ε but not k. Since the time interval is finiteand ∇gk(0) → 0 as k → ∞, this implies |∇gk(t)| ≤ C, independent of k.Proceeding by induction on q: Suppose |∇j gk(t)| is bounded independent ofk, for j = 1, . . . , q, so that (8.1) again gives∣∣∣∣ ∂∂t∇q+1gk

∣∣∣∣ ≤ C (1 +∣∣∇q+1gk

∣∣) ,which implies a bound independent of k on Z×[a+ε, b). Note that derivativesin time directions are also bounded, since they can be written in terms ofspatial derivatives via the Ricci flow equation. It follows by the Arzela-Ascolitheorem (in the form of Corollary 8.13) that there is a subsequence whichconverges in C∞ on Z× [a+ ε, b− ε]. A diagonal subsequence argument thenproduces a subsequence which converges in C∞ on compact sets of M×(a, b)to a complete solution g(t) of Ricci flow, thus proving that the correspondingsubsequence of (Mk, gk(t), Ok) converges to (M, g(t), O) in the sense of thetheorem.

8.5 Blowing Up of Singularities

We will apply the compactness results of this chapter, in particular The-orem 8.10, to a solution of the Ricci flow with a finite maximal time ofexistence T . As the curvature explodes in this situation, we need to choosea sequence of times ti T and rescale of the metric to make the curvaturesbounded. By doing this we hope that the limiting manifold will tell us aboutthe nature of the singularity, and hopefully some desirable topological in-formation. We pursue this idea in detail.

Suppose that a solution of the Ricci flow (M, g(t)) exists on a maximaltime interval t ∈ [0, T ) with finite final time T <∞. By Theorem 7.4,

lim suptT

|R|( · , t) =∞

so the maximum value of |R| on M explodes to +∞ as t T . In which case,choose points Oi ∈M and times ti T such that

|R|(Oi, ti) = sup(x,t)∈M×[0,ti]

|R|(x, t),

and also set Qi := |R|(Oi, ti). Applying the parabolic rescaling of the Ricciflow discussed in Section 3.1.3, we define

gi(t) := Qi g(ti +Q−1i t)

so that (M, gi(t)) satisfies Ricci flow on the time interval [−tiQi, (T − ti)Qi].For each i and times t ≤ 0, note that


|R(gi(t))| =1

Qi|R(g(ti +Q−1

i t))| ≤ 1

by the definition of Qi. Also observe, for each i and times t > 0, that

supM|R(gi(t))| ≤

1

1− C(n)t

by the doubling-time estimate discussed in Section 6.2.2. Therefore gi(t) isdefined with

supi

supM×(a,b)

|R(gi(t))| <∞

for any a < 0 and some b = b(n) > 0. In which case, the sequence(M, gi(t), Oi), t ∈ (a, b) has uniform bounded geometry. So by Theorem 8.10,we can pass to a subsequence j such that (M, gj(t), Oj) converges to a com-plete pointed solution to the Ricci flow (M∞, g∞(t), O∞), for all t ∈ (a, b),provided we can establish a suitable lower bound on the injectivity radius.

Remark 8.15. It is precisely such a bound that is missing from our analysis.Historically this has been a major difficulty, except in special circumstances.9

However, as we shall see in the next two chapters, this issue has been elegantlyresolved by the work of Perel’man [Per02].

9 One of the cases which can be handled is where the sectional curvatures are positive:Klingenberg [Kli59] proved that an even dimensional simply connected manifold withpositive sectional curvatures has injectivity radius equal to its conjugate radius, whichis at least π/

√Kmax by the Rauch comparison theorem. He also proved this for odd

dimensions provided the sectional curvatures are globally 1/4-pinched [Kli61], andAbresch and Meyer [AM94] extended this result to allow global pinching with someexplicit pinching ratio below 1/4.

Chapter 9

The F-Functional and Gradient Flows

After Ricci flow was first introduced, it appeared for many years that therewas no variational characterisation of the flow as the gradient flow of a geo-metric quantity. In particular, Bryant and Hamilton established that theRicci flow is not the gradient flow of any functional on Met — the space ofsmooth Riemannian metrics — with respect to the natural L2 inner product(with the exception of the two-dimensional case, where there is indeed suchan ‘energy’). Considering the prominent role variational methods have playedin geometric analysis, pde’s and mathematical physics, it seemed surprisingthat such a natural equation as Ricci flow should be an exception. One of themany important contributions Perel’man made was to elucidate a gradientflow structure for the Ricci flow, not on Met but on a larger augmented space.Part of this structure was already implicit in the physics literature [Fri85]. Inthis chapter we discuss this structure, at the centre of which is Perel’man’sF-functional [Per02]. The analysis will provide the ground work for the proofof a lower bound on injectivity radius at the end of Chapter 10.

9.1 Introducing the Gradient Flow Formulation

We introduce ‘the gradient flow’ associated to an energy functional in generalterms. The concept naturally arises in branches of physics, pde’s, numericalanalysis and related areas.

Definition 9.1. If H is a Hilbert space with a smooth functional E : H → R,the gradient vector field ∇E : H → H is given at each u ∈ H by the uniquevector ∇E(u) ∈ H, such that

〈∇E(u), v〉 = dE(u)(v) (9.1)

for all v ∈ H.

A consequence of (9.1) is that ‖dE(u)‖ = ‖∇E(u)‖ for any u ∈ H. Moreover,∇E defines a gradient flow Φ given by the ode

139

140 9 The F-Functional and Gradient Flows

d

dtΦu(t) = −∇E(Φu(t))

Φu(0) = u

for any fixed u ∈ H. To interpret Φ, we observe that flow lines, with respectto the graph gr(E) = (u,E(u)) ∈ H × R, are paths of steepest decent.Indeed, for any time dependent u = u(t) ∈ H, we have

d

dtE(u(t)) = dE(u)(u) = 〈∇E(u), u〉 ≥ −‖∇E(u)‖‖u‖

with equality holding if and only if u = −λ∇E(u) for λ > 0.The archetypical gradient flow in pde is the one associated to the Dirichlet

energy functional

E(u) =1

2

ˆRn|∇u|2dx =

1

2‖∇u‖2L2

with the usual L2-inner product. Indeed, by Proposition 1.69(c), if u is smoothin x and t then

d

dtE(u) =

ˆRn

⟨ ∂∂t∇u,∇u

⟩dx = −

ˆRn

∂u

∂t∆udx = 〈−∆u, u〉 .

Thus formally the gradient flow is the standard heat equation ∂u∂t = ∆u. Note

that this example (in common with most other examples arising in pde andcalculus of variations) does not quite fit into the gradient flow frameworkdefined above, since the energy E is not even defined on the Hilbert spaceL2(Rn). We can make sense of E on the smaller space L2 ∩ C∞, but onthis space the inner product is not complete, and E is not differentiable inthe Frechet sense (see Appendix A — note that Proposition A.3 does notapply here, since the derivative −∆u is not continuous with respect to theL2 metric):

E(u+ v)− E(u)− δv(u) =1

2

ˆ|∇(u+ v)|2 − |∇u|2 + 2v∆udx

=1

2

ˆ|∇v|2 dx+

ˆ(v∆u+∇v · ∇u) dx

=1

2

ˆ|∇v|2

which is not even bounded as ‖v‖L2 → 0 in L2 ∩ C∞, let alone o(‖v‖L2).However, E is Gateaux differentiable on this space, and the derivative δvu isa continuous linear map with respect to the L2 metric for any smooth u, sothe gradient vector is still well defined.

9.2 Einstein-Hilbert Functional 141

Remark 9.2. When examining the gradient flow formulation of the Ricci flowthere is a natural L2-inner product, induced by the fibrewise product onSym2 T ∗M , given by

〈h, h′〉 =

ˆM

gikgj`hijh′k`dµ(g) (9.2)

referred to as the canonical Riemannian metric on the space of Riemannianmetrics Met.1 Note that this metric, by construction, is invariant under theaction of the diffeomorphism group; however, as in the previous example theinner product is not complete.

9.2 Einstein-Hilbert Functional

Let (M, g) be a closed2 Riemannian manifold. One of the most natural func-tionals one can construct on Met is the so-called Einstein-Hilbert functionalE : Met→ R, which is the integral of the scalar curvature:

E(g) =

ˆM

Scal dµ.

By computing the variation of E at g, in direction h = g, we see that

δhE(g) =

ˆM

δhScal(g) dµ+

ˆM

Scal δhdµ(g)

=

ˆM

−∆trgh+ δ2h− 〈h,Ric〉+Scal

2trgh dµ

where we recall the variation equations Proposition 3.10 and 3.11. Now as Mis closed,

´M∆trghdµ = 0, so by the divergence theorem (i.e. Theorem 1.68)

we see that´Mδ2h dµ =

´Mgijgpq∇2

q,jhpi dµ =´M

div V dµ = 0, where V =

V q∂q = (gpqgij∇jhip)∂q. Thus the variation of E is

δhE(g) =

ˆM

⟨Scal

2g − Ric, h

⟩dµ.

It is important to note that the Scal2 g-term is due to variation of the volume

element dµ. From the formula above, (twice) the gradient flow of E is givenby

∂

∂tgij = 2(∇E)ij = Scal gij − 2Rij .

1 Note that Met is an infinite dimensional cone (in the vector space Sym2T∗M) andso is highly non-compact. Arbitrary sequences of Riemannian metrics can degeneratein complicated ways; however there are two rather trivial but nonetheless importantsources of non-compactness, namely those of diffeomorphism invariance and scaling(cf. Chapter 8).2 That is, compact without boundary.


We note that this equation looks similar to the Ricci flow, but the extra termmeans that this equation is not parabolic: From equation (4.2) the symbol isgiven by

(σ[Scal g − 2Ric]ξ)(h) = |ξ|2h+ trh ξ ⊗ ξ− ξ ⊗ h](ξ)− h](ξ)⊗ ξ +

(h(ξ, ξ)− |ξ|2trh

)g,

where h] is the linear operator corresponding to h (obtained by raising anindex using the metric). Choosing ξ = en, this maps h = ξ ⊗ ξ − |ξ|2g to(2− n) times itself, but maps h = e1 ⊗ e2 + e2 ⊗ e1 to itself. Of course thereis also a kernel which can be removed using DeTurck’s trick, but no suchtrick can make the symbol have definite real part. Therefore the gradientflow is not parabolic. Such equations do not generally have solutions even fora short time — that is, there do not exist paths of steepest descent for thisfunctional.

9.3 F-Functional

To overcome the problems associated with the Einstein-Hilbert functional,Perel’man considers a functional F on the enlarged space Met × C∞(M),defined by

F(g, f) :=

ˆM

(Scal + |∇f |2)e−fdµ. (9.3)

We shall follow the physics literature and call f the dilaton. It is importantto note that F can also be written as F(g, f) =

´M

(Scal + ∆f)e−fdµ on aclosed manifold M .3

Remark 9.3. There are some elementary symmetry properties associated withF . The first of these is that F is diffeomorphism invariant: If ϕ ∈ Diff(M)then F(ϕ∗g, f ϕ) = F(g, f). The second is the scaling behaviour: For anyscalars b and c > 0 we have F(c2g, f + b) = en−2e−bF(g, f).

Proposition 9.4 (Variation of F). On a closed manifold M , the variationof F is equal to

δ(h,k)F(g, f) = −ˆM

〈Ric + Hess(f), h〉 e−fdµ

+

ˆM

(1

2trgh− k

)(2∆f − |∇f |2 + Scal

)e−fdµ.

Proof. The variation of F in direction (h, k) is defined by Appendix A to beδ(h,k)F(g, f) = d

ds

∣∣s=0F(g + sh, f + sk). This implies that

3 Since ∆e−f = (|∇f |2 −∆f)e−f implies that´M|∇f |2e−fdµ =

´M∆fe−fdµ.

9.3 F-Functional 143

δ(h,k)F(g, f) =

ˆM

(δ(h,k)(Scal + |∇f |2)(g, f)

)e−fdµ

+

ˆM

(Scal + |∇f |2)(δ(h,k)(e

−fdµ)(g, f)).

Now by Proposition 3.11 we find that δ(h,k)(e−fdµ)(g, f) = (

trgh2 −k)e−fdµ(g),

and by (3.9) that

δ(h,k)|∇f |2(g, f) = −gikgj`hk`∇if∇jf + 2gij∇if∇jk= −hij∇if∇jf + 2 〈∇f,∇k〉 .

Putting this together gives the variation

δ(h,k)F(g, f) =

ˆM

(−∆trgh+ δ2h− 〈h,Ric〉

− hij∇if∇jf + 2 〈∇f,∇k〉)e−fdµ

+

ˆM

(Scal + |∇f |2

)( trgh

2− k)e−fdµ. (9.4)

Now since ∆e−f = (|∇f |2 − ∆f)e−f , Proposition 1.69(b) implies that´M−(∆trgh)e−fdµ = −

´M

trgh∆e−fdµ =

´M

trgh(∆f − |∇f |2)e−fdµ, and

Proposition 1.69(c) implies that´M〈∇f,∇k〉 e−fdµ =

´M

(∆e−f )k dµ =´M

(|∇f |2 −∆f)ke−fdµ. Putting these two identities together gives

ˆM

(−∆trgh+ 2 〈∇f,∇k〉) e−fdµ = 2

ˆM

( trgh

2− k)(∆f − |∇f |2

)e−fdµ.

Also, we find by the divergence theorem that

ˆM

δ2h e−fdµ =

ˆM

divV e−fdµ = −ˆM

V q∇qe−fdµ,

where V = V q∂q = (gpqgij∇jhip)∂q. Now by using the identity

∇j(gpqgijhip∇qe−f ) = V q∇qe−f + gpqgijhip∇j∇qe−f ,

we see that´M∇j(gpqgijhip∇qe−f )dµ = 0 by the divergence theorem again,

so thatˆM

δ2h e−fdµ =

ˆM

gpqgijhip∇j∇qe−fdµ

=

ˆM

hqj∂jf∂qfe−f + (−∂j∂qf + Γ rjq∂rf)e−fdµ

=

ˆM

(hij∇if∇jf − 〈Hess(f), h〉

)e−fdµ.


Therefore we find thatˆM

(−∆trgh+ δ2h− 〈h,Ric〉 − hij∇if∇jf + 2 〈∇f,∇k〉

)e−fdµ

=

ˆM

−〈Ric + Hess(f), h〉 e−fdµ+

ˆM

( trgh

2− k)(

2∆f − 2|∇f |2)e−fdµ.

Combining this with (9.4) yields the desired result.

Corollary 9.5 (Measure-preserving Variation of F). For variations(h, k) satisfying δ(h,k)e

−fdµ(g, f) = 0, the variation

δ(h,k)F(g, f) = −ˆM

〈Ric + Hess(f), h〉 e−fdµ.

9.4 Gradient Flow of Fm and Associated CoupledEquations

With Proposition 9.4 in mind, Perel’man formulates an appropriate gradientflow, in the form of a coupled system of equations, that can be related to theRicci flow. To see this, first fix a smooth positive background measure4 dωon M , and define a smooth graph X : Met→Met× C∞(M) by letting

X : g 7→(g, log

dµ(g)

dω

).

The resulting composition Fm = F X : Met → R is a functional on Metthat modifies F . It takes the form

Fm(g) =

ˆM

(Scal +

∣∣∣∣∇ logdµ

dω

∣∣∣∣2 )dω=

ˆM

(Scal + |∇f |2)dω,

where now f := log dµdω .5 From Corollary 9.5, the variation of Fm is

δhFm(g) = −ˆM

hij(Rij +∇i∇jf)dω,

4 Here we take a measure to mean a positive n-form. Note if dm is any positive n-form,then for any open U ⊂ M we can define m(U) :=

´Udm. Hence m : B(M)→ [0,∞)

is a positive measure (in the strict sense) defined on the Borel σ-algebra B(M).5 The quotient of two n-forms makes sense: For if dm1 and dm2 are two positiven-forms, we have that dm1 = ϕ1dω and dm2 = ϕ2dω, where ϕ1, ϕ2 are the corres-ponding Randon-Nykodym derivatives with respect to the background measure dω.In which case we define dm1

dm2:= ϕ1

ϕ2.

9.4 Gradient Flow of Fm and Associated Coupled Equations 145

since dω is fixed and dω = e−fdµ by the definition of f . So by (9.2) andProposition A.3, the gradient vector field of Fm (if it exists) is given by

∇Fm(g) = −(Rij +∇i∇jf).

Hence (twice) the positive gradient flow of Fm on Met is

∂

∂tgij = 2∇Fm(g) = −2(Rij +∇i∇jf),

and the associated evolution equation for f = log dµdω equal to

∂f

∂t= −∆f − Scal,

since ∂f∂t = 1

2gij ∂gij

∂t = −(Scal + gij∇i∇jf).6 In which case we consider thecoupled modified Ricci flow :

∂

∂tg = −2(Ric + Hess(f)) (9.5a)

∂f

∂t= −Scal−∆f (9.5b)

In general, with Section 9.1 in mind, any gradient flow of Fm is non-decreasing along the flow lines, with ∂

∂tFm = ‖2∇Fm‖2L2 . This now gives

Perel’man’s monotonicity formula for the gradient flow of Fm.

Proposition 9.6. If (g(t), f(t)) is a solution to the coupled modified Ricciflow (9.5), then

d

dtFm(g(t)) = 2

ˆM

∣∣Rij +∇i∇jf∣∣2dω.

9.4.1 Coupled Systems and the Ricci Flow. Remarkably the gradientflow (9.5) is (up to diffeomorphism) equivalent to Ricci flow. This is achievedsimply by performing a time-dependent diffeomorphisms (similar to that seenin Section 9.5) that transforms the coupled modified system into the Ricciflow.

On an intuitive level, if the diffeomorphism is generated by flowing alongthe time-dependent vector field V (t), then the new equations for g and fbecome gij = −2(Rij +∇i∇jf) + LV g and f = −∆f − Scal + LV f . Using

6 Note that if ω1(t) and ω2(t) are time-dependent n-forms, then

∂

∂tlog

ω1

ω2=

∂ω1

∂t

ω1−

∂ω2

∂t

ω2.


these together with the fact that L∇fg = 2∇∇f and L∇ff = |∇f |2,7 weconsider following coupled Ricci flow :

∂

∂tg = −2Ric (9.6a)

∂f

∂t= −∆f + |∇f |2 − Scal. (9.6b)

We get a solution to this system first by solving g = −2Ric forwards in timethen by solving f = −∆f + |∇f |2 − Scal backwards in time. The rest of thissection is devoted to confirming this intuition formally.

9.4.1.1 Converting the Gradient Flow to a Solution of the Ricci Flow.Given a solution (g(t), f(t)) to the gradient flow (9.5), we show that thereis a solution (g(t), f(t)) to the coupled Ricci flow (9.6) by flowing along thegradient of f .

Lemma 9.8. Let (g(t), f(t)), t ∈ [0, T ], be a solution to the system (9.5).Define the one-parameter family of diffeomorphism Φ(t) ∈ Diff(M) by

d

dtΦ(t) = ∇g(t)f(t), Φ(0) = idM . (9.7)

Then the pullback metric g(t) = Φ(t)∗g(t) and dilaton f(t) = f Φ(t) satisfythe system (9.6).

Proof. By the theory of time-dependent ode’s, (9.7) always has solution (cf.[Lee02, p. 451]). In which case8

∂

∂tg =

∂

∂t

(Φ∗g

)= Φ∗

(∂g∂t

)+ Φ∗

(L∇g f g

)= −2Φ∗

(Ric(g)

)= −2Ric(g).

So we find that

7 To see this, we first observe:

Proposition 9.7. For any 1-form ω and X,Y vector fields,

(Lω]g)(X,Y ) = (∇ω)(X,Y ) + (∇ω)(Y,X).

Proof. By compatibility of g, X(g(ω], Y )) = g(∇Xω], Y ) + g(ω],∇XY ). Henceg(∇Xω], Y ) = X(ω(Y )) − ω(∇XY ) = (∇ω)(X,Y ). However on the other hand,since ∇g = 0, we have Lω]g(X,Y ) = g(∇Xω], Y ) + g(X,∇Y ω]).

Thus when ω = (df)] we have that (L∇fg)(X,Y ) = (∇∇f)(X,Y ) + (∇∇f)(Y,X) =2Hess(f)(X,Y ), since the Hessian is symmetric.8 Note, if ψ is a solution to the ode: ∂

∂tψ(x, t) = X(ψ(x, t), t). Then

∂

∂tψ∗g =

∂g

∂t+ g(∇iX, ∂j) + g(∂i,∇jX).

9.4 Gradient Flow of Fm and Associated Coupled Equations 147

∂

∂tf =

∂

∂t

(f Φ

)=∂f

∂t Φ+

⟨∇ f Φ, ∂Φ

∂t

⟩g

=(− ∆f − ¯Scal

) Φ+

∣∣∇ f Φ∣∣2g

= −∆f − Scal +∣∣∇f ∣∣2,

where ∆ and ∇ are with reference to g(t).

9.4.1.2 Converting the Ricci Flow to a Solution of the Gradient Flow. Wenow show the converse. Given a solution (g(t), f(t)) to the coupled Ricci flow(9.5), we show that there is a solution (g(t), f(t)) to the gradient flow (9.6)by flowing backwards along the gradient of f . To do this, we need to solve abackwards heat equation for f .

Lemma 9.9. Let g(t), t ∈ [0, T ), be a solution of the Ricci flow and let fTbe an arbitrary function on M .

1. Then there exists a unique solution to the backwards heat equation

∂f

∂t= −∆f + |∇f |2 − Scal, t ∈ [0, T ] (9.8a)

f(T ) = fT (9.8b)

2. Furthermore, given a solution f(t) to (9.8), define the one-parameterfamily of diffeomorphism Ψ(t) ∈ Diff(M) by

d

dtΨ(t) = −∇g(t)f(t), Ψ(0) = idM . (9.9)

Then the pullback metric g(t) = Ψ∗(t)g(t) and the pullback dilaton f(t) =f Ψ(t) satisfy (9.5).

Proof. 1. Re-parametrise time by τ = T − t and set u = e−f . Note that

∂u

∂τ= −∂u

∂t=∂f

∂tu = (∆f + |∇f |2 − Scal)u

= ∆u− Scalu,

and so u satisfies∂u

∂τ= ∆u− Scalu.

As this is a linear parabolic equation (forwards) in time with initial data atτ = 0, there exists a unique solution on [0, T ].

2. One can verify that g and f satisfy (9.5) by the same procedure as inLemma 9.8 except that here we flow along −∇f rather than ∇ f .


9.4.2 Monotonicity of F from the Monotonicity of Fm. The diffeo-morphism invariance of all quantities under consideration implies the mono-tonicity formula for the Ricci flow:

Proposition 9.10. If (g(t), f(t)) is a solution to (9.6) on a closed manifoldM , then

d

dtF(g(t), f(t)) = 2

ˆM

∣∣Rij +∇i∇jf∣∣2e−fdµ.

Proof. As (g(t), f(t)) is a solution to (9.6), Lemma 9.9 implies that g(t) =Ψ∗(t)g(t) and f(t) = f(t) Ψ(t) are a solution to (9.5).

By Remark 9.3, F is invariant under diffeomorphisms. Thus F(g, f) =F(g, f), and so d

dtF(g, f) = ddtF(g, f). Moreover, by Proposition 9.6 we see

that

d

dtF(g, f) = 2

ˆM

∣∣Rij + ∇i∇j f∣∣2ge−fdµ(g)

= 2

ˆM

∣∣Rij +∇i∇jf∣∣2 e−fdµ(g).

Chapter 10

The W-Functional and LocalNoncollapsing

The F-functional provides a gradient flow formalism for the Ricci flow, dis-cussed in Chapter 9. We hope to be able to use this to understand the sin-gularities of Ricci flow, but the F-functional is not yet enough to do this,because it does not behave well under the scaling transformations neededin the blow-up analysis. To overcome this, Perel’man introduced the W-functional which includes a positive scale factor τ . The advantage of thisfunctional is that it can be related to aspects of the local geometry, in par-ticular volume ratios of balls with radius on the order of

√τ . As discussed

at the end of Chapter 8, the missing ingredient in the singularity analysisis a suitable lower bound on the injectivity radius on the scale determinedby the curvature, and Perel’man was able to prove such a bound using theW-functional. The injectivity estimate is proved here in Section 10.4.

10.1 Entropy W-functional

On a closed n-dimensional manifold M , define Perel’man’s entropy W-functional W : Met× C∞(M)× R+ → R by

W(g, f, τ) =

ˆM

(τ(Scal + |∇f |2) + f − n

)u dµ, (10.1)

where u := (4πτ)−n2 e−f and τ > 0 is the scale parameter. By inspection we

note that this is related to F-functional by

W(g, f, τ) =1

(4πτ)n/2

(τ F(g, f) +

ˆM

(f − n)e−fdµ). (10.2)

In a similar way to that of Chapter 9, we look to find a gradient flowfor W that includes the Ricci flow which makes W monotone. With this, weprove the local noncollapsing result for the Ricci flow. Thereafter we provethe desired lower injectivity bounds.

149

150 10 The W-Functional and Local Noncollapsing

Remark 10.1. Like F , the functional W is diffeomorphism-invariant: If Φ ∈Diff(M) then W(Φ∗g, Φ∗f, τ) = W(g, f, τ), where Φ∗g is the pullback met-ric and Φ∗f = f Φ. The scaling properties of W are rather nicer thanthose of F : Under the scaling transformation (g, f, τ) 7→ (cg, f, cτ),we haveW(cg, f, cτ) =W(g, f, τ).

Proposition 10.2 (Variation of W). On a closed manifold M , the vari-ation of W is equal to

δ(h,k,ζ)W(g, f, τ) =

ˆM

⟨Ric + Hess(f)− 1

2τg,−τh+ ζg

⟩u dµ

+

ˆM

τ(1

2trgh− k −

n

2τζ)(

Scal + 2∆f − |∇f |2 +f − n− 1

τ

)u dµ.

Proof. We calculate the variation δ(h,k,ζ)W(g, f, τ) of W at (g, f, τ) in thedirection (h, k, ζ) via a two step process. This is done by separating out thevariation in the scale parameter so that

δ(h,k,ζ)W(g, f, τ) = δ(h,k,0)W(g, f, τ) + δ(0,0,ζ)W(g, f, τ).

Now to compute δ(h,k,0)W(g, f, τ) with τ fixed, we look to (10.2). By Pro-position 9.4 we have that

δ(h,k,0)

( τ

(4πτ)n/2F(g, f)

)(g, f, τ) = −

ˆM

τ〈Ric + Hess(f), h〉u dµ

+

ˆM

τ(1

2trgh− k

)(2∆f − |∇f |2 + Scal

)u dµ,

and by direct computation

δ(h,k,0)

( 1

(4πτ)n/2

ˆM

(f − n)e−fdµ)

(g, f, τ)

=

ˆM

(k +

(1

2trgh− k

)(f − n

))u dµ.

To compute δ(0,0,ζ)W(g, f, τ) with g and f fixed, we see directly that

δ(0,0,ζ)W(g, f, τ) =

ˆM

(ζ(1− n

2

)(Scal + |∇f |2

)− nζ

2τ(f − n)

)u dµ.

By combining all of the terms, we find that the variation of W equals

10.2 Gradient Flow of W and Monotonicity 151


ˆM

[⟨Ric + Hess(f),−τh+ ζg

⟩+ τ(1

2trgh− k

)(2∆f − |∇f |2 + Scal +

f − nτ

)+ k + ζ(|∇f |2 −∆f)− nζ

2τ(f − n)− nζ

2(Scal + |∇f |2)

]u dµ.

Now absorb − n2τ ζ into the first bracket of the terms on the second line to get


ˆM

[⟨Ric + Hess(f),−τh+ ζg

⟩+ τ(1

2trgh− k −

n

2τζ)(

Scal + 2∆f − |∇f |2 +f − nτ

)+ k + (n− 1)ζ(∆f − |∇f |2)

]u dµ.

Also absorb − 12τ g into the angled bracket terms, together with the fact that

〈−12τ g,−τh+ ζg〉 = 1

2 trgh− n2τ ζ, so that finally


ˆM

[⟨Ric + Hess(f)− 1

2τg,−τh+ ζg

⟩+ τ(1

2trgh− k −

n

2τζ)(

Scal + 2∆f − |∇f |2 +f − n− 1

τ

)+ (n− 1)ζ(∆f − |∇f |2)

]u dµ.

The desired result now follows since the last term of the latter equationvanishes because

´M

(∆f − |∇f |2)e−fdµ =´M∆e−fdµ = 0.

Corollary 10.3 (Measure-preserving Variation of W). For variations(h, k, ζ) satisfying δ(h,k,ζ)u dµ(g, f, τ) = 0, the variation


ˆM

⟨Ric + Hess(f)− 1

2τg,−τh+ ζg

⟩u dµ.

10.2 Gradient Flow of W and Monotonicity

In this section we want to formulate an appropriate gradient flow for W thatmakes the functional monotone. Whereas in Section 9.4 it was possible to dothis formally for the F-functional, here we derive the gradient flow for Wheuristically.

We do this first by fixing the measure dm = (4πτ)−n/2e−fdµ so that thevariation δ(h,k,ζ)dm(g, f, τ) vanishes (i.e. − n

2τ ζ − k + 12 trgh = 0). By solving

this for f we find that

f = logdµ

dm− n

2log(4πτ).


Now by taking the gradient flow for the metric gij as done in (9.5), we obtainthe following coupled gradient flow:

∂

∂tg = −2(Ric + Hess(f)) (10.3a)

∂f

∂t= −∆f − Scal +

n

2τ(10.3b)

dτ

dt= −1 (10.3c)

where the last condition τ = −1 is imposed in order to ensure monotonicity,since

d

dtW =

ˆM

(Rij +∇i∇jf −

1

2τgij)(− τ gij + τ gij

)dm

= 2τ

ˆM

∣∣Rij +∇i∇jf −1

2τgij∣∣2dm

whenever dm = u dµ is fixed, gij = −2(Rij +∇i∇jf) and τ = −1.By performing the same diffeomorphism change that was outlined in Sec-

tion 9.4.1, we obtain the following coupled system of equations:

∂

∂tg = −2Ric (10.4a)

∂f

∂t= −∆f + |∇f |2 − Scal +

n

2τ(10.4b)

dτ

dt= −1 (10.4c)

which includes the Ricci flow. It now follows, by a similar argument to thatof Section 9.4.2, that we have the following monotonicity result for the W-functional:

Proposition 10.4. If (g(t), f(t), τ(t)) be a solution to the coupled system(10.4) on a closed manifold M , then

d

dtW(g(t), f(t), τ(t)) =

ˆM

2τ∣∣∣Ric + Hess(f)− 1

2τg∣∣∣2u dµ. (10.5)

10.2.1 Monotonicity of W from a Pointwise Estimate. We notethat there is an alternative approach to proving the entropy monotonicity ofW based on the conjugate heat operator.

By defining the heat operator = ∂∂t−∆ acting on C∞(M×[0, T )) we see,

by evaluating ddt

´vw, that the conjugate heat operator ∗ = − ∂

∂t −∆+Scalis conjugate to in the following sense.

Lemma 10.5. If g(t), t ∈ [0, T ) is a solution to the Ricci flow and v, w ∈C∞(M × [0, T )), then

10.3 µ-Functional 153

ˆ T

0

(ˆM

(v)w dµ

)dt =

[ˆM

vw dµ

]T0

+

ˆ T

0

(ˆM

v (∗w) dµ

)dt.

Now by defining

w :=(τ(R+ 2∆f − |∇f |2) + f − n

)u

so that

W =

ˆM

w dµ,

as´M

(∆f − |∇f |2)u dµ =´M∆udµ = 0, we see that the monotonicity of W

follows immediately from the following proposition since

d

dtW =

d

dt

ˆM

w dµ = −ˆM

∗w dµ

by Lemma 10.5 with v = 1.

Proposition 10.6 ([Per02, Proposition 9.1], [Top06, p. 77]). Suppose (g, f, τ)evolve according to (10.4). Then the function w satisfies

∗w = −2τ∣∣∣Ric + Hess(f)− 1

2τg∣∣∣2u.

10.3 µ-Functional

We now look at the functional µ : Met× R+ → R defined by

µ(g, τ) = infW(g, f, τ) : f ∈ C∞(M) compatible with g and τ,

where we say that the tuple (g, f, τ) is compatible if

(4πτ)−n2

ˆM

e−f dµ =

ˆM

u dµ = 1.

As we shall see, the functional µ plays an important role in proving the localnoncollapsing result of the next section. Although before we can do so, weneed to check that µ is monotone and bounded from below, and that theinfimum is attained.

Remark 10.7. µ is a homogeneous function of degree 0, i.e. µ(cg, cτ) = µ(g, τ)for any scalar c, and has the diffeomorphism invariance property: µ(Φ∗g, τ) =µ(g, τ) for any Φ ∈ Diff(M), inherited from W.

Proposition 10.8 (µ is Bounded Below). For any given g and τ > 0on a closed manifold M , there exists c ∈ R such that W(g, f, τ) ≥ c for allcompatible f ∈ C∞(M). Consequently µ(g, τ) ≥ c.


Proof. By our scaling property µ(g, 1) = µ(τg, τ), so without loss of general-ity let τ = 1.

Let w =√u = (4π)−n/4e−f/2 > 0 with

´w2dµ = 1. From this we find that

f = −2 logw − n2 log 4π and ∇f = −2∇w/w. Hence we can write W(g, f, 1)

in terms of w:

W(g, f, 1) =

ˆM

(4|∇w|2 +

(Scal− 2 logw − n

2log 4π − n

)w2)dµ

=: H(g, w). (10.6)

Since M is closed, Scal − n − n2 log 4π ≥ infx∈M Scal − n − n

2 log 4π >C > −∞. So the only problem term is the w2 logw one. Fortunately, the LogSobolev inequality allows this to be bounded by a Dirichet type term. It istypically stated (for instance see [CLN06, p. 184]) as follows.

Lemma 10.9 (Log Sobolev Inequality on a Manifold). Let (M, g) beclosed Riemannian manifold. For any a > 0 there exists a constant C(a, g)such that if ϕ > 0 satisfies

´ϕ2dµ = 1, then

ˆM

ϕ2 logϕdµ ≤ aˆM

|∇ϕ|2dµ+ C(a, g).

From the Lemma we finally get

H(g, w) ≥ 2

ˆM

|∇w|2dµ+ C − C(1, g) ≥ C − C(1, g). (10.7)

Proposition 10.10 (Existence of a Smooth Minimiser). For any smoothmetric g on a closed M with τ > 0, the infimum of W over all compatible fis attained by a smooth compatible minimiser f∞.

Proof. Once again, without loss of generality, let τ = 1 and define H(g, w) asin (10.6) above. We will use direct methods in the calculus of variationsto show that H has a minimizer. From the estimate (10.7), any minim-izing sequence wk of compatible functions for H(g, ·) (that is, functionswhich are positive and satisfy

´Mw2 dµ = 1) has bounded Dirichlet energy,

and there exists a subsequence with a weak limit w in W 1,2. Since the Di-richlet energy is weakly lower semicontinuous (see [Dac04, p. 82]), we have´|∇w|2 ≤ lim infk→∞

´|∇wk|2. By the Rellich compactness theorem,1 the

sequence also converges in Lp for p < 2nn−2 , so

´w2 = 1, and

´Scalw2

k con-

verges to´

Scalw2.

1 If Ω ⊂ Rn is a regular bounded domain with 1 ≤ p < n and 1 ≤ q < p∗,the bounded sets on W 1,p(Ω) are precompact in Lq(Ω). In particular, if (uk) is asequence of functions in W 1,p(Ω) such that ‖uk‖W 1,p ≤ C, where C is independentof k, then there is a subsequence of (uk) which converges in Lq(Ω) (cf. [Eva98, p.272] or [Jos08, p. 549]).

10.3 µ-Functional 155

We want to show that the term involving the integral of w2 logw in thedefinition of H also converges. To see this, set ρ = w2 logw, and note that∇ρ = (2w logw + w)∇w. Since |w logw| ≤ c1 + c2w

1+ε/2 for any ε > 0, wefind thatˆ ∣∣∇ρ∣∣ ≤ ˆ ∣∣(w + 2w logw)∇w

∣∣≤( ˆ|2c1 + w|2

) 12( ˆ|∇w|2

) 12

+ 2c2

( ˆ|w|2+ε

) 12(ˆ|∇w|2

) 12

.

Trivially (´|∇w|2)1/2 is bounded by ‖w‖W 1,2 and using Sobolev inequalities2

one can show the other terms are also bounded by ‖w‖W 1,2 as well.Using the Rellich compactness theorem again, we have that the sequence

ρk is precompact in L1, so passing to a subsequence we have´ρk →

´ρ. It

follows that H(w) ≤ limn→∞H(wk) = infH, so the limit w is a minimizer ofH and is compatible. Hence by definition we have H(g, w) = µ(g, 1).

The limit w is clearly non-negative. The first variation formula shows thatit is a weak solution of the Euler-Lagrange equation for H, which is theelliptic equation ∆w + (2 logw + n + n

2 log 4π − Scal + µ)w/4 = 0. We wishto prove that w is smooth and positive, using techniques from pde theory.This is slightly subtle due to the presence of the logarithmic nonlinearity.

We first show that w has continuous derivatives up to second order: Since

|w logw| ≤ C(ε) + εw1+ε, and w ∈ L2nn−2 (for n > 2), we have that w logw ∈

Lp for any p < 2nn−2 . But then Lp estimates (such as [GT83, Theorem 9.11])

imply that w ∈ W 2,p, and hence w ∈ Lq for any q < 2nn−6 (or w ∈ L∞ if

n < 6). But then Lp estimates imply w ∈W 2,q, so that w ∈ Lq for q < 2nn−10

(or w ∈ L∞ if n < 10). Continuing in this way, we find that w ∈ L∞ forn < 2 + 4k after k iterations. In fact we have more, since if k is large enoughso that n < 2 + 4k, then we have w ∈ W 2,q for q > n/2, which implies thatw ∈ C0,α by the Sobolev embedding theorem [GT83, Corollary 7.11].

We now show that w logw is in C0,β for any β < α: Since 0 ≤ w ≤ Kfor some K, we have for any ε > 0 a constant C(ε) such that | logw| ≤C(ε) + w−ε. But then we have (writing ws = (1 − s)w(x) = sw(y) andassuming w(y) > w(x)) that

|(w logw)(x)− (w logw)(y)| =∣∣∣∣ˆ 1

0

(1 + logws) ds(w(y)− w(x))

∣∣∣∣≤ˆ 1

0

(C(ε) + w−εs ) ds(w(y)− w(x))

= C(ε)|w(y)− w(x)|+ 1

1− ε|w(y)1−ε − w(x)1−ε|.

2 That is, ‖w‖Lp∗ ≤ C‖w‖W 1,p for any p∗ = np/(n−p) and 1 ≤ p < n (cf. [Eva98, p.265]).


But now we observe that

w(y)1−ε − w(x)1−ε = (1− ε)ˆ 1

0

w−εs ds(w(y)− w(x)),

so we have (since w is Holder continuous with exponent α) that

|(w logw)(x)− (w logw)(y)| ≤ C(ε)|w(y)− w(x)|+ 1

1− ε|w(y)− w(x)|1−ε

≤ Cd(y, x)α(1−ε).

Thus w logw is Holder continuous with exponent α(1− ε), for any ε > 0, asclaimed. Schauder estimates [GT83, Theorem 6.2] then imply that w ∈ C2,β .In particular w is a classical solution of the equation.

A strong maximum principle (precisely, one such as is proved in [Vaz84,Theorem 1]) implies that w has a positive bound below, so that w logw isa smooth function of w, and so is C2,β . Higher regularity now follows bySchauder estimates, so w ∈ C∞(M).

Proposition 10.11 (Monotonicity of µ). If (g(t), τ(t)), for t ∈ [0, T ), isa solution to

∂g

∂t= −2Ric

∂τ

∂t= −1

on a closed manifold M with τ(t) > 0. Then

µ(g(t2), τ(t2)) ≤ µ(g(t1), τ(t1))

for all times 0 ≤ t1 ≤ t2 ≤ T .

Proof. For any t0 ∈ (0, T ], let f solve (10.4b) backwards in time on [0, t0]with

f∣∣t=t0

= arg minW(g(t0), f , τ(t0)) : f ∈ C∞(M) compatible with g and τ.

By Proposition 10.4, the monotonicity of W implies that

d

dtW(g(t), f(t), τ(t)) ≥ 0,

for all t ∈ [0, t0]. We note that compatibility is preserved by the flow (10.4)since

10.4 Local Noncollapsing Theorem 157

d

dt

ˆM

(4πτ)−n/2e−fdµ =

ˆ ( n2τ− ∂f

∂t+

1

2trg

∂g

∂t

)u dµ

=

ˆ (∂u∂t− Scalu)dµ

=

ˆ(−∗u−∆u)dµ = 0.

Hence for all times t we have that

µ(g(t), τ(t)) ≤ W(g(t), f(t), τ(t))

≤ W(g(t0), f(t0), τ(t0))

= µ(g(t0), τ(t0))

where the last equality is by construction.

In particular, by letting τ(s) = −s + r2 + t2, where t1 = 0 and t2 = t weget the following useful inequality.

Corollary 10.12. If g(t), t ∈ [0, T ), is solution to the Ricci flow on a closedmanifold M . Then for all t ∈ [0, T ) and r > 0 we have

µ(g(0), r2 + t) ≤ µ(g(t), r2). (10.8)

10.4 Local Noncollapsing Theorem

In order to prove the desired injectivity bounds, it is enough to work withvolume ratios. This is both analytically convenient and necessary since it israther difficult to work with the injectivity radius directly. We show thatthere exists a lower bounds on the volume collapsing ratio under the Ricciflow by establishing an upper bound on µ in terms of various local geometricquantities. From this we prove a stronger version of Perel’man’s ‘local non-collapsing’ result [Per02, Sect. 4] where only a pointwise bounds on the scalarcurvature Scal is required rather than the full curvature tensor R.

Proposition 10.13 ([Top05]). Let (M, g) be a closed Riemannian manifold.Then for any point p and r > 0 we have

µ(g, r2) ≤ logVolB(p, r)

rn+

(36 + r2

B(p,r)

|Scal|dµ

)VolB(p, r)

VolB(p, r/2). (10.9)

Remark 10.14. Note that the first term on the right-hand side of the inequal-ity is the desired volume ratio. We shall look to bound the other terms underreasonable conditions on the initial data.

Proof. Choose τ = r2 and let w =√u so that

´w2dµ = 1. Note that this

implies that f = −2 logw − n2 log(4πr2) and ∇f = −2∇w/w. By taking the

infimum over all f ∈ C∞(M) compatible with g and τ , we get


µ(g, r2) ≤ˆM

(r2(Scalw2 + 4|∇w|2) + f − n

)w2dµ. (10.10)

We now make a judicious choice of compatible f so that the right-hand sideof (10.10) reflects the local geometry at a point p with respect to the metricg. In particular let

f(x) = c− log φ

(dg(x, p)

r

)2

or alternatively

w(x)2 = (4πr2)−n/2φ

(dg(x, p)

r

)2

e−c

where c = c(n, g, x, r) is chosen so that´w2dµ = 1, dg(x, p) is the distance

between x and p with respect to the metric g, and φ : [0,∞) → [0, 1] is thesmooth cut-off function distributed so that φ(y) = 1 for y ∈ [0, 1/2]; φ(y) = 0for y ∈ [1,∞) with a slope chosen so that |φ′| ≤ 3 for 1/2 ≤ y ≤ 1.

Claim. The constant c satisfies the following inequality:

1

VolB(p, r)≤ (4πr2)−n/2e−c ≤ 1

VolB(p, r/2). (10.11)

Proof of Claim. As |φ| ≤ 1 and suppw ⊂ B(p, r), then

1 =

ˆw2dµ ≤ (4πr2)−n/2e−c VolB(p, r).

Furthermore, as φ(y) = 1 for 0 ≤ y ≤ 1/2 we get

ˆM

w2dµ ≥ˆB(p,r/2)

w2dµ = (4πr2)−n/2e−c VolB(p, r/2).

We now estimate each of the terms in (10.10) separately.

Term 1. By letting ψ(x) = φ(dg(x, p)/r) we note that |∇ψ| ≤ 1r sup |φ′| ≤ 3/r

and as the gradient of ψ is supported on B(p, r)/B(p, r/2) we have by (10.11)that

4r2

ˆM

|∇w|2dµ = 4r2

ˆM

(4πr2)−n/2e−c|∇ψ|2dµ

≤ 4r2

VolB(p, r/2)

ˆM

|∇ψ|2dµ

≤ 36VolB(p, r)

VolB(p, r/2).

Term 2. As φ ≡ 0 outside B(p, r) then (10.11) implies that


r2

ˆM

Scalw2dµ = r2

ˆM

Scal (4πr2)−n/2e−cψ2dµ

≤ r2

VolB(p, r)

ˆB(p,r)

|Scal|dµ.

Term 3. As the support suppw ⊂ B(p, r) and log(4π)−n/2 < 0 we have that

ˆM

fw2dµ =

ˆM

(−n2

log(4πr2)− logw2)w2dµ

= log(4πr2)−n/2ˆM

w2dµ−ˆM

w2 logw2dµ

= log(4π)−n/2 + log r−n −ˆB(p,r)

w2 log2 dµ

≤ log r−n + log VolB(p, r)

where the last line follows from Jensen’s inequality.3

Motivated by the right-hand side of (10.9), we define

νr(g) = infτ∈(0,r2]

µ(g, τ)

MR(p, r) = sup0<s≤r

s2

B(p,s)

|Scal|dµ.

So by (10.9) we have that

VolB(p, s)

sn≥ eµ(g,s2) exp

(−(36 +MR(p, s))

VolB(p, s)

VolB(p, s/2)

). (10.12)

Remark 10.15. Since s2fflB(p,s)

|Scal|dµ → 0 as s → 0, the quantity MR is a

well defined finite number for all r > 0. Trivially, if r1 ≤ r2 we getMR(p, r1) ≤MR(p, r2).

We now look to bound the volume ratios by νr and MR.

Corollary 10.16. If (M, g) is a closed manifold and 0 < s ≤ r then

VolB(p, s)

sn≥ eνr(g) exp

(− 3n(36 +MR(p, r))

).

3 Which states that if on a manifold N , ϕ is s convex function on R and v ∈ L1(N)then

N

ϕ v dµ ≥ ϕ(

N

v dµ).

In particular, if ϕ(x) = x log x and v ≥ 0 with´Nv dµ = 1 then

ˆN

v log v dµ ≥ − log VolN.


Proof. Firstly if VolB(p,s)VolB(p,s/2) ≤ 3n, that is if at the point p the volume doubling

property holds at scale s. Then by (10.12) and νr(g) ≤ µ(g, s2), the desiredestimate follows.

Now suppose VolB(p,s)VolB(p,s/2) ≥ 3n, then it is clear that

VolB(p, s/2k)

VolB(p, s/2k+1)→ 2n

as k → ∞.4 So there is a k > 0 such that VolB(p,s/2k)VolB(p,s/2k+1)

≤ 3n. However

VolB(p,s/2k)VolB(p,s/2k+1)

> 3n for all 0 ≤ i < k (i.e. it is the first such k where ratio

below 3n). Now using Proposition 10.13 with radius of s2k

we find that

µ(g, (s/2k)2

)≤ log

VolB(p, s/2k)(s

2k

)n + 3n(36 +MR(p, s/2k)).

Hence we find that

VolB(p, s)

sn≥(3

2

)nVolB(p, s/2)

( s2 )n

≥(3

2

)nkVolB(p, s/2k)(s

2k

)n≥(3

2

)nkeνr(g)e−3n(36+MR(p,r))

from which the result now follows.

Corollary 10.17. Let (M, g) be a closed manifold with 0 < s ≤ r. If Scal ≤K(n)r−2 in B(p, r), then MR(p, r) ≤ K(n) so that

VolB(p, s)

sn≥ eνr(g) exp

(− 3n(36 +K(n))

). (10.13)

We are now in a position to prove the following local noncollapsing the-orem:

Theorem 10.18 (Local Noncollapsing). Let (M, g(t)), for t ∈ [0, T ), be asolution to the Ricci flow on a closed manifold with T <∞ and let ρ ∈ (0,∞).There exists a constant κ = κ(n, g(0), T, ρ) > 0 such that for p ∈M , t ∈ [0, T )and r ∈ (0, ρ] with

Scal ≤ 1

r2

in Bg(t)(p, r), the volume ratio

4 This is intuitively clear since small balls will, up to first order, have volumes closeto that of balls of Rn (cf. Theorem 1.65).


Volg(t)Bg(t)(p, s)

sn≥ κ

for all 0 < s < r.

Proof. By (10.8) we have

ν√ρ2+T

(g(0)) ≤ νr(g(t))

for r ∈ (0, ρ] and t ∈ [0, T ). So by (10.13) the result follows.

10.4.1 Local Noncollapsing Implies Injectivity Radius Bounds.From Theorem 10.18 we can now deduce a positive lower bound on the

injectivity radius. This will complete our discussion on the blowing up atsingularities that started in Section 8.5 of Chapter 8. The main theorem isthe following:

Theorem 10.19. There exists ρ > 0 and K = K(n) > 0 such that if (M, g)is a closed Riemannan manifold satisfying |R| ≤ 1 then there exists p ∈ Msuch that

VolB(p, r)

rn≤ K

rinj(M)

for all r ∈ (0, ρ].5

In the work of Perelman [Per02] this point was considered too obvious towarrant mention. Indeed the idea is rather standard in comparison geometry:For example, predecessors to Hamilton’s compactness theorem, such as theresults of Greene-Wu [GW88] and Peters [Pet87], had no explicit assumptionon injectivity radius, instead assuming an upper bound on diameter and alower bound on volume. They then inferred a lower bound on injectivityradius from a result of Cheeger [Che70, Corollary 2.2], which is similar tothe above result but somewhat trickier since it uses a global rather thanlocal lower bound on volume. Here we present an argument similar to that in[Top05, Lemma 8.4.1], which is itself an adaptation of the original argument ofCheeger (see also [HK78], where a similar result is proved by a quite differentargument). We give a sketch of the main argument only.

We need only consider the case where the injectivity radius is small (onthe scale of the curvature), since otherwise there is nothing to prove. In sucha setting one can apply the following lemma of Klingenberg:

Lemma 10.20. If M is a compact Riemannian manifold with sectionalcurvature satisfying sect(M) ≤ 1 and inj(M) ≤ π then there exists a closedunit speed geodesic γ : R/(λZ)→M with λ = 2 inj(M).

5 Recall that the injectivity radius inj(p) of a point p is defined to be the su-premum of all r > 0 such that expp is an embedding when restricted to Br(0) (i.e.inj(p) = supr > 0 : expp is defined on dr(0) ⊂ TpM and is injective), and that theinjectivity radius of a Riemannian manifold M is inj(M) = infp∈M inj(p).


Now we show that for any point p in γ, the volume of B(p, r) is small ifthe injectivity radius is small. We observe that VolB(p, r) ≤ VolB(γ, r),where B(γ, r) = q ∈ M : d(q, γ(s)) ≤ r for some s. Now any pointq ∈ B(γ, r) can be reached by following a geodesic from a point of γ inan orthogonal direction for a distance at most r (see Figure 10.1). LetE1(s), . . . , En−1(s), En(s) = γ′(s) be an orthonormal frame for Tγ(s)M ob-tained by parallel transport, for 0 ≤ s < λ. Then define f : [0, λ)×Bn−1

r (0)→M by

f(s, V 1, . . . , V n−1) = expγ(s)

( n−1∑i=1

V iEi(s)).

That is, f(s, V ) is obtained by following the geodesic from γ(s) in directionV iEi(s) orthogonal to γ′(s) for time 1, and so travelling distance less than r.The map f is a smooth, so we can write by the change of variables formula

VolB(γ, r) ≤ˆ

[0,λ)×Bn−1r (0)

|detDf |.

The Jacobian Df can be expressed in terms of Jacobi fields, the behaviourof which is controlled once curvature is controlled.6 In particular, for r smallcompared to 1 we have |detDF | comparable to 1, so the right-hand side iscomparable to λVol(Bn−1

r (0)) = cnλrn−1. Dividing through by rn implies

that VolB(p, r)/rn ≤ Cnλ/r, for some Cn > 0, which gives the result.

Fig. 10.1 Points withindistance r of γ are ob-tained by following ortho-gonal geodesics from γ fordistance at most r.

g

p

10.5 The Blow-Up of Singularities and LocalNoncollapsing

We are now in a position to complete the blow-up of singularities resultby using the local noncollapsing theorem to obtain the desired injectivitybound discussed in Section 8.5. We follow a similar discussion presented in[Top05, Section 8.5].

As before, suppose there is a solution (M, g(t)) of the Ricci flow on amaximal time interval t ∈ [0, T ) with T < ∞. By Theorem 7.4 there exists

6 The estimate required is provided by the Rauch comparison theorem, see for ex-ample [dC92, Theorem 2.3].

10.5 The Blow-Up of Singularities and Local Noncollapsing 163

points Oi ∈M and times ti T such that

|R|(Oi, ti) = supM×[0,ti]

|R|(x, t).

With this sequence, define the blow-up metrics

gi(t) = Qig(ti +Q−1i t),

where Qi = |R|(Oi, ti). As discussed in Section 8.5, for all a < 0 and someb = b(n) > 0, the curvature of gi is bounded. In particular,

supM×(a,b)

|Scal(gi(t))| < C

for sufficiently large i = i(a) and some constant C = C(n) < ∞. Now if0 < r < 1/

√C, then |Scal(gi(0))| ≤ r−2. So if 0 < r ≤ 1/

√CQi, then

|Scal(g(ti))| ≤ r−2.By Theorem 10.18 for all p ∈ M , 0 < r ≤ 1/

√CQi and sufficiently large

i, there is a lower bound

VolBg(ti)(p, r)

rn> κ

where κ = κ(n, g(0), T ) > 0. Returning to the blow-up flows gi, we find forall p ∈M that

VolBgi(0)(p, r)

rn> κ

for all 0 < r ≤ 1/√C. So by Theorem 10.19, fix r = min 1√

C, ρ > 0 so that

inj(M, gi(0)) ≥ r

K

VolBgi(0)(p, r)

rn>

κ

Kr > 0.

In particular κr/K is a positive bound depending only on n, g(0) and T .Therefore the compactness theorem (i.e. Theorem 8.10) gives the following:

Theorem 10.21 (Blow-up of Singularities). Suppose (M, g(t)) is a solu-tion to the Ricci flow on a maximal time interval [0, T ) with finite final timeT <∞. Then there exist sequences Oi ∈M and ti T with

|R|(Oi, ti) = supM×[0,ti]

|R|(x, t)→∞

such that by defining gi(t) := Qig(ti + Q−1i t), where Qi = |R|(Oi, ti), there

exists b = b(n) > 0 and a complete Ricci flow (M∞, g∞(t)), for t ∈ (−∞, b),and a point O∞ such that

(M, gi(t), Oi) −→ (M∞, g∞(t), O∞)


as i→∞. Moreover |R(g∞(0))|(O∞) = 1 and |R(g∞(t))| ≤ 1 for t ≤ 0.

10.6 Remarks Concerning Perel’man’s Motivation FromPhysics

It is remarked by Perel’man [Per02, p. 3] that:

‘The Ricci flow has also been discussed in quantum field theory, as an approxim-ation to the renormalisation group (RG) flow for the two-dimensional nonlinearσ-model. . . this connection between the Ricci flow and the RG flow suggests thatRicci flow must be gradient-like; the present work confirms this expectation.’

In this section we expand on this comment. By doing so we see, at least inpart, where the motivation for the F and W functionals originate from.

As [Gro99] explains, the method of renormalisation group is a methoddesigned to describe how dynamics of a physical system change as the scale(i.e. distance or energies) at which we probe changes.

Physics is heavily scale dependent, for instance in fluid dynamics each scaleof distance has a different theory. For instance at length scales of 1cm, classicalcontinuous mechanics (Navier-Stokes equations) is appropriate. However atlength scales 10−13 − 10−18cm quantum chromodynamics (quarks) is a moresuitable theory.

Moreover, physics at larger scales decouples from the physics at a smallerscale. The theory at a larger scale remembers only finitely many parametersfrom the theories at smaller scales and throws the rest of the details away.Passing from smaller scales to larger scales involves averaging over irrelevantdegrees of freedom. For instance, it is possible to reconstruct the thermody-namics of a gas from molecular theory (by averaging over all configurations)but it is impossible to reconstruct the behaviour of molecules from the mac-roscopic behaviour of the gas its self. This decoupling is the reason why weare able to do physics. The aim of renormalisation group method is to explainhow this decoupling takes place and why exactly information is transmittedfrom scale to scale through finitely many parameters.

In the context of particle physics, renormalisation is a process of makingsense of ultraviolet and infrared divergences arising in the Feynman diagramintegrals. It is done in a two stage process. The first step is regularisation.It consists in introducing a cut-off which makes the integrals converge butdepend on the cut-off. There are many types of cut-off schemes, for instancemomentum cut-off, dimensional regularisation and so on. The integrals willusually tend to infinity as the cut-off goes to infinity. Because of this we needa second step called renormalisation. This step consists in making finitelymany parameters of the Lagrangian depend on the regularisation, so thatthey go to infinity as the regularisation goes to infinity but all the correlationfunctions remain finite.

The behaviour of a quantum field theory under renormalisation is governedby the so-called β-function. Formally, it is a vector field on the space of

10.6 Remarks Concerning Perel’man’s Motivation From Physics 165

dimensionless parameters for a particular quantum theory. The differentialequation associated to this vector field is called the renormalisation groupequation. Fixed points of this equation correspond the ultraviolet and infraredlimits of the theory. In the case of a single coupling constant g, the vectorfield generating the renormalisation flow is written as

µ∂

∂µ+ β(g)

∂

∂g

where µ is the scale parameter.7 When there are n such couplings g1, . . . , gn,the renormalisation group equation for gi takes the form

µdgi∂µ

= βi(g1, . . . , gn). (10.14)

In general we know nothing about the flow except that the point gi = 0 (thefree theory) is a fixed point of this flow. It is conjectured that topologically,the flow behaves as a gradient flow.

Conjecture 10.22. There exists a function F (g1, . . . , gn) such that its totalderivative with respect to µ is positive at any nonsingular point of the flow(10.14).8

The conjecture prohibits various complicated dynamical patterns which couldoccur in a multi-dimensional dynamical systems. Thus it implies that anytrajectory is either driven to a fixed point or to infinity.

According to [Gro99, p. 578], the intuition behind the conjecture is theWilsonian point of view that the (backwards) renormalisation group flowcomes from erasing degrees of freedom. The meaning of F is ‘the measureof the number of degrees of freedom’. Unfortunately it is unknown how tomake mathematical sense of this idea in general but is proved in special case2-dimensional field theories.

For instance, in the case of 2-dimensional Bosonic non-linear σ-models, weare interested in mappings φ = φ(σ, τ) : Σ →M where Σ is a 2-dimensionalRiemann surface with metric hab and M is a D-dimensional closed Rieman-nian manifold with metric gµν . The action S is of the form

7 A coupling constant is a coefficient in a Lagrangian that measures the strength ofa particular interaction among elementary fields.8 By taking a logarithmic scale λ = log µ, equation (10.14) can be re-written as

dgi

∂λ= βi(g1, . . . , gn).

The choice of the scale parameter µ is just a convention.


S =1

4πα′

ˆΣ

(gµν(φ)

∂φµ

∂xa∂φν

∂xb

+ εabBµν(φ)∂φµ

∂xa∂φν

∂xb+ α′Φ(φ)R(2)

)hab√h dσdτ

where R(2) is the world-sheet Ricci scalar with respect to hab, Φ is the so-called dilaton field, Bµν is a 2-form antisymmetic tensor field and εab is theantisymmetric tensor (taking values ε01 = −ε10 = 1 and ε00 = ε11 = 0).The constant α′ is needed to make action dimensionless. In this case thecouplings for S are gµν , Bµν and Φ. To first order the respective β-functions(see [GSW88, Sect. 3.4] and [Pol98, Sect. 3.7]) are

βgµν = −α′(Rµν + 2∇µ∇νΦ) +α′

4H λρµ Hνλρ +O(α′2)

βBµν = α′(

1

2∇λHλ

µν −Hλµν∇λΦ

)+O(α′2)

βΦµν =26−D

6+ α′

(1

2∇2Φ−∇µΦ∇µΦ+

1

24HµνρH

µνρ

)+O(α′2)

where Hµνρ = ∂µBνρ + ∂ρBµν + ∂µBρµ or simply H = dB.From this system of couplings, the formulas (9.5) and (10.3) seem to be

strongly motivated by this example. Furthermore, Perel’man’s comments in-dicate his emphasis on the gradient flow approach (even though it is notused strongly in many of the key arguments in his work on the geometrisa-tion conjecture, as the more robust reduced volume functional is employedinstead). For further discussion on the connection between Perel’man’s workand renormalisation group (RG) flows see [OSW06,Car10].

Chapter 11

An Algebraic Identity for CurvatureOperators

In this chapter and the next we look at one of the most important recentdevelopments in the theory of Ricci flow: The work of Bohm and Wilking[BW08] which gives a method for producing whole families of preserved con-vex sets for the Ricci flow from a given one. This remarkable new methodhas broken through what was an enormous barrier to further applications ofRicci flow: In particular the proof of the differentiable sphere theorem reliesheavily on this work.

The problem to be dealt with is the following: There are numerous ex-amples of convex cones in the space of curvature tensors which are knownto be preserved by the Ricci flow. Examples which have been known for along time include the cone of positive curvature operators, and the cone of2-positive curvature operators. We will see other examples in Chapter 13,such as the cone defined by the positive isotropic curvature condition. Onewould like to be able to find a whole family of preserved convex cones in-terpolating between the given cone and the ‘degenerate’ cone consisting ofconstant positive sectional curvature operators. If this can be done, then onecan argue using the maximum principle (Theorem 6.15) that solutions of theRicci flow which have their curvature in the given cone at the initial timewill evolve to have constant curvature as they approach their maximal timeof existence. In the next chapter we will give the details of the constructionof the family of cones and the argument required for their application (seeSection 6.5.3 for a discussion of how this works in the simple case of Ricciflow for 3-manifolds).

In the present chapter we discuss a fundamental identity, Theorem 11.32,due to Bohm and Wilking. It is the basis for their cone construction andis the result of a detailed and delicate analysis of how the various parts ofthe curvature tensor (first discussed in Section 3.5) combine to produce thereaction terms in the evolution equation for curvature seen in Theorem 5.21.

167

168 11 An Algebraic Identity for Curvature Operators

11.1 A Closer Look at Tensor Bundles

We begin with an examination of the framework needed to apply the vectorbundle maximum principle to the evolution equation for the curvature tensor.

In Section 6.5.3 we discussed the construction of subsets of the bundleof symmetric 2-tensors which are convex in the fibre and invariant underparallel transport, from any O(n)-invariant convex subset of the vector spaceof symmetric n×n matrices. In that situation further simplification is possiblesince symmetric matrices are diagonalisable, and the question reduces toconvex subsets of Rn invariant under interchange of coordinates.

In higher dimensions the curvature tensor is no longer simply a symmetric2-tensor, but lives in a vector bundle with considerably more complicatedstructure. We will investigate this structure in detail in the next section,however before doing so we discuss the construction of suitable subsets ofvector bundles in a more general context.

11.1.1 Invariant Tensor Bundles. In three dimensions we appliedthe vector bundle maximum principle on the bundle of symmetric 2-tensorsof the spatial tangent bundle S, which we know is a parallel subbundle ofthe bundle of 2-tensors, and so (by the arguments of Section 1.5.3) inherits acanonical metric and compatible connection. In order to carry out this step inhigher dimensions we consider an important method of constructing parallelsubbundles of tensor bundles.

Suppose we have a vector bundle E over a manifold M , with a metricg and compatible connection ∇ supplied. Then the construction given inSection 1.5.3 provides metrics and compatible connections on each of thetensor bundles T pq (E). Let O(E) be the orthonormal frame bundle of E, so

that the fibre of O(E) at a point p ∈ M consists of the isometries from Rk(with the standard inner product) to Ep (with inner product gp). Recall thatO(k) acts on O(E) by

O(k)×O(E)→ O(E); (O, Y ) 7→ Y O,

where Y O(u) = Y (Ou) for each u ∈ Rk. For each frame Y ∈ O(E)x thereexists a dual frame Y∗ for E∗x, defined by (Y∗(ω))(Y (u)) = ω(u) for all ω ∈(Rk)∗ and u ∈ Rk. There is a corresponding action of O(k) on the dualframes, defined by (Y∗)

O = (Y O)∗.As noted in Section 5.2.1, a (p, q)-tensor T at x gives rise to a function T

from O(E)x to T pq (Rk), defined by

(T (Y ))(v1, . . . , vp, ω1, . . . , ωq) = T (Y (v1), . . . , Y (vp), Y∗(ω1), . . . , Y∗(ωq))

for any v1, . . . , vp ∈ Rk and ω1, . . . , ωq ∈ (Rk)∗. The function T is equivariantwith respect to the action of O(k) on T pq (Rk) defined by

11.1 A Closer Look at Tensor Bundles 169

O(k)× T pq (Rk)→ T pq (Rk); (O, T ) 7→ TO,

where TO(v1, . . . , vp, ω1, . . . , ωq) = T (Ov1, . . . ,Ovp,O∗ω1, . . . ,O∗ωq) for allv1, . . . , vp ∈ Rk and ω1, . . . , ωq ∈ (Rk)∗, and O∗ is the dual transformationon (Rk)∗ corresponding to O, defined by (O∗ω)(Ov) = ω(v), for all ω and v.Here equivariance means that

T (Y O) = (T (Y ))O

where the action of O on the left-hand side is on the frame Y ∈ O(E), andthat on the right-hand side is on T (Y ) ∈ T pq (Rk).

An invariant subset K of T pq (Rk) is a subset which is invariant under the

action of O(k) — in the sense that T ∈ K implies that TO ∈ K for everyO ∈ O(k). In particular, a vector subspace V of T pq (Rk) which is invariant iscalled an invariant subspace.1 From any invariant subset we can construct asubset K(E) of T pq (E) as follows:

K(E) ∩ T pq (E)x =T ∈ T pq (E)x : ∀Y ∈ O(E)x, T (Y ) ∈ K

.

In particular from an invariant subspace V we construct a subbundle V (E)of T pq (E), by

V (E)x =T ∈ T pq (E)x : ∀Y ∈ O(E)x, T (Y ) ∈ V

.

We call V (E) the invariant subbundle corresponding to the invariant subspaceV . The main result we need is the following:

Proposition 11.1. If K is an invariant subset of T pq (Rk), then the subsetK(E) of T pq (E) is invariant under parallel transport.

Proof. The argument is essentially the same as that used in Section 6.5.3.1for the case of the 2-tensor bundle. Let σ : I → M be a smooth curve, andlet T be a parallel section of σ∗(T pq (E)) such that T0 ∈ K(E)σ(0). That is,

for any frame Y0 ∈ O(E)σ(0), we have T0(Y0) ∈ K. Fix any such frame, anddefine a frame Ys ∈ F (E)σ(s) for each s ∈ I by parallel transport, i.e. for

each u ∈ Rk,σ∇∂s (Ts(u)) = 0.

Then Ys ∈ O(E)σ(s) since the compatibility of ∇ gives

∂

∂sg(Ys(u), Ys(v)) = g(σ∇∂s(Ys(u)), Ys(v)) + g(Ys(u), σ∇∂s(Ys(v))) = 0.

1 Note that the action of O(k) on Tpq (Rk) defined above is an orthogonal representa-

tion, i.e. a Lie group homomorphism from O(k) to O(Tpq (Rk)). Representation theory

tells us that Tpq (Rk) decomposes as an orthogonal direct sum of irreducible invariantsubspaces. An arbitrary invariant subspace can be written as a direct sum of theseirreducible ones.


Note that the dual frame Y ∗s is also parallel by definition of the dual connec-tion. But we also have, for all v1, . . . , vp ∈ Rk and ω1, . . . , ωq ∈ (Rk)∗,

∂

∂s

(Ts(Ys)

)(v1, . . . , vp, ω1, . . . , ωq)

=∂

∂s(Ts (Ys(v1), . . . , Ys(vk), Y ∗s (ω1), . . . , Y ∗s (ωq)))

= 0,

since Ts, Ys(vi) and Y ∗s (ωj) are all parallel. Therefore Ts(Ys) is constant, andso is in K for all s ∈ I, and so Ts ∈ K(E) for all s.

The particular case where K is a subspace immediately gives the following:

Corollary 11.2. Invariant subspaces of T pq (E) are parallel.

It follows from the results of Section 1.8.7 that invariant subbundles inheritmetrics and compatible connections from the tensor bundle.

Example 11.3. There are no nontrivial invariant subbundles of E itself: Givenany non-zero vector v in Rk, there is an O(k) element which takes v to anyother given vector of the same length, so the linear span of the orbit of vunder the action is all of Rk.

Example 11.4. Since the metric tensor is invariant, the operations of raisingand lowering indices are also, and the invariant subspaces of T pq (Rk) are in

one-to-one correspondence with those of T p+q0 (Rk). In particular, E∗ has nonontrivial invariant subspaces.

Example 11.5. The invariant subbundles of the bundle of 2-tensors on E aregiven by finding the invariant subspaces of the action of O(n) on T 2

0 (Rk)(i.e. the space of k × k matrices) with the action (O,M) 7→ OTMO. Wehave already found one invariant subspace, namely the subspace of sym-

metric matrices: In this case(OTMO

)T= OTMT

(OT)T

= OTMO ifM = MT . Similarly the subspace of antisymmetric matrices is preserved.2

Another invariant subspace is the trace-free matrices, M : tr(M) = 0,since tr

(OTMO

)= tr

(MOOT

)= tr(M) for O ∈ O(k). And finally, there is

the subspace consisting of multiples of the identity matrix, which is invari-ant since OT IkO = OTO = Ik. This gives a decomposition of T 2

0 (Rk) as thefollowing direct sum of invariant subspaces:

T 20 (Rk) =

∧2(Rk)⊕ Sym2

0(Rk)⊕ RIk

where∧2

(Rk) is the antisymmetric matrices, and Sym20(Rk) is given by the

intersection of the symmetric matrices with the traceless ones.

2 We get this for free: The orthogonal complement of an invariant subspace is alwaysinvariant.

11.1 A Closer Look at Tensor Bundles 171

11.1.2 Constructing Subsets in Invariant Subbundles. We nowhave a method constructing subsets of invariant subspaces which are invariantunder parallel transport, since if K is an invariant subset of T pq (Rk) whichis contained in an invariant subspace V , then by Proposition 11.1 K(E) is asubset of V (E) which is invariant under parallel transport. Furthermore, ifK is also convex then the subset K(E) is convex in the fibres of K(V ), sinceany frame Y ∈ O(E)x gives a linear isomorphism of V (E)x to V which takesK(E)x to K, and linear isomorphisms preserve convexity.

Unlike the situation discussed in Section 6.5.3, the action of O(k) on highertensor spaces does not allow reduction to a simple form such as the diagonalmatrices, so the construction of invariant convex subsets of T pq (Rk) is not asstraightforward. However, there is a recipe which will prove quite convenientand which fits in nicely with our formulation of the maximum principle: Letc ∈ R and let ` be an arbitrary nontrivial linear function on T pq (Rk) (that

is, an element of the dual space(T pq)∗ ' T qp (Rk)). Then we can construct an

invariant convex set K`,c in T pq (Rk) as follows:

K`,c =⋂

O∈O(k)

T ∈ T pq (Rk) : `(TO) ≤ c

.

This is an intersection of half-spaces, hence convex, and is invariant by con-struction. An arbitrary invariant convex set may be constructed by takingintersections of sets of this form.

Example 11.6. The cone of positive definite matrices in Sym2(Rk) can beexpressed in exactly the form above: Define `(T ) := −T (e1, e1), and choosec = 0. Then we have

K`,c =⋂

O∈O(k)

T ∈ Sym2(Rk) : T (Oe1,Oe1) ≥ 0

=T ∈ Sym2(Rk) : ∀v ∈ Rk, T (v, v) ≥ 0

.

The other cones constructed in Section 6.5.3 may be similarly described:The cone of positive Ricci curvature (for k = 3) corresponds to `(T ) =−T (e1, e1) − T (e2, e2) and c = 0; and the cones of pinched Ricci curvaturecorrespond to `(T ) = εT (e3, e3)− T (e1, e1)− T (e2, e2) and c = 0.

11.1.3 Checking that the Vector Field Points into the Set. Toapply the maximum principle there is one further ingredient required: Wemust check that the vector field in the reaction-diffusion equation points intothe set. We will discuss the specific situation corresponding to Ricci flowin the next section, but here we discuss more generally a situation wherechecking this condition can be reduced to a rather concrete question.

Let E be a vector bundle over M ×R with a given metric and compatibleconnection. Let u be a section of an invariant tensor bundle V (E) which


satisfies the reaction-diffusion equation of the form

∇∂tu = ∆u+ F (u).

Recall that for any orthonormal frame Y ∈ O(E), we have a map which takeT ∈ V (E) to T (Y ) in the invariant subspace V . Assume that there exists avector field ψ : V → V , equivariant in the sense that (ψ(T ))O = ψ(TO) forall T ∈ V and O ∈ O(k). We require that the vector field F maps to ψ, inthe sense that for any u ∈ V (E)(x,t) and Y ∈ O(E)(x,t),

F (u)(Y ) = ψ(u(Y )).

In this case it is straightforward to check the following:

Proposition 11.7. Let K be a convex invariant subset of V , and K(E)the corresponding subset of V (E). Then the vector field F on V (E) points(strictly) into K(E) if and only if the vector field ψ on V points (strictly)into K.

Proof. The definition of F pointing into K is given in Definition 6.12. LetsK be the support function of K, so that

sK(`) = sup`(w) : w ∈ K

for each ` ∈ V ∗. Also let s be the support function of K(E), so that

s(x, t, `) = sup`(w) : w ∈ K(E)(x,t)

for each (x, t, `) ∈ V (E)∗. These are related as follows: Fix (x, t) ∈ M × R,and let Y ∈ F (E)(x,t). Then for any ` ∈ V (E)∗(x,t) we have ˜(Y ) ∈ V ∗, and

s(x, t, `) = sup`(w) : w ∈ K(E)(x,t)= sup(˜(Y ))(w(Y )) : w(Y ) ∈ K= sup(˜(Y ))(w) : w ∈ K= sK(˜(Y )).

From the definitions of tangent and normal cones in Appendix B it followsthat

Nv(Y )K = ˜(Y ) : ` ∈ NvK(x,t)

andTv(Y )K = z(Y ) : z ∈ TvK(x,t).

It follows that ψ(v(Y )) = (F (v))(Y ) is in (the interior of) Tv(Y )K if and onlyif F (v) is in (the interior of) TvK(x,t), and the Proposition is proved.

Remark 11.8. As we will see below, the evolution equation for the curvaturetensor has exactly this property. This is a drastic simplification: To check

11.2 Algebraic Curvature Operators 173

that the vector field points into the set we now have only to consider an ex-plicit vector field on a fixed finite-dimensional vector space, and ask whetherit points into a given convex invariant set. In effect the maximum principlehas removed the geometry from the analysis, and all that remains is a veryconcrete and rather algebraic question. This fact has been well understoodsince at least the early 1990s, so it is perhaps surprising that useful applic-ations of these ideas in dimensions n ≥ 4 have only been possible recently(following the appearance of [BW08]).

11.2 Algebraic Curvature Operators

The (spatial) curvature tensor can be considered a (4, 0)-tensor on the spatialtangent bundle — that is, a section of the tensor space T 4

0 (S). However, thesymmetries of the curvature tensor mean that the curvature tensor in fact liesinside an invariant subbundle, which we call the bundle of algebraic curvatureoperators Curv(S), corresponding to an invariant subspace Curv of the vectorspace T 4

0 (Rn) defined to be the set of all R ∈ T 40 (Rn) such that

R(u, v, w, z) +R(v, u, w, z) = 0

R(u, v, w, z) +R(u, v, z, w) = 0

R(u, v, w, z)−R(w, z, u, v) = 0

R(u, v, w, z) +R(v, w, u, z) +R(w, u, v, z) = 0

for all u, v, w, z ∈ Rn. One can check directly that this subspace is O(n)-invariant (in fact GL(n)-invariant).

Since elements of Curv are antisymmetric in the first and last pairs ofarguments, and symmetric under interchange of the first pair with the lastpair, it is natural to interpret them as symmetric bilinear forms acting on then(n+1)

2 -dimensional space∧2

(Rn), with each antisymmetric pair of arguments

in Rn corresponding to a single∧2

(Rn)-valued argument: That is, Curv ⊂Sym2(

∧2 Rn). Accordingly, the curvature tensor can be viewed as a self-

adjoint linear endomorphism of∧2

(S), defined by the formula

g(R(x, y)u, v) = 〈R(x ∧ y), u ∧ v〉 = R(x ∧ y, u ∧ v),

for all x, y, u, v ∈ S(p,t) ' Rn, where 〈·, ·〉 is defined by (C.2).3

By letting (ei)ni=1 denote an orthonormal basis for TpM , the curvature op-

erator R maps ei∧ej to 12

∑p,q Apqep∧eq under our adopted summation con-

vention discussed in Appendix C. By definition Ak` = 〈R(ei ∧ ej), ek ∧ e`〉 =Rijk`, in which case the curvature operator can be written in the form

R(ei ∧ ej) =∑k<`

Rijk`ek ∧ e` (11.1)

3 Note our definition here differs in sign convention from, say [Pet06, p. 36].


with respect to the orthonormal basis (ei ∧ ej)i<j . Likewise one can ex-press R when interpreted as a symmetric bilinear form, i.e. an element ofSym2(

∧2T ∗pM), by R =

∑i<j,k<`Rijk`(e

i ∧ ej)⊗ (ek ∧ e`).4

Now consider a different orthonormal basis (σα)Nα=1 for∧2

TpM with cor-responding dual (ϕα). With respect to this basis, the curvature R (as a bi-linear form) can be expressed as R = Rαβϕ

α ⊗ ϕβ , where Rαβ = R(σα, σβ).Since ϕα =

∑i<j ϕ

αije

i∧ej and σα =∑i<j σ

ijα ei∧ej we find the components

of R, with respect to the two orthonormal bases, are related by

Rijk` = Rαβϕαijϕ

βk` and Rαβ = Rijk`σ

ijα σ

k`β .

11.2.1 Interpreting the Reaction Terms. We first observe that theevolution equation for the curvature tensor is exactly of the form describedin Section 11.1.3, in the sense that the reaction terms are described by aninvariant vector field. Recall the evolution equation derived in Theorem 5.21:

∇∂tRijkl = ∆Rijkl + 2(Bijkl −Bijlk +Bikjl −Biljk).

We seek to understand the quadratic reaction terms

Qijk` = Bijk` −Bij`k +Bikj` −Bi`jk, (11.2)

which we introduced in Chapter 3.5 The first observation is that these termsare indeed invariant, since they are given simply by contracting with themetric. Thus according to Proposition 11.7 we need only consider the corres-ponding vector field Q on the invariant subspace Curv. By appealing to theinherent Lie algebra structure associated with the curvature operator we willrewrite these as the sum of two natural quadratic vector fields on Curv.

To achieve this formally, first identify U ⊗U ' gl(U) where the isomorph-

ism given by (C.1), with U =∧2 Rn. Then identify

∧2 Rn ' so(n) with theisomorphism given by (C.3). The quadratic curvature terms, referred to asthe squared and sharp products respectively, are defined by:

R2αβ = RαλRλβ

R#αβ =

1

2cγηα cδθβ RγδRηθ

where cβγα are the structure constants for so(n) with respect to the basis (ϕα).One can easily show that if R ≥ 0 then R2 ≥ 0 and R# ≥ 0.

4 Every symmetric bilinear form β : V × V → R defines a mapping V → V ∗; v 7→(w 7→ β(v, w)) and visa versa. Here we have implicitly identified V ' V ∗ to conformto our adopted summation convention.5 Recall by (3.2) that Bijk` = RipjqRpkq` where, in this algebraic context, wesuppress the index raising.


Remark 11.9. The sharp product is a contraction with the structure constantsand the squared product is ‘natural’ in the sense that:

R2 = (Rαβϕα ⊗ ϕβ)(Rγδϕ

γ ⊗ ϕδ)= RαβRγδδ

βγϕα ⊗ ϕδ

= RαλRλβϕα ⊗ ϕβ

where the second equality is due to (C.1).

With these identifications, we find that the vector field Q on Curv can bewritten in the form Q(R) = R2 + R#. This is an immediate consequence ofthe following two lemmas.

Lemma 11.10. For any R ∈ Curv,

R2ijk` = Bijk` −Bij`k.

Proof. Since Bijk` = Bji`k = Bk`ij we find that

RijpqRk`pq = (Ripqj +Riqjp)(Rkpq` +Rkq`p)

= (Riqjp −Ripjq)(Rkq`p −Rkp`q)= B`kji −Bjik` −Bij`k +Bijk`

= 2(Bijk` −Bij`k).

In which case ∑p,q

RijpqRpqk` =∑p,q

Rαβϕαijϕ

βpqRγδϕ

γpqϕ

δk`

= RαβRγδϕαijϕ

δk`

∑p,q

ϕβpqϕγpq

= 2RαβRγδϕαijϕ

δk`δ

βγ

= 2R2ijk`

since δβγ =⟨ϕβ , ϕγ

⟩= 1

4ϕβijϕ

γk`

⟨ei ∧ ej , ek ∧ e`

⟩= 1

2

∑p,q ϕ

βpqϕ

γpq.

Lemma 11.11. For R ∈ Curv,

R#ijk` = Bikj` −Bi`jk.

Proof. From (C.5) we find that


Bikj` −Bi`jk = RipkqRjp`q −Rip`qRjpkq= Rαβϕ

αipϕ

βkqRγδϕ

γjpϕ

δ`q −Rαβϕαipϕ

β`qRγδϕ

γjpϕ

δkq

= ϕαipϕγjp(ϕ

β`qϕ

δqk − ϕδ`qϕ

βqk)RαβRγδ

= ϕαipϕγjp[ϕ

β , ϕδ]`kRαβRγδ

= ϕαipϕγjpc

βδη ϕ

η`kRαβRγδ

=1

2(ϕαipϕ

γjp − ϕ

γipϕ

αjp)ϕ

η`kc

βδη RαβRγδ

=1

2cγαθ cβδη RαβRγδϕ

θijϕ

ηk` = R#

ijk`

Remark 11.12. We note that neither R2 nor R# satisfy the first Bianchi iden-tity. However, their sum does. In particular:

Proposition 11.13. For any R ∈ Curv, Q(R) ∈ Curv.

Proof. The symmetry and antisymmetry identities are trivial. To check theBianchi symmetry, we see that

2(Qijk` +Qik`j +Qi`jk) = RijpqRk`pq + 2RipjqR`pkq − 2RipjqRkp`q

+RikpqR`jpq + 2RipkqRjp`q − 2RipkqR`pjq

+Ri`pqRjkpq + 2Rip`qRkpjq − 2Rip`qRjpkq

= RijpqRk`pq − 2RipjqRk`pq

+RikpqR`jpq − 2RipkqR`jpq

+Ri`pqRjkpq − 2Rip`qRjkpq

since R satisfies the first Bianchi identity. Thus

2(Qijk` +Qik`j +Qi`jk) = (Rijpq −Ripjq +Riqjp)Rk`pq

+ (Rikpq −Ripkq +Riqkp)R`jpq

+ (Ri`pq −Rip`q +Riq`p)Rjkpq

= 0.

Example 11.14. To get a feel for the quadratic structure, in particular thatof the problematic sharp term, we look at the special case when n = 3. Insuch a situation N = n(n− 1)/2 = 3, and the Lie algebra so(3) ' R3 — viathe isomorphism ι defined in Section 6.5.3 — so that [X,Y ]i = (X × Y )i =εijkXjYk (cf. Example C.1). In which case (as described in Section 6.5.3)the curvature operator becomes the tensor Λ ∈ Sym2(R3), and the structureconstants

cαβγ = (ϕα × ϕβ) · ϕγ = εαβγ

equal elementary alternating 3-tensors (i.e. the volume form). Thus we have


Λ#αβ =

1

2cγηα cδθβ Λγδ =

1

2εαγηεβδθΛγδΛηθ

and we claim that:

Claim. The sharp term equals the cofactor matrix. That is,

Λ#αβ = (adjΛ)βα.

To see this, note that Λ−1αβ = (2 detΛ)−1εαpqεβabΛapΛbq.

6 So by Lemma 3.13

we find that Λ# = detΛ · tΛ−1 = t(adjΛ). Therefore if

Λ =

a b cd e fg h k

we see that Λ2 corresponds to the usual matrix product and

Λ# = detΛ · tΛ−1 =

ek − fh fg − dk dh− egch− bk ak − cg bg − ahbf − cd cd− af ae− bd

.

In particular if Λ = diag(a, e, k), then Λ# = diag(ek, ak, ae).

11.2.2 Algebraic Relationships and Generalisations. To presentthe results of Bohm and Wilking we need to extend the above squared andsharp products a little. By setting V = Rn, we define the circle and sharpoperators ,# : S2(

∧2V ∗)× S2(

∧2V ∗)→ S2(

∧2V ∗) to be

R S =1

2(RS + SR)

(R#S)αβ = (R#S)(σα, σβ) =1

2cγηα cδθβ RγδSηθ

respectively.

Remark 11.15. From the antisymmetry of cαβγ , R#S = S#R. Moreover, in

what is to come we shall let I denote the identity element in S2(∧2

V ∗).However, one should note R#I is not equal to R in general.

Now with these definitions in place, Lemma 11.10 and 11.11 can easily beextended to the following result.

Lemma 11.16. For R,S ∈ S2(∧2

V ∗),

(R S)ijk` =1

2RijpqSpqk`

(R#S)ijk` = RipkqSjp`q −Rip`qSjpkq.6 Since (detΛ)εijk = εpqrΛipΛjqΛkr.


Furthermore, the quadratic term defines a bilinear operator Q : S2(∧2

V ∗)×S2(∧2

V ∗)→ S2(∧2

V ∗) by polarisation:

Q(R,S) = R S +R#S. (11.3)

Related to this is the operator Q : S2(∧2

V ∗) → S2(∧2

V ∗) which is takento be Q(R) = Q(R,R) = R2 +R#.

11.2.2.1 Huisken’s Trilinear Form. There is a trilinear form on S2(∧2

V ∗)defined by

tri(R,S, T ) = 2 〈Q(R,S), T 〉 = 2 tr((R S +R#S)T

).

The trilinear form is symmetric in all three components, since it is straight-forward to check that

tr((R#S)T

)=

1

2

N∑α,β,γ=1

〈[R(σα), S(σβ)], T (σγ)〉〈[σα, σβ ], σγ〉 ,

which is clearly symmetric in all three components.The vector bundle maximum principle (together with Corollary 6.16) al-

lows us to produce curvature conditions preserved by the Ricci flow pde fromsets preserved by flow of the vector field Q on Curv. So it is of great interestto construct functions which are monotone under this flow.

Proposition 11.17. The ode ddtR = Q(R) is the gradient flow of

P (R) =1

3tr(R3 +RR#) =

1

6tri(R,R,R).

Proof. Let ∂∂tR = S and observe — by the total symmetry of tr(R#S)T —

that

∂

∂tP (R) =

1

3tr(SR2 +RSR+R2S + (R#S + S#R) R+R# S)

= tr((R2 +R#)S

)=⟨R2 +R#, S

⟩.

What is more, this can be improved to give a scaling-invariant monotonefunction:

Proposition 11.18. Under the ode ddtR = Q(R), the radial projection R =

R/|R| onto the unit sphere evolves in the direction of the gradient of the

scaling-invariant function P (R). In particular P (R) = P (R)/|R|3 is strictlyincreasing except at points where Q(R) = λR for some λ ∈ R.

Proof. We directly compute:


d

dtP (R) =

1

|R|3⟨Q(R),

d

dtR⟩− 6P (R)

|R|5⟨R,

d

dtR⟩

=1

|R|3(|Q(R)|2 − tri(R,R,R)

|R|2〈R,Q(R)〉

)=

1

|R|3∣∣∣Q(R)− 〈Q(R), R〉

|R|2R∣∣∣2,

since 〈Q(R), R〉 = tri(R,R,R).

In particular, this says that the super-level sets of P (R) are cones in thespace of curvature operators, and the vector field Q(R) points into them.The bad news is, however, that these cones are in general not convex (seeFigure 11.1) so that the maximum principle unfortunately does not apply.

Fig. 11.1 Level sets forP (R) restricted to theplane R : trR = 1for n = 3. The centrepoint corresponds tocurvature of S3, andthe triangle correspondsto the cone of positivesectional curvature, withthe vertices correspondingto the curvature of S2×R.

11.2.2.2 The Wedge Product. The and # products enable one to con-struct a new element in S2(

∧2V ∗) from two given elements. It is important to

note there is another method of constructing elements in S2(∧2

V ∗) from twoelements of S2(V ). Following [BW08, p. 1082], we define the wedge product

A ∧B :∧2

V →∧2

V of two elements A,B ∈ gl(n,R) ' V ⊗ V by

(A ∧B)(x ∧ y) :=1

2

(A(x) ∧B(y) +B(x) ∧A(y)

). (11.4)

It is easy to see that A ∧ B = B ∧ A if A,B ∈ S2(V ). By using the wedge,

there is a natural inclusion map id∧ : S2(V )→ S2(∧2

V ∗) given by

A 7→ A ∧ id.

Note that I = id∧ id, so Iijk` = δikδj`− δi`δjk is the curvature tensor on thestandard sphere. When restricted to the space of symmetric 2-tensors, thewedge product is (up to a constant) equal to the Kulkarni-Nomizu product,namely:


Fig. 11.2 Top and side views of the level sets of P (R) restricted to the unit sphere,showing the lines where one of the sectional curvatures vanishes. The intersectionpoints of these lines are the curvature operators of S2 × R and H2 × R, and thecurvature of S3 and of H3 are at the ‘top’ amd ‘bottom’ of the sphere.

〈(A ∧B)(ei ∧ ej), ek ∧ e`〉 =1

2〈AipBjqep ∧ eq +BipAjqep ∧ eq, ek ∧ e`〉

=1

2(AikBj` +Aj`Bik −Ai`Bjk −AjkBi`)

=1

2(A?B)ijk`. (11.5)

It is advantageous to define the Ricci operator Rc : S2(∧2

V ∗) → S2(V )by

〈Rc(R)(ei), ej〉 =

n∑k=1

〈R(ei ∧ ek), ej ∧ ek〉 = Rikjk,

for all R ∈ S2(∧2

V ∗). Note that Rc(R) = Ric as expected. Similarly, thescalar operator scal is defined by

scal(R) := tr Rc(R) = Rikik,

for all R ∈ S2(∧2

V ∗), where scal(R) = Scal as expected.It is interesting to note:

Proposition 11.19. The operator 2 id∧ is the adjoint of Rc.

Proof. Using the trace norms on S2(V ) and S2(∧2

V ∗) we compute:

11.3 Decomposition of Algebraic Curvature Operators 181

〈2 id∧(A), R〉 = (2A ∧ id)αλRλα

=1

4(A? id)ijk`Rk`ij

=1

4(Aikδj` +Aj`δik −Ai`δjk −Ajkδi`)Rk`ij

= AikRicki

= 〈A,Rc(R)〉

For future computations we also observe: If A,B ∈ S2(V ) then

〈Rc(A ∧B)(ei), ej〉 =∑k

〈(A ∧B)(ei ∧ ek), ej ∧ ek〉

=1

2(AijBkk +BijAkk −AikBkj −BikAkj)

=1

2(Aijtr(A)−AikBkj +Bijtr(A)−BikAkj),

which implies

Rc(A ∧B) =1

2A(tr(B) id−B

)+

1

2B(tr(A) id−A

).

In particular we find that

Rc(A ∧A) = tr(A)A−A2 and Rc(A ∧ id) =n− 2

2A+

tr(A)

2id. (11.6)

So if tr(A) = 0 then

Rc(A ∧A) = −A2 and Rc(A ∧ id) =n− 2

2A. (11.7)

11.3 Decomposition of Algebraic Curvature Operators

The space of curvature operators can be decomposed in a directly analogousway to Section 3.5. By explicitly identifying

∧2V ∗ ' so(n), there is a natural

decomposition of

S2(so(n)) = 〈I〉 ⊕ 〈Ric〉 ⊕ 〈W 〉 ⊕

∧4(Rn)

into O(n)-invariant, irreducible, pairwise inequivalent subspaces, whenevern ≥ 4. The non-trivial aspect of the statement is the irreducibility of thedecomposition, however this plays only a minor role in our discussion. Adetailed algebraic proof of this fact can be found in [GW09, Sect. 10.3] (onecould also consult [Bes08, p. 45]).

An endomorphism R ∈ S2(so(n)) is satisfies the first Bianchi identity ifand only if it has no component in the last factor, so we have


Curv = 〈I〉 ⊕ 〈Ric〉 ⊕ 〈W 〉 .

Now given a curvature operator R ∈ Curv let RI , RRic , RW denote the

projections onto 〈I〉, 〈Ric〉 and 〈W 〉 respectively, so that

R = RI +RRic +RW .

Here, naturally, 〈I〉 denotes the subspace generated by the identity, 〈Ric〉

denote the traceless Ricci subspace and 〈W 〉 denotes the Weyl part. For suchan R, also define

λ =scal(R)

nand σ =

‖Rc0(R)‖2

n,

where Rc0(R) = Rc(R)− scal(R)n id.

In a direct comparison with (3.14) and (3.15), an algebraic curvature op-erator can be decomposed as

R =λ

n− 1id ∧ id +

2

n− 2Rc0(R) ∧ id +RW (11.8)

or

R =−scal(R)

(n− 1)(n− 2)id ∧ id +

2

n− 2Rc(R) ∧ id +RW .

Proposition 11.20. For any A ∈ S2(V ) with tr(A) = 0,

A ∧A = − tr(A2)

n(n− 1)id ∧ id− 2

n− 2(A2)0 ∧ id + (A ∧A)W

where (A2)0 is the trace-free part of A2.

Proof. Using (11.8) with (11.7) one finds

A ∧A =tr(−A2)

n(n− 1)id ∧ id +

2

n− 2

(−A2 − scal(−A2)

nid)∧ id + (A ∧A)W ,

from which the result follows.

11.3.1 Schur’s Lemma. With equation (11.8) in mind, we relate mani-folds with constant sectional curvature to the decomposition of the curvatureendomorphism. We will use Schur’s Lemma in the final argument discussedin Chapter 14, since we seek to deform our manifold into a space form.

Lemma 11.21. A Riemannian manifold M has constant sectional curvature,equal to κ0, if and only if R = κ0 id ∧ id, where R is the curvature of M .

Proof. Suppose, for each p ∈M , that K(Π) = κ0 for all 2-planes Π ≤ TpM .From Section 1.7.6 we have that


R(x, y, x, y) = κ0(|x|2|y|2 − 〈x, y〉2)

= 〈κ0(id ∧ id)(x ∧ y), x ∧ y〉 ,

for any x, y ∈ TpM linearly independent. By (11.5), id ∧ id has the samesymmetry properties as the curvature R, so uniqueness gives the result. Theconverse is immediate.

We shall say that a manifold M is isotropic if, for each p ∈M , the sectionalcurvature Kp(Π) is independent of the 2-plane Π ≤ TpM . The followinglemma of Schur [Sch86] show the distinction between isotropic manifolds andmanifolds of constant curvature is artificial.

Lemma 11.22 (Schur). Let Mn be a connected Riemannian manifold withn ≥ 3. If Kp(Π) = f(p), for all 2-planes Π ≤ TpM and p ∈ M , then fmust be constant. In other words, all isotropic manifolds M have constantsectional curvature.

Proof. For each p ∈M , our hypothesis implies that

R∣∣p(x, y, x, y) = f(p)(|x|2|y|2 − g(x, y)2)

for all x, y ∈ TpM linearly independent. So by Lemma 11.21 we must haveR = 1

2f g ? g. In which case 2∇R = (∇f) g ? g + f (∇g) ? g + f g ? (∇g) =(∇f) g ? g, since ∇g ≡ 0. Using the second Bianchi identity for R gives

0 = (Uf)(g(W,X)g(Z, Y )− g(Z,X)g(W,Y )

)+ (Xf)

(g(W,Y )g(Z,U)− g(Z, Y )g(W,U)

)+ (Y f)

(g(W,U)g(Z,X)− g(Z,U)g(W,X)

)for all X,Y,W,Z,U ∈X (M).

If X ∈ TpM is an arbitrary tangent vector to the manifold, then it ispossible to choose Y and Z such that X,Y, Z are orthogonal (since dimM ≥3). By setting U = Z at p, the above equation implies that g(X(f)Y −Y (f)X,W ) = 0 for all W . Since X and Y are linearly independent, we haveX(f) = Y (f) = 0. Since X is arbitrary, the derivative of f vanishes, and fis locally constant. Since M is connected, f is constant.

11.3.2 The Q-Operator and the Weyl Subspace. An important stepof Bohm and Wilking is to establish the relationship between Q, defined by(11.3), and the various curvature subspace discussed in the above.

Proposition 11.23. For any R ∈ Curv,

Rc(R2 +R#)ij = Rc(R)k`Rikj`

scal(R2 +R#) = Rc(R)k`Rc(R)k`.

In particular, if R ∈ 〈W〉 then Q(R) = R2 +R# ∈ 〈W〉.


Proof. From Lemma 11.10 and 11.11,

(R2 +R#)ijkj = Qijkj = Bijkj −Bijjk +Bikjj −Bijjk= RpiqjRpkqj − 2RpiqjRpjqk +RipkqRpjqj .

By the first Bianchi identity,∑p,j

RiqpjRkrpj =∑p,j

(RipqjRkrpj +RijpqRkrpj)

=∑p,j

(RipqjRkrpj +RipjqRkrjp) = 2∑p,j

RipqjRkrpj .

In which case,

RpiqjRpjqk = RipqjRkqpj =1

2RiqpjRkqpj =

1

2RqipjRqkpj .

Thus (R2 +R#)ijkj = RipkqRpjqj = RipkqRc(R)pq.Now if R ∈ 〈W 〉 then Rc(R) = 0, so Rc(R2 + R#)ik = Ripkq

Rc(R)pq = 0and scal(R2 +R#) = 0. Hence R2 +R# ∈ 〈W 〉.

Proposition 11.24. If R ∈ 〈Ric〉 and S ∈ 〈W 〉 then Q(R,S) ∈ 〈Ric

〉.

Proof. For S,W ∈ 〈W 〉 and R ∈ 〈Ric〉, it suffices to show:

tri(S,R,W ) = 0

tri(S,R, I) = 0

To prove the first, recall that tri is totally symmetric, so that tri(S,R,W ) =tri(W,S,R). By Proposition 11.23, (S + W )2 + (S + W )# ∈ 〈W 〉 and so

WS + SW + 2W#S ∈ 〈W 〉. Therefore as R ∈ 〈Ric〉,

0 = 〈WS + SW + 2W#S,R〉 = tr((WS + SW + 2W#S)R

)= tri(W,S,R).

To prove the second, we find that

tri(S,R, I) = tri(S, I,R) = 2tr((S + S#I)R

)= 0,

where the last equality is due to Lemma 11.25 below.Since the identities can be rewritten as

〈SR+RS + 2R#S,W 〉 = 0

〈SR+RS + 2R#S, I〉 = 0

one must have SR+RS + 2S#R ∈ 〈Ric〉. The result now follows.


11.3.3 Algebraic Lemmas of Bohm and Wilking. For the compu-tation purposes of the next section, it is necessary to examine the followinglemmas of [BW08, Sect. 2].

Lemma 11.25. For R ∈ Curv,

R+R#I = (n− 1)RI +n− 2

2RRic = Ric ∧ id.

Remark 11.26. The lemma implies Q(R, I) = R+R#I has no Weyl part, i.e.(R+R#I)W = 0.

Proof. By Lemma 11.16,

(R#I)ijk` = RipkqIjp`q −Rip`qIjpkq= Ripkpδj` −Ri`kj −Rip`pδjk +Rik`j

= Ricikδj` − Rici`δjk −Rijk`

since Iijk` = 〈I(ei ∧ ej), ek ∧ e`〉 = 〈ei ∧ ej , ek ∧ e`〉 = δikδj` − δi`δjk. Usingthis we find that

(Ric ∧ id)ijk` =1

2(Ricikδj` + Ricj`δik − Rici`δjk − Ricjkδi`)

=1

2

((R+R#I)ijk` − (R+R#I)jik`

)= Rijk` + (R#I)ijk`,

where the last equality is due to

(R#I)ijk` = (R#I)αβϕαijϕ

βk` =

1

2cγηα cδθβ RγδIηθϕ

αijϕ

βk`

= −(R#I)αβϕαjiϕ

βk` = −(R#I)jik`.

We say a curvature operator R is of Ricci type if R = RI + RRic , i.e.

RW = 0.

Lemma 11.27. If R ∈ Curv be a curvature operator of Ricci type, then

R2 +R# =1

n− 2Ric∧ Ric

+2λ

n− 1Ric∧ id− 2

(n− 2)2(Ric 2)0 ∧ id

+

(λ2

n− 1+

σ

n− 2

)I.

Proof. Recall that RI = λn−1I and R0 := RRic

= 2n−2Ric

∧ id, so that


R2 +R# = (RI +R0)2 + (RI +R0)#

= R20 +R#

0 +R2I +R#

I + 2(RIR0 +RI#R0)

= R20 +R#

0 +λ2

(n− 1)2(I + I#I) +

2λ

n− 1(R0 +R0#I).

As the last two summands are known by Lemma 11.25, it suffices to show:

Claim 11.28.

R20 +R#

0 =1

n− 2Ric∧ Ric− 2

(n− 2)2(Ric 2)0 ∧ id +

σ

n− 2I

To prove this claim, choose an orthonormal basis (ei)ni=1 of eigenvectors of

Ric

with corresponding eigenvalues (λi)ni=1. The curvature operator R0 is

diagonal with respect to (ei ∧ ej)i<j , with eigenvalues given by

R0(ei ∧ ej) =1

n− 2

(Ric

(ei) ∧ ej + ei ∧ Ric

(ej))

=λi + λjn− 2

ei ∧ ej .

On the left-hand side of the claim this implies that

R20(ei ∧ ej) =

(λi + λjn− 2

)2

ei ∧ ej .

Furthermore, with

(R0)ijk` = 〈R0(ei ∧ ej), ek ∧ e`〉 =λi + λjn− 2

(δikδj` − δjkδi`)

and Lemma 11.16, one has

R#0 (ei ∧ ej) =

1

2(R#

0 )ijk`ek ∧ e`

=1

2(R0)ipkq(R0)jp`qek ∧ e` − (k ↔ `)

=1

2

∑p,q

λi + λpn− 2

λj + λpn− 2

(δikδpq − δiqδpk)(δj`δpq − δjqδp`)ek ∧ e`

− (k ↔ `)

=1

2

(∑p

λi + λpn− 2

λj + λpn− 2

ei ∧ ej

− λi + λjn− 2

λj + λjn− 2

ei ∧ ej −λi + λin− 2

λj + λin− 2

ei ∧ ej)− (k ↔ `)

=∑p 6=i,j

λi + λpn− 2

λj + λpn− 2

ei ∧ ej .

11.4 A Family of Transformations for the Ricci flow 187

Meanwhile, on the right-hand side one finds that((Ric 2)0 ∧ id

)(ei ∧ ej) =

1

2

((Ric 2)0(ei) ∧ ej + ei ∧ (Ric

2)0(ej))

=λ2i + λ2

j − 2σ

2ei ∧ ej ,

since (Ric

2)0(ei) = (Ric

2(ei)− σ)ei = (λ2i − σ)ei. From this, we see that(

1


(n− 2)2(Ric 2)0 ∧ id +

σ

n− 2I

)(ei ∧ ej)

=

(λiλjn− 2

+nσ − λ2

i − λ2j

(n− 2)2

)ei ∧ ej .

Since λk is the k-th eigenvalue of the traceless Ricci tensor Ric

, the sum∑k λk is the trace (which is zero). Hence∑

k 6=i,j

λk = −λi − λj∑k 6=i,j

λ2k = nσ − λ2

i − λ2j .

Thus we find that(λi + λjn− 2

)2

+∑p 6=i,j

λi + λpn− 2

λj + λpn− 2

=1

(n− 2)2

((n− 2)λiλj + nσ− λ2

i − λ2j

)from which the claim is proved.

11.4 A Family of Transformations for the Ricci flow

So far we have established the relationship between curvature conditions pre-served by the Ricci flow and convex subsets of the vector space Curv whichthe vector field Q points into. We call such a set a preserved set , and inparticular a cone which is preserved is called a preserved cone. We seek tofind a whole family of preserved convex cones interpolating between a giveninitial cone C and the ‘degenerate’ cone consisting of constant positive sec-tional curvature operators. This is done by considering an appropriate lineartransformation of Curv which maps the cone C to a new one. We show thatthe new cone is preserved if and only if the vector field Q pulls back by thegiven transformation to a vector field which points into C. So if the originalcone C is preserved by the Ricci flow, in the sense that Q points into it,then it suffices to show that the difference between Q and the pulled back


vector field points into C. It is this difference which we explicitly compute inTheorem 11.32, and hence control in the next chapter.

The question remains as to what transformation will work. Since we wantto map convex O(n)-invariant cones to convex O(n)-invariant cones, it makessense to consider linear maps l : Curv → Curv which are O(n)-equivariant,so that l(G(R)) = G(l(R)) for all G ∈ O(n) and R ∈ Curv. The equivarianceimplies that every eigenspace of ` is O(n)-invariant,7 and hence is a directsum of irreducible components of Curv. The image and kernel of ` on eachcomponent is again invariant, and the components are pairwise inequivalent,so the only possibility is that ` maps each irreducible component either to zeroor to itself. Since the eigenspaces of ` are invariant subspaces, ` is necessarilya real multiple of the identity on each component.

With this insight, a naıve idea would be to consider a linear transformationlc(R) = R + cRI which magnifies only the scalar curvature part. It is easyto see that lc(C) converges to R+I whenever c → ∞ (since 1

c lc(R) → RI asc→∞). However, it is not true in general that the vector field Q(R) pointsinto these cones. It turns out to be important to allow a more general familyof linear operators, to give more freedom to select a family of transformationswhich have the required behaviour (in fact, since scalings of a cone do notchange it, the family defined below are in effect all of the O(n)-invariantlinear transformations of Curv, though it is convenient to choose the overallscaling to leave the Weyl part unchanged):

Definition 11.29. For any a, b ∈ R, define the O(n)-invariant linear trans-formation la,b : Curv→ Curv by

la,b(R) = R+ 2(n− 1)aRI + (n− 2)bRRic . (11.9)

Remark 11.30. Note that each transformation la,b preserves the Weyl partand is a multiple of the identity on each of the other two irreducible com-ponents. By the decomposition (11.8), the transformation can be rewrittenas

la,b(R) = (1 + 2(n− 1)a)RI + (1 + (n− 2)b)RRic +RW .

It is easy to see that l0,0(R) = R and la,b is invertible provided both a 6=− 1

2(n−1) and b 6= − 1(n−2) (we will always assume this is the case). Note if

a→∞ while b/a→ 0, la,b(R)/|la,b(R)| → I/|I| with scal(R) > 0, so la,b(C)converges to the line of positive constant sectional curvature operators.

Bohm and Wilking work with these transformations by bringing the dy-namics back to the original cone C. Thus we need to determine the vectorfield obtained by pulling back the vector field Q from the transformed cone`a,b(C) to C. This is given by a vector field Xa,b defined as follows:

7 Since eigenspaces can be complex, we must use the fact that the complexifications ofour real irreducible representations are irreducible representations of O(n,C), and sohave no nontrivial invariant complex subspaces. See [GW09, Sect. 10.3.2] for details.


Xa,b(R) = l−1a,b

(la,b(R)2 + la,b(R)#

)= (l−1

a,b Q la,b)(R). (11.10)

The difference, denoted by Da,b, between Xa,b and the original vector fieldfor the Ricci flow is defined by

Da,b(R) = (l−1a,b Q la,b)(R)−Q(R)

= Xa,b(R)−Q(R).

From this we find that:

Lemma 11.31. The vector field Q strictly points into la,b(C) (in the senseof Definition 6.12) if and only if the vector field Xa,b strictly points into C.

Proof. Note that ∂(la,bC) = la,b(∂C), so that points in ∂(la,bC) have theform la,bR for some R ∈ ∂C. By definition, Q strictly points into la,b(C) iffor any la,bR ∈ ∂la,b(C) we have Q(la,bR) ∈ Int

(Tla,bRla,b(C)

). Now

Tla,b(R) (la,bC) =⋃h>0

h−1 (la,bC − la,bR)

= la,b

( ⋃h>0

h−1 (C −R))

= la,b (TRC) ,

and the same holds for the interiors. Thus

Q(la,bR) ∈ Int(Tla,bRla,b(C)) if and only if l−1a,b (Q(la,bR)) ∈ Int (TRC)

which says precisely that Xa,b(R) points strictly into C.

The idea now is as follows: Suppose we know that C is preserved by Ricciflow, in the sense that the vector field Q points into C. We want Xa,b =Q + Da,b to point strictly into C, so it suffices to prove that Da,b pointsstrictly into C.

The miraculous discovery of Bohm and Wilking is that the operatorDa,b(R) is independent of the Weyl component of R, and so can be simplycomputed in terms of the eigenvalues of the Ricci curvature of R. This greatlysimplifies the task of understanding how R2 + R# changes under the linearmap la,b.

Theorem 11.32 (Main Formula for Da,b). For any a, b ∈ R,

Da,b(R) =((n− 2)b2 − 2(a− b)

)Ric∧ Ric

+ 2aRic ∧ Ric + 2b2 Ric 2 ∧ id

+tr(Ric

2)

n+ 2n(n− 1)a

(nb2(1− 2b)− 2(a− b)(1− 2b+ nb2)

)I.


Proof. We first establish that Da,b(R) is independent of the Weyl part of R.The precise form of Da,b(R) can then be explicitly computed from the Weylfree part of R.

Claim 11.33. The algebraic curvature operator Da,b(R) is independent ofthe Weyl component of R. That is,

Da,b(R+ S) = Da,b(R)

for all S ∈ 〈W 〉 and R ∈ Curv.

Proof of Claim. First we note that if S ∈ 〈W 〉 then la,b(S) = S, and hence

Da,b(S) = l−1a,bQ(S)−Q(S) = 0 (11.11)

since Q(S) ∈ 〈W 〉 by Proposition 11.23. We view D = Da,b as a quadraticform in R, and let

B(R,S) :=1

4

(D(R+ S)−D(R− S)

)= l−1

a,bQ(la,b(R), S)−Q(R,S)

be the corresponding bilinear form.It suffices to show B(R,S) = 0 for all S ∈ 〈W 〉 and R ∈ Curv: In this case

D(R+S)−D(R) = (D(R) + 2B(R,S) +D(S))−D(R) = 0, since D(S) = 0for any S ∈ 〈W 〉 by (11.11).

We fix S = W ∈ 〈W 〉 and prove B(S, · ) = 0 by considering B(R,S) forR in each of the O(n)-irreducible components of Curv:

1. Suppose R ∈ 〈W 〉. Then by (11.11) we have D(R+ S) = D(R− S) = 0,so B(R,S) = 0.

2. When R = I is the identity, la,b(I) = (1 + 2(n − 1)a)I =: (1 + α)I byDefinition 11.29, so

B(W, I) = l−1a,b ((1 + α)Q(I,W ))−Q(I,W ).

However we have by the polarisation identity

Q(I,W ) =1

4(Q(R+ I)−Q(R− I))

=1

4

((W + I)2 + (W + I)# − (W − I)2 − (W − I)#

)= (W +W#I) ,

which vanishes by Lemma 11.25.

3. It remains to consider the case when R ∈ 〈Ric〉. From the definition of

B(W,R) it suffices to prove that Q(W, l(R)) = l(Q(W,R) To do this,

note that since R ∈ 〈Ric〉, we have l(R) = (1 + (n − 2)b)R =: (1 + β)R

by Definition 11.29, and so


Q(W, l(R)) = (1 + β)Q(W,R) = l(Q(W,R)),

since Q(W,R) ∈ 〈Ric〉 by Proposition 11.24.

This completes the proof that Q(W, · ) = 0, and hence that D(R) is inde-pendent of the Weyl part of R.

Now we can compute the proof of Theorem 11.32 by assuming that RW =0, so that R = RI +RRic

. Let α = 2(n− 1)a, β = (n− 2)b and R0 = RRic =

2n−2Ric

∧ id so that

la,b(R) =(1 + α)λ

n− 1I + (1 + β)R0.

Using this with Lemma 11.25 yields

Da,b(R) = l−1a,b

( (1 + α)2λ2

(n− 1)2(I2 + I#I) +

2(1 + α)(1 + β)λ

n− 1(R0 + I#R0)

+ (1 + β)2(R20 +R#

0 ))−Q(R)

= l−1a,b

( (1 + α)2λ2

n− 1I +

2(1 + α)(1 + β)λ

n− 1Ric∧ id

+ (1 + β)2Q(R0))−Q(R).

By Proposition 11.20, with A = Ric

, we have

Ric∧ Ric

=−σn− 1

I − 2

n− 2(Ric 2)0 ∧ id + (Ric

∧ Ric

)W . (11.12)

So by Claim 11.28 we get

Q(R0) =1


(n− 2)2(Ric 2)0 ∧ id +

σ

n− 2I

=σ

n− 1I − 4

(n− 2)2(Ric 2)0 ∧ id +

1

n− 2(Ric∧ Ric

)W .

In this form we know how `−1a,b acts on each term. Also, by Lemma 11.27 we

have

Q(R) =( λ2

n− 1+

σ

n− 2

)I +

2λ

n− 1Ric∧ id

+1


(n− 2)2(Ric 2)0 ∧ id.

Using these formulas for Q(R0) and Q(R) with the fact that l−1a,b(R) =

11+α

λn−1I + 1

1+βR0 results in


Da,b(R) =(1 + α)2λ2

n− 1l−1a,b(I) +

2(1 + α)(1 + β)λ

n− 1l−1a,b(Ric

∧ id)

+ (1 + β)2l−1a,b(Q(R0))−Q(R)

=(1 + α)λ2

n− 1I +

2(1 + α)λ

n− 1Ric∧ id

+(1 + β)2

1 + α

σ

n− 1I − 4(1 + β)

(n− 2)2(Ric 2)0 ∧ id +

(1 + β)2

n− 2(Ric∧ Ric

)W

−( λ2

n− 1+

σ

n− 2

)I − 2λ

n− 1Ric∧ id

− 1

n− 2Ric∧ Ric

+2

(n− 2)2(Ric 2)0 ∧ id.

Substituting (Ric∧ Ric

)W from (11.12) and simplifying gives

Da,b(R) =αλ2

n− 1I +

2αλ

n− 1Ric∧ id +

2β + β2

n− 2Ric∧ Ric

+2β2

(n− 2)2(Ric 2)0 ∧ id +

( (1 + β)2

1 + α− n− 1

n− 2+

(1 + β)2

n− 2

) σ

n− 1I

=α

n− 1

(Ric ∧ Ric− Ric

∧ Ric

) +2β + β2

n− 2Ric∧ Ric

+2β2

(n− 2)2(Ric 2)0 ∧ id +

( (1 + β)2

1 + α− 1 +

2β + β2

n− 2

) σ

n− 1I

since Ric ∧ Ric− Ric∧ Ric

= λ2I + 2λRic∧ id. Substituting a and b back in

for α and β together with (Ric

2)0 = Ric

2 − σid and tr(Ric

2) = nσ gives thedesired equation.

Corollary 11.34 (Eigenvalues of Da,b and Rc(Da,b)). Suppose (ei)ni=1 is

an orthonormal basis of eigenvectors corresponding to the eigenvalues (λi)ni=1

of Ric

. Then ei ∧ ej, for i < j, is an eigenvector of Da,b(R) corresponding tothe eigenvalue

dij =((n− 2)b2 − 2(a− b)

)λiλj + 2a(λ+ λi)(λ+ λj) + b2(λ2

i + λ2j )

+σ

1 + 2(n− 1)a

(nb2(1− 2b)− 2(a− b)(1− 2b+ nb2)

). (11.13)

Furthermore, ei is an eigenvector of Rc(Da,b(R)) with eigenvalue

ri = −2bλ2i + 2aλ(n− 2)λi + 2a(n− 1)λ2

+σ

1 + 2(n− 1)a

(n2b2 − 2(n− 1)(a− b)(1− 2b)

). (11.14)

Remark 11.35. Notice that λi + λ are the eigenvalues of the Ricci tensor Ric.


Proof. Choose an orthonormal basis (ei)ni=1 of eigenvectors of Ric

with cor-

responding eigenvalues (λi)ni=1. By (11.4) we find that (Ric

∧Ric

)(ei ∧ ej) =

λiλjei ∧ ej , (Ric

2 ∧ id)(ei ∧ ej) = 12 (λ2

i +λ2j )ei ∧ ej , and (Ric∧Ric)(ei ∧ ej) =

(λi+ λ)(λj + λ)ei∧ej . From this it easily follows that ei∧ej is an eigenvectorof Da,b(R) with eigenvalue dij given by (11.13).

For the second expression, using (11.6), (11.7) and (11.5) we find that

Rc(Ric ∧ Ric) = (n − 1)λ2id + (n − 2)λRic− Ric

2, Rc(id ∧ id) = (n − 1)id,

Rc(Ric∧Ric

) = −Ric

2, and Rc(Ric

2∧ id) =(n2 − 1

)Ric

2 + nσ2 id. From which

we have

Rc(Da,b(R)) = −2bRic 2 + 2(n− 2)aλRic

+ 2(n− 1)aλ2 id

+2(n− 1)b+ (n− 2)2b2 − 2(n− 1)a(1− 2b)

1 + 2(n− 1)aσ id

So by the same diagonalisation argument for Ric

, (11.14) follows.

Chapter 12

The Cone Construction of Bohm andWilking

12.1 New Invariant Sets

In this section the remarkable formulae derived in the previous section, par-ticularly the identities (11.13) and (11.14), will be applied to construct afamily of cones preserved by the Ricci flow. We follow the argument presen-ted by Bohm and Wilking who applied it to produce a family of preservedcones interpolating between the cone of positive curvature operators and theline of constant positive curvature operators. The construction applies muchmore generally, so that given any preserved cone satisfying a few conditions,there is a family of cones linking that one to the ray of constant positivecurvature operators. As we will see, this is a crucial step in proving thatsolutions the Ricci flow converge to spherical space forms.

Definition 12.1 (Pinching Family of Convex Cones). We call a con-tinuous family C(s) ⊂ Curv of top-dimensional closed convex cones, para-metrised by s ∈ [0,∞), a pinching family (with respect to the vector fieldQ(R) = R2 +R#) if

1. C(s) is an O(n)-invariant cone for each s ≥ 0;2. Each R ∈ C(s)\0 has positive scalar curvature;3. Q(R) strictly points into1 C(s) at every point R ∈ ∂C(s)\0, for alls > 0;

4. C(s) converges (in compact sets in the Hausdorff topology) to the one-dimensional cone R+I as s→∞.

The motivation for making this definition comes from previous work, par-ticularly that by Hamilton [Ham82b] in which he proved that the cones

C(s)s∈[0,1) =R ∈ S2(so(3)) : Ric ≥ s tr Ric

3id

1 Recall that this means Q(R) is in the interior of the tangent cone TRC(s).

195

196 12 The Cone Construction of Bohm and Wilking

are preserved by Ricci flow, so that compact manifolds with positive Riccimaintain a bound on the ratio of the eigenvalues of the Ricci curvature aslong as the solution exists. This provides a continuous family of closed convexcones with C(0) equal to the cone of 3-dimensional curvature operators withnonnegative Ricci curvature, and C(1) equal to the cone of constant positivecurvature.

12.1.1 Initial Cone Assumptions. We will assume that we have aninitial preserved closed convex cone C(0) which is O(n)-invariant, containedin the cone of positive sectional curvature, and contains the cone of positivecurvature operators. We will show that:

Theorem 12.2. There exists a pinching family C(s), for 0 ≤ s < ∞, ofclosed convex cones starting at the cone C(0).

The construction uses the identities proved in the previous chapter toproduce a pinching family of cones, given by applying the operators la,b (forcarefully chosen a and b) to the intersection of C(0) with a cone of operatorswith pinched Ricci tensor (defined by a pinching ratio p). The definition ofthis family is in two stages: The first increases b to a critical value, and thesecond increases a to infinity (thus, as discussed in Remark 11.30, the familyof cones approaches the line of positive constant curvature as s→∞).

Lemma 12.3. For s ∈ [0, 12 ], let

a =(n− 2)s2 + 2s

2 + 2(n− 2)s2, b = s, p =

(n− 2)s2

1 + (n− 2)s2.

Then the vector field Q points strictly into the cone

C(s) = la,b

(R ∈ Curv : R ∈ C(0),Ric ≥ p tr Ric

n

)for 0 < s ≤ 1

2 .

Proof. By Lemma 11.31, we must prove that the vector field Xa,b pointsstrictly into the untransformed cone C(0) ∩ Cp at any non-zero boundarypoint R, where

Cp =R ∈ Curv : Ric ≥ p tr Ric

n

.

That is, we must check that Xa,b(R) is in the interior of the tangent coneTR(C(0) ∩ Cp). By Theorem B.7, it suffices to check that Xa,b(R) ∈ TRC(0)for R ∈ ∂C(0) ∩ Cp, and that Xa,b(R) ∈ TRCp for R ∈ ∂Cp ∩ C(0). Weconsider these two cases in turn:

1. The boundary of C(0): Suppose R ∈ ∂C(0) ∩ Cp. By assumption C(0)contains the cone C+ of positive curvature operators, and hence R + C+ ⊂C(0), and so

12.1 New Invariant Sets 197

C+ ⊂ TRC(0) =⋃h>0

C(0)−Rh

.

Since we know by assumption that Q points into C(0), it suffices to provethat Da,b strictly points into C(0), and for this it suffices to prove that Da,b

is in C+. We can do this by checking the positivity of the eigenvalues of Da,b

(given by Corollary 11.34).As R ∈ Cp, we have the following estimate for the eigenvalues of Ric0:

λi ≥ −(1− p)λ.

Next observe with our parametrisation that

2(a− b) =1− 2b

1 + (n− 2)b2(n− 2)b2 and (n− 2)b2 + 2b =

2a

1− p.

In which case Corollary 11.34 implies that

dij =( 2a

1− p− 2a

)λiλj + 2a(λi + λ)(λj + λ) + b2(λ2

i + λ2j )

+(1− 2b)σ

1 + 2(n− 1)a

(nb2 − (n− 2)b2

1 + (n− 2)b2(1− 2b+ nb2)

)= 2a

( 1

1− pλiλj + (1− p)λ2 + (λi + λj)λ

)+ 2apλ2 + b2(λi + λ2

j )

+2(1− 2b)b2σ

1 + 2(n− 1)a

1− (n− 2)b

1− (n− 2)b2

>2a

1− p(λi + (1− p)λ)(λj + (1− p)λ)

≥ 0

since 0 ≤ b ≤ 1/2 and λi + λ ≥ pλ.2. The boundary of Cp: The cone Cp may be given as an intersection of

half-spaces in the following way (compare the general construction in Section11.1.2):

Cp =⋂

O∈O(n)

R ∈ Curv : `(RO) ≤ 0

where `(R) = p scal(R)

n −Rc(R)(e1, e1). By symmetry we can assume that weare working at a boundary point R with `(R) = 0 and R ∈ C(0), and byTheorem B.7 it is sufficient to prove that `(Xa,b(R)) < 0 (for R 6= 0).

Since `(RO) ≤ 0 for all O we have λi + λ ≥ pλ, so that λi ≥ −(1 − p)λ,with equality holding for i = 1 (corresponding to the e1 direction). We haveto show that

Rc(Xa,b(R))11 > pscal(Xa,b(R))

n.


Firstly, by Proposition 11.23 we have

scal(Q(R)) =∑k

|Rickk|2 =∑k

(λk + λ)2 = n(σ + λ2). (12.1)

As∑λi = 0 and

∑λ2i = nσ, we find from Corollary 11.34 that

scal(Da,b(R)) =∑i

ri

= −2bnσ + 2an(n− 1)λ2

+nσ

1 + 2(n− 1)a

(n2b2 − 2(n− 1)(a− b)(1− 2b)

)= 2n(n− 1)aλ2 − nσ +

n(1 + (n− 2)b)2

1 + 2(n− 1)aσ.

Combining this result with (12.1) gives

scal(Xa,b(R))

n= (1 + 2(n− 1)a)λ2 +

(1 + (n− 2)b)2

1 + 2(n− 1)aσ. (12.2)

Note that this equation holds for any a 6= − 12(n−1) and is independent of the

choice of parametrisation.Now by Proposition 11.23 with Rickk ≥ pλ we get

Rc(R2 +R#)ii =∑k

RickkRikik ≥ pλRicii ≥ p2λ2.

Also, from the previous step our given parametrisation implies that

dij =2a

1− p(λi + (1− p)λ)(λj + (1− p)λ) + 2apλ2 + b2(λ2

i + λ2j )

+σ

1 + 2(n− 1)a

(nb2(1− 2b)− 2(a− b)(1− 2b+ nb2)

)≥ 2apλ2 + b2(λ2

i + λ2j )

+σ

1 + 2(n− 1)a

(nb2(1− 2b)− 2(a− b)(1− 2b+ nb2)

).

In which case we find that


Rc(Xa,b)ii ≥ p2λ2 +∑j 6=i

dij

≥ p2λ2 + 2a(n− 1)pλ2 + (n− 2)b2λ2i + nb2σ

+(n− 1)σ

1 + 2(n− 1)a

(nb2(1− 2b)− 2(a− b)(1− 2b+ nb2)

)= p2λ2 + 2a(n− 1)pλ2 + (n− 2)b2λ2

i

+( (1 + (n− 2)b)2

1 + 2(n− 1)a+ 2b− 1

)σ.

By combining this result with (12.2) we find that

Rc(Xa,b)ii − pscal(Xa,b)

n≥ p(p− 1)λ2 + (n− 2)b2λ2

i

+(

(1− p) (1 + (n− 2)b)2

1 + 2(n− 1)a+ 2b− 1

)σ.

From the stated parametrisation of p = p(b) and a = a(b) it is straightforwardto check that

p(p− 1)λ2 + (n− 2)b2λ2i = (n− 2)b2(λ2

i − (1− p)2λ2)

(1− p) (1 + (n− 2)b)2

1 + 2(n− 1)a+ 2b− 1 =

2nb2

nb+ 1.

Hence we get

Rc(Xa,b)ii − pscal(Xa,b)

n≥ (n− 2)b2(λ2

i − (1− p)2λ2) +2nb2

nb+ 1

≥ 2nb2

nb+ 1> 0

since λi = −(1− p)λ.

Lemma 12.4. For s ∈ [1/2,∞), let

a =1 + 2s

4, b =

1

2, p = 1− 4

n+ 4s.

Then the vector field Q(R) strictly points into the cone C(s) = la,b (C(0) ∩ Cp)at points R ∈ ∂C(s) \ 0.

Remark 12.5. Notice that lims→∞ la,b(R) = 2(n − 1)RI . Consequently thecones of the lemma converge to R+I for s→∞.

Proof. The proof is very similar to the previous one: We have two cases tocheck, the first for R ∈ (∂C(0)) ∩Cp, and the other for R ∈ C(0) ∩ ∂Cp. Forconvenience we define u = s− 1

2 ≥ 0.


Fig. 12.1 The trajectoryof the cone parameterisedby s in the (a, b)-planewhen n = 89. 0.1 0.2 0.3 0.4 0.5

b

0.2

0.4

0.6

0.8

1

a

1. The boundary of C(0): As before it is sufficient to prove that the ei-genvalues of Da,b are positive. Using Corollary 11.34 with a = (1 + u)/2 andb = 1/2, we find that

dij =(n− 2

4−u)λiλj+(1+u)(λ+λi)(λ+λj)+

1

4(λ2i +λ2

j )−σnu

4n+ 4(n− 1)u.

We observe that:

Claim. The quantity σ = 1n

∑λ2i is bounded above in terms of λ:

σ ≤ 16(n− 1)λ2

(n+ 2 + 4u)2. (12.3)

Proof of Claim. Since λi+λ ≥ pλ for all i, consider the optimisation problemfor σ = σ(λ1, . . . , λn) under the constraint

∑λi = 0. As the function σ has

a strictly positive Hessian, it achieves its maximum on the boundary of theset

(λ1, . . . , λn) : λi + λ ≥ pλ for all i, and∑

λi = 0

and furthermore at the vertices. Therefore the extremal points of σ are at(−(1 − p)λ, . . . , (n − 1)(1 − p)λ) and permutations thereof. Hence σ ≤ (n −1)(1− p)2λ2.

In addition, as λi + λ ≥ pλ and 1− p = 4n+2+4u , we also have

λi +4λ

n+ 2≥ λi +

4λ

n+ 2 + 4u≥ 0. (12.4)

From these inequalities we find that


dij =n+ 2

4λiλj + (1 + u)λ2 + (1 + u)(λi + λj)λ

+1

4(λ2i + λ2

j )−σnu

4n+ 4(n− 1)u

=n+ 2

4

(λi +

4λ

n+ 2

)(λj +

4λ

n+ 2

)+

1

4(λ2i + λ2

j )

+ λ2(n− 2

n+ 2+ u)

+ u(λi + λj)λ−σnu

4n+ 4(n− 1)u

≥(n− 2

n+ 2+ u− 8u

n+ 2 + 4u− 4(n− 1)nu

(n+ (n− 1)u)(n+ 2 + 4u)2

)λ2

≥(n− 2

n+ 2+ u

n− 6 + 4u

n+ 2 + 4u− 4(n− 1)u

(n+ 2)(n+ 2 + 4u)

)λ2

=(

4u2 +(n− 6− 4

n+ 2

)u+ n− 2

) λ2

n+ 2 + 4u.

To get the first inequality we discarded the first two terms, and used theinequality (12.4) for the fourth term and (12.3) for the last. The secondinequality is due to n

(n+(n−1)u)(n+2+4u) ≤1

n+2 for n ≥ 1 and u ≥ 0. Moreover,

when n = 3 we have 4u2 + (n − 6 − 4n+2 )u + n − 2 = 4u2 − 19

5 u + 1 > 0.

Also for fixed u ≥ 0, the function 4u2 + (n− 6− 4n+2 )u+ n− 2 is monotone

increasing in n. Therefore dij > 0 for all n ≥ 3 and u ≥ 0.2. The boundary of Cp: By the same argument as in the previous case, it

suffices to show that:

Rc(Xa,b)ii = Rc(Da,b)ii + Rc(R2 +R#)ii > pscal(Xa,b)

n.

By Corollary 11.34, with a = (1 + u)/2 and b = 1/2, we find that

ri = −λ2i + (u+ 1)λ(n− 2)λi + (u+ 1)(n− 1)λ2 +

σn2

4n+ 4(n− 1)u.

Also, Proposition 11.23 with Rickk ≥ pλ implies that Rc(R2 + R#)ii =∑k RickkRikik ≥ pλRicii ≥ p2λ2 and by (12.2) we also have

scal(Xa,b)

n= (n+ (n− 1)u)λ2 +

n2σ

4n+ 4(n− 1)u.

In which case we may suppose that λi = −(1− p)λ, so it suffices to show

ri + p2λ2 > p(

(n+ (n− 1)u)λ2 +n2σ

4n+ 4(n− 1)u

)or equivalently


0 ≤ p2λ2 − (1− p)2λ2 − (u+ 1)λ2(n− 2)(1− p) + (u+ 1)(n− 1)λ2

+σn2

4n+ 4(n− 1)u− p(

(n+ (n− 1)u)λ2 +n2

4n+ 4(n− 1)uσ).

Since σ ≥ 0 we can neglect the terms containing σ. Dividing by λ2 gives

p2 − (1− p)2 + (u+ 1) + (u+ 1)p(n− 2)− p(n+ (n− 1)u) = u(1− p)

which is clearly positive.

12.2 Generalised Pinching Sets

Hamilton [Ham86, p. 163] introduced the notion of a pinching set, which heused to prove that solutions of the Ricci flow on four-manifolds with positivecurvature operator converge to space-forms. His definition is as follows:

Definition 12.6 (Pinching Set). A subset Z ⊂ Curv is a pinching set if

1. Z is closed and convex;2. Z is O(n)-invariant;3. Z is preserved by the ode d

dtR = R2 +R#;4. there exist δ > 0 and K <∞ such that

|R| ≤ K|R|1−δ

for all R ∈ Z, where R = R− 1N trR is the trace-free part of R.

This definition is closely analogous to the conditions he used in his paperon three-manifolds [Ham82b], where he proved that sets of the form

R : Ric ≥ εScal g ∩ R : |Ric0|2 ≤ CScal2−γ

are preserved by the Ricci flow for suitable γ depending on ε (see Section 6.5.3for a different construction of pinching sets for this situation). The importantpoint is that if the scalar curvature becomes large then the traceless partbecomes relatively small, so that the curvature is close to that of a constantsectional curvature space.

12.2.1 Generalised Pinching Set Existence Theorem. Bohm andWilking [BW08, Theorem 4.1] observed that a weaker notion of pinchingset suffices for applications. The theorem below (which is similar to thatof Bohm and Wilking but modified following ideas of Brendle and Schoen[BS09a, Sect. 3]) still provides a set which guarantees pinching to constantsectional curvatures where the scalar curvature is large. The result is a usefultool, because it proves the existence of such a pinching set simply from theexistence of a suitable family of cones. We will apply it in the final argument(see Section 14.1) to the family of cones constructed above.

12.2 Generalised Pinching Sets 203

Theorem 12.7. Let C(s)s∈[0,∞) be a continuous family of closed convexO(n)-invariant cones in Curv of maximal dimension, contained in the half-space of curvature operators with positive scalar curvature. Suppose that forany s > 0 and any R ∈ ∂C(s)\0, the vector Q(R) = R2 +R# is containedin the interior of the tangent cone of C(s) at R. Then if K is any compact setcontained in the interior of C(0), there exists a closed convex O(n)-invariantset F ⊂ C(0) with the following properties:

1. Q(R) is in the tangent cone of F for every R ∈ ∂F ;2. K ⊂ F ;3. For each s > 0 there exists ρ(s) such that F + ρ(s)I ⊂ C(s).

Proof. By the continuity of the family C(s), given any compact K in theinterior of C(0) we have K ⊂ C(s0) for some s0 > 0. Adapting an idea ofBrendle and Schoen, we will construct a set F of the form

F = C(s0) ∩∞⋂i=1

R : R+ 2ihI ∈ C(si),

where h > 0 and (si) is an increasing sequence approaching infinity. ThusF is produced by a sequence of intersections with translated copies of thecones C(si). The idea is to choose (si) in such a way that each intersectiononly changes F where the scalar curvature is large, and the vector field Q(R)points into the new part of the boundary at each stage (that is, into theboundary of the translated cone at points of large scalar curvature).

The set F is manifestly convex, and condition (3) of the theorem holdsautomatically. Also, since each set C(s) is O(n)-invariant, so is F . We firstchoose s0 > 0 and h > 0 such that

K ⊂ C(s0) ∩ scal(R) ≤ h.

Lemma 12.8. For any s ≥ s0 there exists N(s) ≥ 1 (non-decreasing in s)such that if s ∈ [s0, s] and R ∈ ∂C(s) with scal(R) ≥ N(s), then Q(S) is inthe tangent cone to C(s) at R for every S with |S −R| ≤ 2|I|.

Proof of Lemma. The set Z = (s,R) : s ∈ [s0, s], R ∈ ∂C(s), scal(R) = 1is compact. We claim:

Claim. There exists r > 0 such that (s,R) ∈ Z, |S −R| ≤ r implies Q(S) isin the tangent cone to C(s) at R.

Proof of Claim. If not, there exists a sequence (si, Ri, Si) with (si, Ri) ∈ Zand |Si − Ri| → 0, but Q(Si) not in TRiC(si). Thus for each i there existsa unit norm linear function ì with ì(Ri) = supC(si) ì so that ` is in thenormal cone NRiC(si), but ì(Q(Si)) > 0 (see Definition B.4 in Appendix B).By compactness, after passing to a subsequence we have si → s ∈ [s0, s], and(by continuity of the family C(s)) Ri → R ∈ ∂C(s) ∩ scal(R) = 1. Bycompactness of the set of unit norm linear functions we also have ì → `, and


`(R) = supC(s) ` so that ` is in the normal cone to C(s) at R. Since |Si−Ri| →0 we also have Si → R, and by continuity of Q we have `(Q(R)) ≥ 0. But thisis a contradiction to the assumption that Q(R) is in the interior of TRC(s)(equivalently `(Q(R)) < 0 for every ` ∈ NRC(s)).

We claim the lemma holds with N(s) = 2|I|/r. For if R ∈ ∂C(s)with scal(R) ≥ N(s), then (s,R/scal(R)) ∈ Z and |S − R| ≤ 2|I| gives|S/scal(R)−R/scal(R)| ≤ 2|I|/scal(R) ≤ 2|I|/N(s) = r. Hence Q(S/scal(R))is in the tangent cone to C(s) at R/scal(R). The result follows since Q(S) =scal(R)2Q (S/scal(R)) and the tangent cone to C(s) at R is the same as thetangent cone to C(s) at R/scal(R).

Lemma 12.9. There exists a non-increasing function δ : [s0,∞)→ R+ suchthat whenever s ∈ [s0, s] and R+I ∈ C(s) with scal(R) ≤ N(s), then R+2I ∈C(s+ δ(s)).

Proof of Lemma. The set Z = (s,R+ 2I) : s ∈ [s0, s], scal(R) ≤ N(s), R+I ∈ C(s) is a compact set in the interior of (s,A) : s ∈ [s0, s], A ∈ C(s).By continuity of the family C(s), there exists δ > 0 such that Z ⊂ (s,A) :s ∈ [s0, s], A ∈ C(s+ δ).

We now construct the sequence si inductively by taking si+1 = si + δ(si)for each i ≥ 1. Note that since δ is a non-increasing positive function,limi→∞ s(i) =∞ (otherwise we would have δ(s) = 0 for s > limi→∞ si).

Let

Fj = C(s0) ∩j⋂i=1

R : R+ 2ihI ∈ C(si).

We prove that for each j,

Fj+1 ∩ scal(R) ≤ 2jN(sj)h = Fj ∩ scal(R) ≤ 2jN(sj)h. (12.5)

To see this we must show that the set on the right is contained in that on theleft. If R ∈ Fj ∩ scal(R) ≤ 2jN(sj)h then R+ 2jhI ∈ C(sj) and scal(R) ≤2jN(sj)h. Therefore R/(2jh) + I ∈ C(sj) and scal(R/(2jh)) ≤ N(sj), so byLemma 12.9,

R/(2jh) + 2I ∈ C(sj + δ(sj)) = C(sj+1)

and R+ 2j+1hI ∈ C(sj+1). Thus R ∈ Fj+1 as required.Now equation (12.5) says that as j increases above an index i, the part of

Fj with scal(R) ≤ 2iN(si)h does not change. It follows that F ∩ scal(R) ≤2iN(si)h = Fi ∩ scal(R) ≤ 2iN(si)h for each i. In particular F is locallythe intersection of finitely many closed sets, so is closed. Also, we have

K ⊂ C(s0) ∩ scal(R) ≤ h = F ∩ scal(R) ≤ h ⊂ F.

It remains to prove that Q(R) is in the tangent cone of F for every R ∈ ∂F .In such a case R is in ∂C(s0) or in ∂(C(si)−2ihI) for finitely many values ofi, where necessarily scal(R) ≥ 2i−1N(si)h by the inclusion (12.5). We must

12.2 Generalised Pinching Sets 205

show that Q(R) lies in TR(C(si)− 2ihI

)= TR+2ihIC(si) for each such i.

This can be done by letting

R′ = 21−ih−1(R+ 2ihI) and S = 2i−1h−1R,

so that scal(R′) ≥ N(si) and |S − R′| = 2|I|. Now by Lemma 12.8, Q(S) isin the tangent cone to C(si) at R′, and hence Q(R) ∈ TR

(C(si)− 2ihI

)as

required.

Chapter 13

Preserving Positive Isotropic Curvature

The condition of positive curvature on totally isotropic 2-planes was firstintroduced by Micallef and Moore [MM88]. They were able to prove thefollowing sphere theorem:

Theorem 13.1. Let M be a compact simply connected n-dimensional Rieman-nian manifold which has positive curvature on totally isotropic two-planes,where n ≥ 4. Then M is homeomorphic to a sphere.

One nice feature of their condition is that it is implied by strict pointwise 1/4-pinching; so in particular this theorem implies that compact simply connectedmanifolds with pointwise 1/4-pinched sectional curvatures are homeomorphicto spheres, thus refining the classical Berger-Klingenberg-Rauch result.

Recently Brendle and Schoen [BS09a] used this condition, with the ma-chinery of Bohm and Wilking (as seen in Chapters 11 and 12), to prove thedifferentiable pointwise 1/4-pinching sphere theorem. One of the nontrivialaspects of their proof involves showing that positive isotropic curvature is pre-served by the Ricci flow in all dimensions n ≥ 4. In regard to this, Brendleand Schoen [BS09a, p. 289] remarked that:

‘This is a very intricate calculation which exploits special identities and inequal-ities for the curvature tensor arising from the first and second variations appliedto a set of four orthonormal vectors which minimise the isotropic curvature.After this paper was written, we learned that H. Nguyen [Ngu08] has inde-pendently proved that positive isotropic curvature is preserved under the Ricciflow.’

In this chapter we will prove that positive isotropic curvature and positivecomplex sectional curvature are preserved by the Ricci flow. We also show inCorollary 13.13 that positive isotropic curvature on M × R2 is sufficient forpointwise 1/4-pinching.

207

208 13 Preserving Positive Isotropic Curvature

13.1 Positive Isotropic Curvature

Given a real vector space V , we consider its complexification VC which extendsscalar multiplication to include multiplication by complex numbers. Formallythis is achieved by letting VC = V ⊗R C be the tensor product of V with thecomplex numbers C. To make VC into a complex vector space, we definecomplex scalar multiplication by λ(v ⊗ µ) := v ⊗ (λµ) for all λ, µ ∈ C andv ∈ V . We define the complex conjugate of v ⊗ λ to be v ⊗ λ. By the natureof the tensor product, every vector v ∈ VC can be written uniquely in theform v = v1 ⊗ 1 + v2 ⊗ i, where v1, v2 ∈ V . Furthermore, we can regard VCas the direct sum of two copies of V , that is VC ' V ⊕ iV . Note that thecomplex dimension dimC(VC) = dimR(V ) is equal to the real dimension ofV . It is a common practice to drop the tensor product symbol and just writev = v1 + iv2. The archetypical example is the complexification of Rn, whichto no surprise is simply Cn.

Given a real inner product space V = (V, 〈·, ·〉), one can naturally ex-tend the real inner product to be complex-linear in both arguments (notHermitian) by defining

(x+ iy) · (u+ iv) := 〈x, u〉 − 〈y, v〉+ i 〈y, u〉+ i 〈x, v〉 . (13.1)

On the other hand there is also a natural extension of the inner product onV to a Hermitian inner product 〈〈·, ·〉〉 by defining

〈〈x+ iy, u+ iv〉〉 = (x+ iy, u− iv) = 〈x, u〉+ 〈y, v〉+ i 〈y, u〉 − i 〈x, v〉 .

Note that 〈〈U, V 〉〉 = U · V .Other tensors which act on V can also be extended in various ways, with

each argument extending to be either linear or conjugate linear. In particular,if R is an algebraic curvature operator and V = Rn, then it is natural toextend R as a Hermitian bilinear form acting on

∧2VC. In particular, for any

two-dimensional complex subspace Π of VC, R defines a complex sectionalcurvature of Π, as follows:

KC(Π) = R(Z,W, Z, W ),

where W and Z are an orthonormal basis for Π with respect to the Hermitianinner product 〈〈·, ·〉〉. Note that the symmetries of R imply that KC(Π) is realand independent of the choice of orthonormal basis. Writing Z = X+ iY andW = U + iV , and using the first Bianchi identity, we find that

KC(Π) = R(X,U,X,U) +R(X,V,X, V )

+R(Y,U, Y, U) +R(Y, V, Y, V )− 2R(X,Y, U, V ). (13.2)

A complex subspace U of VC is said to be isotropic, with respect to thecomplex form defined in equation (13.1), if there is at least one non-zero

13.1 Positive Isotropic Curvature 209

vector u ∈ U such that u · v = 0 for all v ∈ U . The subspace is calledtotally isotropic if this is true for every vector u (equivalently, the restrictionof the complex form (13.1) to U vanishes). In particular, we say a complexvector z 6= 0 is isotropic if z · z = 0. Note that if z = x + iy then z · z =‖x‖2−‖y‖2 + 2i 〈x, y〉, so z is isotropic if and only if x⊥ y and ‖x‖ = ‖y‖. Asubspace is totally isotropic precisely when it is composed entirely of isotropicvectors.

We are interested in totally isotropic 2-planes, i.e. two-dimensional com-plex subspaces, of RnC = Cn. Every such plane can be spanned by two vectorsZ = X + iY and W = U + iV where X,Y, U, V are all orthonormal (so thereare no such planes for n ≤ 3).

In this chapter we are interested in the following curvature conditions:

Definition 13.2 (PCSC Condition). We say R ∈ Curv has positive com-plex sectional curvature (PCSC) if all complex sectional curvatures of R arepositive. This defines an invariant closed convex cone in Curv:

CPCSC =R ∈ Curv : R(X,Y, X, Y ) ≥ 0 for all X,Y ∈ RnC

=

⋂X,Y ∈RnC

R ∈ Curv : `X,Y (R) ≥ 0

where `X,Y (R) := R(X,Y, X, Y ). We say a Riemannian manifold M haspositive complex sectional curvature if R ∈ Int (CPCSC(TpM)) for all pointsp ∈M .

Definition 13.3 (PIC Condition). We say R ∈ Curv has positive isotropiccurvature (PIC) if the complex sectional curvatures of all totally isotropic 2-planes in RnC are positive. This again defines an invariant closed convex cone:

CPIC = R ∈ Curv : `X,Y (R) ≥ 0 if X ·X = X · Y = Y · Y = 0 (13.3)

=⋂

O∈O(n)

R ∈ Curv : ˜(RO) ≥ 0

(13.4)

where ˜ := `e1+ie2,e3+ie4 , so by (13.2) we have

˜(R) = R1313 +R2424 +R1414 +R2323 − 2R1234. (13.5)

We say that a Riemannian manifold M has positive isotropic curvature ifR ∈ Int (CPIC(TpM)) for all points p ∈M .

Remark 13.4. By applying the vector bundle maximum principle, we willshow that both of these conditions are preserved by the Ricci flow, by provingthat the vector field Q on Curv points into the corresponding invariant cones.Moreover — with an eye to the differentiable sphere theorem — we will re-late the positive isotropic curvature condition to the pointwise 1/4-pinchingcondition as well.


13.2 The 1/4-Pinching Condition and PIC

In this section we relate the positive isotropic curvature condition with the1/4-pinching condition needed for the differentiable sphere theorem. This isachieved indirectly by working with the PIC condition on M × R2 and therelated cone CPIC2

.

13.2.1 The Cone CPICk . To avoid certain unpleasant analytical aspects,Brendle and Schoen [BS09a, Sect. 3] consider the PIC condition on M × R2

rather than that on M .1

The curvature operator Rk of M × Rk is an element of Curv(TM × Rk)given by the natural injection π∗k : Curv(Rn) → Curv(Rn+k) defined by

Rk(u, v, w, z) := (π∗kR)(u, v, w, z) = R(πu, πv, πw, πz)

for all u, v, w, z ∈ Rn+k, where πk : Rn+k → Rn is the orthogonal projection.We define CPICk to be the cone of all curvature operators R which producepositive isotropic curvature on M × Rk, i.e.

CPICk :=R ∈ Curv(Rn) : Rk ∈ CPIC(Rn+k)

= (π∗k)

−1 (CPIC(Rn+k)

).

Clearly, CPICk is a closed, convex, O(n)-invariant cone in Curv.

Remark 13.5. The same construction with PIC replaced by PCSC is ratherdull: Since Rk(X,Y, X, Y ) = R(πkX,πkY, πkX, πkY ), we have that Rk ∈CPCSC(Rn+k) if and only if R ∈ CPCSC(Rn). However with the PIC condition,the cone does change:

CPICk =R ∈ Curv : `πkX,πkY (R) ≥ 0 if X ·X = X · Y = Y · Y = 0

=

⋂(X,Y )∈Ak

R ∈ Curv : `X,Y (R) ≥ 0

where Ak is the set of all (X,Y ) ∈ RnC × RnC such that there exists (X, Y ) ∈Rn+k

C ×Rn+kC with πk(X) = X, πk(Y ) = Y and X ·X = X ·Y = Y ·Y = 0. Note

that Ak increases with k, so CPICk decreases with k (i.e. CPICk ⊃ CPIC` , fork < `). We will obtain a clearer understanding of these cones in Section 13.4.3.

In order to apply the maximum principle with these curvature conditions,we need to prove that the Ricci flow reaction vector field Q points into CPICk

(i.e. that CPICk is a preserved cone). One would expect that this conditionshould relate closely to whether the cone CPIC itself is preserved: If (M, g0)

1 At the time of writing, nobody knows how to handle Ricci flow on PIC manifoldsin dimensions higher than 4. One reason for the difficulty is that the curvature ofthe manifold Sn−1 × R lies strictly inside CPIC, so that singularities can form andsurgery arguments are required (see [Ham97,CZ06,CTZ08] for the case n = 4).

13.2 The 1/4-Pinching Condition and PIC 211

is a Riemannian manifold for which M × Rk has PIC, and g(t) is a solutionof Ricci flow on M with initial data g0, then the product metrics on M ×Rk— or on M × (S1)k — also evolve by Ricci flow (cf. Section 3.1.1.3). Thusif we know that CPIC is a preserved cone, the metric on M × (S1)k remains

PIC, and so R remains in CPICk . This strongly suggests that it should besufficient to prove that PIC is preserved. The following simple observationsconfirm this:

Lemma 13.6. For R ∈ Curv, π∗k(Q(R)) = Q(π∗k(R)).

Proof. Since Q is O(n+ k)-invariant, we can choose to compute it in a basiswith e1, . . . en an orthonormal basis for Rn and en+1, . . . , en+k an orthonormalbasis for Rk. Then for 1 ≤ a, b, c, d ≤ n+ k we have that

B(π∗kR)abcd =∑

1≤p,q≤n+k

RapbqRcpdq

=∑

1≤p,q≤n+k

R(πkea, πkep, πkeb, πkeq)R(πkec, πkep, πked, πkeq)

=∑

1≤p,q≤n

R(πkea, ep, πkeb, eq)R(πkec, ep, πked, eq)

= (π∗kB(R))abcd .

Since Q is a sum of such terms, the result follows.

Lemma 13.7. For any R ∈ ∂CPICk ,

TRCPICk = (π∗k)−1 (TRkCPIC

).

Proof. The map π∗k is a linear isomorphism from Curv(Rn) to the subspaceL =

⋂v∈0×Rk

R ∈ Curv(Rn+k) : R(v, ·, ·, ·) = 0

. In particular, π∗k maps

CPICk to L ∩ CPIC. Therefore

π∗k(TRCPICk

)= π∗k

( ⋃h>0

CPICk −Rh

)=⋃h>0

L ∩ CPIC − π∗kRh

= L ∩( ⋃h>0

CPIC − Rkh

)= L ∩ TRkCPIC.

Theorem 13.8. If CPIC(Rn+k) is a preserved cone, then so is CPICk(Rn).

Proof. For R ∈ ∂CPICk , we have

π∗k (Q(R)) = Q(R) ∈ L ∩ TRCPIC = π∗k(TRCPICk

),


so Q(R) ∈ TRCPICk since π∗k is injective.

For present purposes our interest is mainly in the cone CPIC2 . It has thefollowing elementary properties:

Proposition 13.9. The cone CPICk contains the cone of nonnegative curvature

operators R : R ≥ 0 for any k, and every R ∈ CPIC2has nonnegative sec-

tional curvature.

Proof. We first show that the cone of nonnegative curvature operators incontained in CPIC:

SupposeR ∈ Curv is a nonnegative curvature operator, and let e1, e2, e3, e4be an orthonormal 4-frame in Rn. By setting ϕ,ψ ∈

∧2 Rn to be

ϕ = e1 ∧ e3 + e4 ∧ e2

ψ = e1 ∧ e4 + e2 ∧ e3

we see that

0 ≤ R(ϕ,ϕ) +R(ψ,ψ)

= R1313 +R1414 +R2323 +R2424 − 2R1234

and so R ∈ CPIC. The inclusion of the non-negative curvature operators inCPICk now follows, since ιk maps non-negative curvature operators to non-negative curvature operators.

Now we show that CPIC2is contained in the cone of positive sectional

curvature operators: If R ∈ CPIC2, let e1, e2 ⊂ Rn be an orthonormal

2-frame, and define

e1 = (e1, 0, 0) e2 = (0, 0, 1)

e3 = (e2, 0, 0) e4 = (0, 1, 0)

which is an orthonormal 4-frame for Rn × R2. As R ∈ CPIC,

0 ≤ R1313 + R1414 + R2323 + R2424 − 2R1234

= R1212

since R(e1, e3, e1, e3) = R(e1, e2, e1, e2). Therefore R has nonnegative sec-tional curvatures.

13.2.2 An Algebraic Characterisation of the Cone CPIC2. By fol-

lowing [BS09a, Sect. 4], we characterise the cone CPIC2by providing a ne-

cessary and sufficient condition for R to have PIC. We use this to show thatall curvature operators R which are 1/4-pinched must lie in the cone CPIC2

,thus relating the 1/4-pinching condition to the PIC condition on M × R2.

To begin we quote the following linear algebra result:

13.2 The 1/4-Pinching Condition and PIC 213

Lemma 13.10 ([Che91, Lemma 3.1], [BS09a, p. 303]). Suppose ϕ,ψ ∈∧2 R4

are bivectors that satisfy ϕ ∧ ψ = 0, ϕ ∧ ϕ = ψ ∧ ψ, and 〈ϕ,ψ〉 = 0. Thenthere exists an orthonormal basis e1, e2, e3, e4 for R4 such that

ϕ = a1 e1 ∧ e3 + a2 e4 ∧ e2

ψ = b1 e1 ∧ e4 + b2 e2 ∧ e3

where a1a2 = b1b2.

Working from this lemma, we can prove the following ‘rotated version’:

Lemma 13.11. Suppose ϕ,ψ ∈∧2 R4 satisfy ϕ ∧ ψ = 0 and ϕ ∧ ϕ = ψ ∧ ψ.

Then there exists an orthonormal frame e1, e2, e3, e4 and θ ∈ R such that

cos θ ϕ+ sin θ ψ = a1 e1 ∧ e3 + a2 e4 ∧ e2

− sin θ ϕ+ cos θ ψ = b1 e1 ∧ e4 + b2 e2 ∧ e3

where a1a2 = b1b2.

Proof. Firstly, define θ to be such that

1

2sin 2θ (|ϕ|2 − |ψ|2) = cos 2θ 〈ϕ,ψ〉 .

Now set ϕ′ = cos θ ϕ + sin θ ψ and ψ′ = − sin θ ϕ + cos θ ψ. By hypothesis,the following quantities both vanish:

ϕ′ ∧ ϕ′ − ψ′ ∧ ψ′ = cos 2θ (ϕ ∧ ϕ− ψ ∧ ψ) + 2 sin 2θ ϕ ∧ ψ = 0

ϕ′ ∧ ψ′ =1

2sin 2θ (ϕ ∧ ϕ− ψ ∧ ψ) + cos 2θ ϕ ∧ ψ = 0.

Moreover, by the definition of θ, we also have that

〈ϕ′, ψ′〉 =1

2sin 2θ (|ψ|2 − |ϕ|2) + cos 2θ 〈ϕ,ψ〉 = 0,

from which the assertion follows by Lemma 13.10.

Proposition 13.12 (Characterisation of CPIC2). R ∈ CPIC2

if and onlyif

R1313 + λ2R1414 + µ2R2323 + λ2µ2R2424 − 2λµR1234 ≥ 0,

for all orthonormal 4-frames e1, e2, e3, e4 and all λ, µ ∈ [−1, 1].

Proof. Suppose R ∈ CPIC. Let e1, e2, e3, e4 ⊂ Rn be an orthonormal 4-frame, and let λ, µ ∈ [−1, 1]. Define

e1 = (e1, 0, 0) e2 = (µe2, 0,√

1− µ2)

e3 = (e3, 0, 0) e4 = (λe4,√

1− λ2, 0)


to be such that the vectors e1, e2, e3, e4 form an orthonormal 4-frame forRn × R2. With this, it follows that

0 ≤ R1313 + R1414 + R2323 + R2424 − 2R1234

= R1313 + λ2R1414 + µ2R2323 + λ2µ2R2424 − 2λµR1234.

To show the reverse implication, suppose e1, e2, e3, e4 is an orthonormal4-frame for Rn×R2. By definition each vector ej is of the form ej = (vj , xj),where vj ∈ Rn and xj ∈ R2. Let V be a 4-dimensional subspace containing

v1, v2, v3, v4, and define ϕ,ψ ∈∧2

V by

ϕ = v1 ∧ v3 + v4 ∧ v2

ψ = v1 ∧ v4 + v2 ∧ v3.

It is clear that ϕ ∧ ϕ = ψ ∧ ψ, ϕ ∧ ψ = 0 and V ' R4. So by Lemma 13.11there exists an orthonormal basis e1, e2, e3, e4 for V such that

ϕ′ = cos θ ϕ+ sin θ ψ = a1 e1 ∧ e3 + a2 e4 ∧ e2

ψ′ = − sin θ ϕ+ cos θ ψ = b1 e1 ∧ e4 + b2 e2 ∧ e3

where a1a2 = b1b2. Using the first Bianchi identity we find that

R(ϕ,ϕ) +R(ψ,ψ) = R(ϕ′, ϕ′) +R(ψ′, ψ′)

= a21R1313 + b21R1414 + b22R2323 + a2

2R2424 − 2a1a2R1234.

By setting λ = b1/a1 and µ = b2/a1, our hypothesis implies that the right-hand side is nonnegative. So it follows that

0 ≤ R(ϕ,ϕ) +R(ψ,ψ)

= R(v1, v3, v1, v3) +R(v1, v4, v1, v4) +R(v2, v3, v2, v3)

+R(v2, v4, v2, v4)− 2R(v1, v2, v3, v4)

= R1313 + R1414 + R2323 + R2424 − 2R1234.

Corollary 13.13. Let R ∈ Curv. If the sectional curvatures of R are 1/4-

pinched, then R ∈ CPIC and so R ∈ CPIC2.

Proof. Scale the metric (if need be) so that sectional curvatures of R liein the interval (1, 4]. Let e1, e2, e3, e4 be an orthonormal 4-frame in Rnand let λ, µ ∈ [−1, 1]. By Berger’s Lemma (see Section 1.7.7) we havethat |R(e1, e2, e3, e4)| ≤ 2. In which case the result now follows by Proposi-tion 13.12, since

13.3 PIC is Preserved by the Ricci Flow 215

R1313 + λ2R1414 + µ2R2323 + λ2µ2R2424 − 2λµR1234

≥ 1 + λ2 + µ2 + λ2µ2 − 4|λµ|= (1− |λµ|)2 + (|λ| − |µ|)2

≥ 0.

13.3 PIC is Preserved by the Ricci Flow

It was first shown by Hamilton [Ham97] that positive isotropic curvature ona 4-manifold is preserved by the Ricci flow. The result for n ≥ 4 was settledindependently by Brendle and Schoen [BS09a], and by Nguyen [Ngu08].

Theorem 13.14 (Brendle, Schoen, Nguyen). Let M be a compact man-ifold of dimension n ≥ 4 with a family of metrics g(t)t∈[0,T ) evolving underRicci flow. If g(0) has positive isotropic curvature, then g(t) has positive iso-tropic curvature for all t ∈ [0, T ).

In this section we prove this theorem using ideas from [BS09a, Ngu08,Ngu10,AN07]. In the remaining sections we will look at alternative argumentsand simplifications.

By the previous considerations the proof requires just one step: We mustprove that CPIC is preserved, in the sense that the vector field Q on Curv isin the tangent cone to CPIC at any boundary point. By Definition 13.3,

CPIC =⋂

O∈O(n)

R ∈ Curv : ˜(RO) ≥ 0

where ˜(R) = R1313 + R2424 + R1414 + R2323 − 2R1234. Since CPIC is anO(n)-invariant cone explicitly presented as an intersection of half-spaces,Theorem B.7 of Appendix B implies that it is enough to check that

˜(Q(R)) ≥ 0

for any R ∈ ∂CPIC for which ˜(R) = 0 and ˜(RO) ≥ 0 for all O ∈ O(n).

Lemma 13.15 (Brendle-Schoen Decomposition). For any R ∈ Curv,

˜(Q(R)) =1

2

((R13pq −R24pq)

2 + (R14pq +R23pq)2)

+(

(R1p1q +R2p2q)(R3p3q +R4p4q)−R12pqR34pq

− (R1p3q +R2p4q)(R3p1q +R4p2q)

− (R1p4q −R2p3q)(R4p1q −R3p2q)).

Proof. By Remark 11.12, Q = Q(R) satisfies the first Bianchi identity (al-though R2 and R# do not). In which case


˜(Q(R)) = Q1313 +Q1414 +Q2323 +Q2424 − 2Q1234

= Q1313 +Q1414 +Q2323 +Q2424 + 2Q1342 + 2Q1423.

So by Lemma 11.16 and the first Bianchi identity we find that

R21313 +R2

1414 +R22323 +R2

2424 + 2R21342 + 2R2

1423

=1

2

((R13pq −R24pq)

2 + (R14pq +R23pq)2)

and

R#1313 +R#

1414 +R#2323 +R#

2424 + 2R#1342 + 2R#

1423

= (R1p1q +R2p2q)(R3p3q +R4p4q)


− (R1p4q −R2p3q)(R4p1q −R3p2q)

− 2R1p2qR3p4q + 2R1p2qR4p3q

= (R1p1q +R2p2q)(R3p3q +R4p4q)


− (R1p4q −R2p3q)(R4p1q −R3p2q)

−R12pqR34pq.

From this lemma it is clear that

1

2

n∑p,q=1

((R13pq −R24pq)

2 + (R14pq +R23pq)2)≥ 0,

so in order to prove Theorem 13.14 all that is needed is to verify that:

Claim 13.16. If R ∈ CPIC with ˜(R) = 0 then

n∑p,q=1

((R1p1q +R2p2q)(R3p3q +R4p4q)−R12pqR34pq


− (R1p4q −R2p3q)(R4p1q −R3p2q))≥ 0. (13.6)

Remark 13.17. We observe there is some redundancy in our description ofCPIC, since not all of the half-spaces given by ˜(RO) ≥ 0 are distinct: ˜(R)computes the complex sectional curvature of R in the plane generated bye1 + ie2 and e3 + ie4, but this is unchanged if we choose a different basis forthe same complex 2-plane. In particular we get the same result if we replacee1+ie2 by e2−ie1 = −i(e1+ie2), or replace e3+ie4 by e4−ie3, or interchangee1 + ie2 with e3 + ie4. Thus for any inequalities we prove for ˜(R) there arecorresponding inequalities for ˜(RO), where O is an O(n) matrix which has


the top-most 4× 4 block given by any of0 1 0 0−1 0 0 00 0 1 00 0 0 1

,

1 0 0 00 1 0 00 0 0 10 0 −1 0

, or

0 0 1 00 0 0 11 0 0 00 1 0 0

.13.3.1 Inequalities from the Second Derivative Test. In this section

we prove Claim 13.16 by applying the second derivative test to the functionZ on O(n), defined by

Z(O) = ˜(RO),

along integral curves in the Lie group O(n) through the identity.Integral curves in O(n) can be conveniently computed by following the

flow of left-invariant vector fields: Given Λ ∈ so(n), the corresponding left-invariant vector field X ∈ Lie(O(n)) is given by XO = O Λ.2 The integralcurve through the identity in direction Λ is given by solving the ode3

d

dsO(s) = O(s) Λ

O(0) = In

Equivalently, if we write vi(s) = (O(s))ei, then this is equivalent to the system

d

dsvi =

d

ds(Oei) = O(Λjiej) = Λijvj , (13.7)

with vi(0) = ei.We observe that Z(O(0)) = ˜(R) = 0 by construction, while Z(O(s)) =

˜(RO(s)) ≥ 0 for every s. So the first and second derivative tests imply that

d

dsZ(O(s))

∣∣∣s=0

= 0

d2

ds2Z(O(s))

∣∣∣s=0≥ 0.

By explicitly evaluating these first and second derivative conditions, we showin Sections 13.3.1.1 that the sums over indices p ≤ 4 < q, q ≤ 4 < p and over1 ≤ p, q ≤ 4 of (13.6) all vanish. In Section 13.3.1.2 we show that the sumover p, q ≥ 5 of (13.6) is nonnegative, thus proving Claim 13.16.

2 Recall that we are identifying O(n) with a subset of End(Rn) ' (Rn)∗ ⊗ Rn,namely the set of linear transformations of Rn which are isometries. The tangentspace TInO(n) to the identity is then the subspace of anti-self-adjoint transformationsso(n) = Λ : ΛT + Λ = 0 '

∧2(Rn) (see [Lee02, Example 8.39]).3 Note that the flow Ψ of the left-invariant vector field X is given by Ψs = Rexp sΛ

(i.e. right multiplication by exp sΛ), so that the path O(s) = Ψs(In) = exp sΛ (see[Lee02, Proposition 20.8(g)]).


13.3.1.1 First Order Terms. We write

Z(O(s)) = ˜(RO(s))

= R(v1, v3, v1, v3) +R(v2, v4, v2, v4) +R(v1, v4, v1, v4)

+R(v2, v3, v2, v3)− 2R(v1, v2, v3, v4),

and differentiate directly using equation (13.7). The first derivative gives

1

2

d

dsZ∣∣∣s=0

= (Rp313 +Rp414 −Rp234)Λ1p

+ (Rp323 +Rp424 −R1p34)Λ2p

+ (R1p13 +R2p23 −R12p4)Λ3p

+ (R1p14 +R2p24 −R123p)Λ4p (13.8)

where we have used

d

dsR(va, vb, vc, vd)

∣∣∣s=0

= ΛajRj234 + ΛbjR1j34 + ΛcjR12j4 + ΛdjR123j .

From the first derivative condition, the right-hand side of (13.8) vanishes forany choice of antisymmetric Λ. Choosing Λ = ep ∧ eq with p ≤ 4 < q gives

p = 1 : 0 =R133q +R144q +R432q (13.9a)

p = 2 : 0 =R233q +R244q +R341q (13.9b)

p = 3 : 0 =R131q +R232q +R124q (13.9c)

p = 4 : 0 =R141q +R242q +R213q (13.9d)

The remaining cases are Λ = ep ∧ eq with 1 ≤ p < q ≤ 4. By Remark 13.17,the path O(s) = exp s(e1 ∧ e2) corresponds to multiplying e1 + ie2 by eis.Similarly, the choice O(s) = exp, s(e3 ∧ e4) multiplies e3 + ie4 by eis. Finally,Λ = e1 ∧ e3 + e2 ∧ e4 and Λ = e1 ∧ e4 − e2 ∧ e3 give changes of basis inthe complex 2-plane. All of these keep Z = ˜(Rexp sΛ) fixed. Thus the onlynontrivial identities are given by choosing Λ equal to e1 ∧ e3 and e1 ∧ e4:

e1 ∧ e3 : R3414 +R3423 +R1223 +R1214 = 0 (13.10a)

e1 ∧ e4 : R3424 +R3134 +R2113 +R1224 = 0. (13.10b)

Lemma 13.18 ([BS09a, Proposition 5]).

4∑p,q=1



− (R1p4q −R2p3q)(R4p1q −R3p2q))

= 0.


Proof. By direct computation we get

4∑p,q=1



− (R1p4q −R2p3q)(R4p1q −R3p2q))

= (R1212 +R3434)(R1313 +R1414 +R2323 +R2424 − 2R1234)

+ 2R1234(R1313 +R1414 +R2323 +R2424 + 2R1342 + 2R1423)

− (R1213 +R1242 +R3413 +R3442)2

− (R1214 +R1223 +R3414 +R3423)2

= (R1212 +R3434 + 2R1234)(R1313 +R1414 +R2323 +R2424 − 2R1234)

− (R1213 +R1242 +R3413 +R3442)2

− (R1214 +R1223 +R3414 +R3423)2.

The expression now vanishes by (13.10a) and (13.10b).

Lemma 13.19 ([BS09a, Proposition 7]). For fixed q ≥ 5, we have

4∑p=1



− (R1p4q −R2p3q)(R4p1q −R3p2q))

= 0.

Proof. Using equations (13.9a) and (13.9b) a direct computation shows that

2∑p=1


)= R212q(R313q +R414q) +R121q(R323q +R424q)

−R121qR341q −R122qR342q

= R212q(R313q +R414q +R342q)

+R121q(R323q +R424q −R341q)

= 0

and


4∑p=3

((R1p3q +R2p4q)(R3p1q +R4p2q)− (R1p4q −R2p3q)(R4p1q −R3p2q)

)= (R133q +R234q)R432q + (R143q +R244q)R341q

+ (R134q −R233q)R431q − (R144q +R243q)R342q

= (R133q +R234q +R144q −R243q)R432q

+ (R143q +R244q −R134q +R233q)R341q

= (R133q +R144q +R432q)R432q

+ (R341q +R244q +R233q)R341q

= 0.

Replacing e1, e2, e3, e4 by e3, e4, e1, e2 (see Remark 13.17) yields

2∑p=1

((R1p3q +R2p4q)(R3p1q +R4p2q) + (R1p4q −R2p3q)(R4p1q −R3p2q)

)= 0

4∑p=3


)= 0.

The result now follows by putting these sums together.

13.3.1.2 Second Order Terms. We now calculate the second derivative ofZ and establish the nonnegativity of the final final sum over indices p, q ≥ 5of (13.6). By grouping similar terms we find that the second order derivativeof Z, calculated from (13.8), is equal to

1

2

d2Z

ds2= (Rj3k3 +Rj4k4)Λ1jΛ1k + (Rj3k3 +Rj4k4)Λ2jΛ2k

+ (R1j1k +R2j2k)Λ3jΛ3k + (R1j1k +R2j2k)Λ4jΛ4k

+ 2(Rjk13 +Rj31k −Rj2k4)Λ1jΛ3k + 2(Rjk14 +Rj41k −Rj23k)Λ1jΛ4k

+ 2(Rjk23 +Rj32k −R1jk4)Λ2jΛ3k + 2(Rjk24 +Rj42k −R1j3k)Λ2jΛ4k

+ (Rk313 +Rk414 −Rk234)Λ1jΛjk + (Rk323 +Rk424 −R1k34)Λ2jΛjk

+ (R1k13 +R2k23 −R12k4)Λ3jΛjk + (R1k14 +R2k24 −R123k)Λ4jΛjk

− 2Rjk34Λ1jΛ2k − 2R12jkΛ3jΛ4k.

By identifying the following coefficients:

ajk = R1j1k +R2j2k bjk = R3j3k +R4j4k

cjk = R3j1k +R4j2k djk = R4j1k −R3j2k

ejk = R12jk fjk = R34jk

we can rewrite the second derivative as


1

2

d2

ds2Z(O(s))

∣∣∣s=0

= 2([R3k1j −R4k2j ]− 2R3j1k)Λ1jΛ3k

+ 2([R4k2j −R3k1j ]− 2R4j2k)Λ2jΛ4k

+ 2([R4k1j +R3k2j ]− 2R4j1k)Λ1jΛ4k

+ 2([R3k2j +R4k1j ]− 2R3j2k)Λ2jΛ3k

+ bjkΛ1jΛ1k + bjkΛ2jΛ2k + ajkΛ3jΛ3k + ajkΛ4jΛ4k

+ 0Λ1jΛjk + 0Λ2jΛjk + 0Λ3jΛjk + 0Λ4jΛjk

− 2fjkΛ1jΛ2k − 2ejkΛ3jΛ4k,

where the zero coefficients are the result of applying (13.9a)–(13.9d). Observethat by making the following frame switch4

e1, e2, e3, e4 7→ e2,−e1, e4,−e3,

the terms ajk, bjk, ejk, fjk remain invariant. In which case, if we denote thenew orthonormal frame by O′, we find that

1

2

d2

ds2Z(O′(s))

∣∣∣s=0

= 2([ R4k2j −R3k1j ]− 2R4j2k)Λ1jΛ3k

+ 2([ R3k1j −R4k2j ]− 2R3j1k)Λ2jΛ4k

+ 2([−R3k2j −R4k1j ] + 2R3j2k)Λ1jΛ4k

+ 2([−R4k1j −R3k2j ] + 2R4j1k)Λ2jΛ3k

+ bjkΛ1jΛ1k + bjkΛ2jΛ2k + ajkΛ3jΛ3k + ajkΛ4jΛ4k

+ 0Λ1jΛjk + 0Λ2jΛjk + 0Λ3jΛjk + 0Λ4jΛjk

− 2fjkΛ1jΛ2k − 2ejkΛ3jΛ4k.

We take the sum of the second derivatives in these two frames to get

1

2

( d2

ds2Z(O(s)) +

d2

ds2Z(O′(s))

)∣∣∣s=0

= −4cjkΛ1jΛ3k − 4cjkΛ2jΛ4k

− 4djkΛ1jΛ4k + 4djkΛ2jΛ3k

+ 2bjkΛ1jΛ1k + 2bjkΛ2jΛ2k

+ 2ajkΛ3jΛ3k + 2ajkΛ4jΛ4k

− 4fjkΛ1jΛ2k − 4ejkΛ3jΛ4k

which is a considerably simpler looking expression involving only ajk, bjk, ejkand fjk terms. Moreover, since Z(O(0)) = Z(O′(0)) = 0 and Z(O) ≥ 0 forall O, the sum

d2

ds2Z(O(s))

∣∣∣s=0

+d2

ds2Z(O′(s))

∣∣∣s=0

4 Note that this the same switch used in [BS09a, Proposition 8] (see also Re-mark 13.17).


is positive semi-definite by construction. Thus the corresponding matrix

L = 2

B −F −C −DF B D −C−CT DT A −E−DT −CT E A

is also positive semi-definite (note ET = −E and FT = −F ). Defining

U =

0 0 I 00 0 0 −I−I 0 0 00 I 0 0

,

the conjugation of L by U is

L∨ = ULUT = 2

A E CT DT

−E A −DT CT

C −D B FD C −F B

.

In which case positive semi-definiteness implies that

0 ≤ 1

4trLL∨

= trAB + trEF − trC2 − trD2

=

n∑j,k=5

ajkbjk −n∑

j,k=5

(ejkfjk + cjkckj + djkdkj

).

Thereforen∑

j,k=5

(ajkbjk − ejkfjk − cjkckj − djkdkj

)≥ 0.

This proves Claim 13.16. Theorem 13.14 follows by the maximum principle.

13.4 PCSC is Preserved by the Ricci Flow

In this section we show directly that nonnegative complex sectional curvatureis preserved under the Ricci flow. This provides an alternative approach toshowing that CPIC2 is preserved without discussing the PIC condition. Wealso use the argument as motivation for the next section where we adapt itto give a notationally simpler proof that PIC is preserved.

13.4.1 The Mok Lemma. The main technical tool in our proof will bethe following simple case of a result originally proved by Mok [Mok88].

13.4 PCSC is Preserved by the Ricci Flow 223

Lemma 13.20. Let A be a non-negative Hermitian form on C2n given by

A =

(A BB∗ C

).

ThentrAC ≥ trBB.

Proof. Firstly, note that if A is non-negative definite, then so is the matrix

B =

(C −B∗−B A

)since (

x y)B

(x∗

y∗

)=(y −x

)A

(y∗

−x∗)≥ 0.

We observe that trAB ≥ 0, since If U ∈ U(2n) diagonalises A, then

trAB = tr (U∗AU)(U∗BU) =∑i

aiibii ≥ 0

where aii = (Uei)∗A(Uei) ≥ 0 and bii = (Uei)∗B(Uei) ≥ 0 for each i. Nowobserve

trAB = tr (AC −BB) + tr (CA−B∗B∗).

Since trCA = trA∗CT = trAC and trB∗B∗ = trBB we have that

trAB = 2tr (AC −BB)

and so the lemma follows.

13.4.2 Preservation of PCSC Proof. It is now straightforward toprove that positive complex sectional curvature is preserved under the Ricciflow.5 To prove that the vector field Q points into CPCSC it suffices to consider(by the characterisation of the cone CPCSC in Definition 13.2) R ∈ CPCSC

and X,Y ∈ RnC for which `X,Y (R) = 0, and prove that `X,Y (Q(R)) ≥ 0.Noting that `X,Y (R) = XiY iXkY `Rijkl, this amounts to the following:

Claim 13.21. If R(X,Y, X, Y ) = XiY iXkY `Rijk` = 0 and all complex sec-tional curvatures are nonnegative, then

XiY jXkY `Qijk` ≥ 0.

Proof. To prove this claim, consider X(t) = X + tW and Y (t) = Y + tZ.Also, define

h(t) := Xi(t)Y j(t)Xk(t)Y `(t)Rijk`.

5 We learnt this argument from unpublished work of Ni and Wolfson.


So we have h(0) = 0 and h(t) ≥ 0 for all t. Hence h′(0) = 0 and h′′(0) ≥ 0.Computing 1

2h′′(t) directly, we find(

W iZjXkY ` +W iY jW kY ` +W iY jXkZ`

+XiZjW kY ` +XiZjXkZ` +XiY jW kZ`)Rijk` ≥ 0.

The inequality holds for arbitrary W and Z in RnC. In particular, the sameinequality with W replaced by iW and Z replaced by iZ gives the following:(

−W iZjXkY ` +W iY jW kY ` +W iY jXkZ`

+XiZjW kY ` +XiZjXkZ` −XiY jW kZ`)Rijk` ≥ 0.

Adding these two inequalities gives:(W iY jW kY ` +XiZjXkZ`

+W iY jXkZ` +XiZjW kY `)Rijk` ≥ 0. (13.11)

Writing this as the positivity of a Hermitian form on RnC × RnC we get

(W Z

)( A BB∗ C

)(W ∗

Z∗

)≥ 0

where

Apq = Y j Y `Rjp`q

Bpq = −Y jXkRpjqk

Cpq = XiXkRipkq.

Then Mok’s lemma implies

0 ≤ tr (AC −BB) = XiY jXkY `RipkqRjp`q −XiY jXkY `Rip`qRjpqk

which is precisely the R# term in XiY jXkY `Qijk`. The remanding squaredterm is trivially nonnegative, since

1

2XiY jXkY `RijpqRpqk` =

1

2

∑p,q

|XiY jRijpq|2 ≥ 0.

This completes the proof.

13.4.3 Relating PCSC to PIC. We can relate the complex sectionalcurvature condition to the cone construction discussed in Section 13.2.

13.5 Preserving PIC Using the Complexification 225

Proposition 13.22. The cone CPCSC = CPICk for k ≥ 2.6

Proof. Certainly CPCSC ⊂ CPICk , since if R ∈ CPCSC then the complexsectional curvature of any 2-plane in Cn+k is proportional to the complexcurvature of its projection onto Cn, which is non-negative.

In order to see the converse, it is enough to show that given Z,W ∈ Cnlinearly independent, there exist extensions Z, W on Cn × C2 such that thecomplex 2-plane they generate is totally isotropic and

KM (Z,W, Z, W ) = KM×R2(Z, W ) (13.12)

To show this, let Z = Z+ue1 +ve2 and W = W +xe1 +ye2 be extensionsof Z,W which span a totally isotropic plane. Here e1, e2 is an orthonormalbasis for the factor R2. Clearly (13.12) is satisfied since the extra R2 part isflat. To show the isotropic condition, one needs Z · Z = W · W = Z · W = 0.By expanding the inner product, this is equivalent to

Z · Z + u2 + v2 = 0

W ·W + x2 + y2 = 0

Z ·W + ux+ vy = 0.

These can be written in the matrix form XXT = A, with

X =

(u vx y

)A =

(a cc b

)where a = −Z · Z, b = −W ·W , and c = −Z ·W . The matrix equation canbe solved since A can be diagonalised by a GL(2,C) transformation.

13.5 Preserving PIC Using the Complexification

In this section we give a proof that PIC is preserved, by working directly withthe description (13.3) of PIC in terms of the complex sectional curvatures,instead of the expression (13.4). While the argument is fundamentally thesame, working with the complexification results in considerable notationalsimplification, and perhaps elucidates the particular choices and combinationswhich were made in the previous approach.7 We write

CPIC = R ∈ Curv : `(R) ≥ 0, for all ` ∈ B

6 This observation is due to Nolan Wallach. A similar argument also identifies CPIC1

the cone of operators which has positive complex sectional curvature on isotropic (notnecessarily totally isotropic) 2-planes (cf. Section 13.1).7 We note that there is a more recent argument announced by Wilking, which provesthat a large family of cones are preserved by the Ricci flow, including in particularCPCSC, CPIC1

and CPIC2.


where B = ù,v : u, v ∈ RnC, u · u = u · v = v · v = 0, and ù,v(R) =R(u, v, u, v). By the maximum principle and the results of Section B.5 inAppendix B, it suffices to show that `(u,v)(Q(R)) ≥ 0 whenever R is an

element of CPIC with ù,v(R) = 0 and ù,v ∈ B. Noting that Q(R) = R2+R#,and that ù,v(R

2) = R2(u, v, u, v) ≥ 0, it suffices to prove that ù,v(R#) ≥ 0.

In order to prove this, we first observe that ` is really defined on a quotientof B: àu+bv,cu+dv = |ad − bc|2ù,v, so we have an action of GL(2,C) whichonly changes ` by a positive factor. Also note that (au + bv, cu + dv) stillsatisfies the totally isotropic conditions. Therefore we can choose u and vso that u · u = v · v = 1 and u · v = 0. It follows that we can choose anorthonormal basis for RnC consisting of e1 = u, e2 = u, e3 = v, e4 = v, and(n− 4) other vectors ep, p > 4, which can be chosen to be real (ep = ep).

Let X and Y be arbitrary vectors in the span of ep : p > 4. We willconsider the smooth family (U(t), V (t)) defined by

U(t) = u+ tX − t2X ·X2

u− t2X · Y2

v

V (t) = v + tY − t2X · Y2

u− t2Y · Y2

v.

A direct computation (using the fact that X ·u = X ·u = X ·v = X ·v = 0, andsimilarly for Y ) shows that (U(t), V (t)) is totally isotropic for every t. SinceR is in CPIC, we have Ù(t),V (t)(R) ≥ 0 for all t, but Ù(0),V (0)(R) = 0, andhence the first derivative vanishes and the second derivative is non-negativewhen t = 0. We compute directly:

d

dt`(U(t),V (t))(R)

= R(U , V, U , V ) +R(U, V , U , V ) +R(U, V, ˙U, V ) +R(U, V, U , ˙V ) (13.13)

= 2<(R(U , V, U , V ) +R(U, V , U , V )

). (13.14)

Evaluating at t = 0 gives

0 = <(R(X, v, u, v) +R(u, Y, u, v)

).

The same equation with X and Y replaced by iX and iY gives

0 = =(R(X, v, u, v) +R(u, Y, u, v)

).

Since X and Y are arbitrary vectors in the span of ep : p > 4, this implies

R(X, v, u, v) = R(u, Y, u, v) = 0 (13.15)

for any X,Y ∈ spanep : p > 4.Now we differentiate equation (13.13) again in t:

13.5 Preserving PIC Using the Complexification 227

d2

dt2`(U(t),V (t))(R) = R(U , V, U , V ) +R(U, V , U , V )

+R(U, V, ¨U, V ) +R(U, V, U , ¨V )

+ 2R(U , V , U , V ) +R(U , V, ˙U, V ) + 2R(U , V, U , ˙V )

+ 2R(U, V , ˙U, V ) + 2R(U, V , U , ˙V ) +R(U, V, ˙U, ˙V ).

Now we take the same with X and Y replaced by iX and iY respectively, andadd. Note that this reverses the signs of U and V , so all the terms involvingthese second derivatives cancel. The terms involving R(U , V , U , V ) and itsconjugate also cancel. Evaluating at t = 0 (so that U = X and V = Y ) weobtain the following:

R(X, v, X, v) +R(u, Y, u, Y )

+R(X, v, u, Y ) +R(u, Y, X, v) ≥ 0 (13.16)

for all X,Y ∈ spanep : p > 4. Note the similarity to equation (13.11). Itfollows by the same argument as used there that∑

p,q>4

(R(u, ep, u, eq)R(v, ep, v, eq)−R(u, ep, v, eq)R(v, ep, u, eq)

)≥ 0.

This is almost the same as R#(u, v, u, v), except that the sum is not over allvalues of p. To fix this problem we look in more detail at the terms which aremissing: These are∑1≤p,q≤4


)+∑p≤4<q


)+∑q≤4<p


).

The second and third sums above are zero by the first derivative condition(13.15): The second sum expands to give (for each q > 4)

R(u, u, u, eq)R(v, u, u, eq) +R(u, v, u, eq)R(v, v, v, eq)

−R(u, u, v, eq)R(v, u, u, eq)−R(u, v, v, eq)R(v, v, u, eq).

The second factor in the first and third terms, and the first factor in thesecond and fourth terms, all vanish by (13.15). The third sum expands sim-ilarly. This leaves only the terms where both p and q are less than or equalto 4, which expand as follows:


R(u, u, u, v)R(v, u, v, u) +R(u, u, u, v)R(v, u, v, v)

+R(u, v, u, u)R(v, v, v, u) +R(u, v, u, v)R(v, v, v, v)

−R(u, u, v, u)R(v, u, u, u)−R(u, u, v, v)R(v, u, u, v)

−R(u, v, v, u)R(v, v, u, u)−R(u, v, v, v)R(v, v, u, v).

Here the second factor in the first and sixth terms, and the first factor inthe fourth and seventh terms, vanish since R(u, v, u, v) = `(u,v)R = 0. Theremaining terms factor to give(

R(u, u, u, v) +R(v, v, u, v))(R(u, v, v, v) +R(u, v, u, u)

).

For the punchline we need to consider two more smooth curves in B: Firstlet U(t) = u+ tv and V (t) = v − tu. Then `U(t),V (t) ∈ B for every t, so

0 =d

dt`(U(t),V (t))R

∣∣∣t=0

= R(v, v, u, v)−R(u, u, u, v)

+R(u, v, v, v)−R(u, v, u, u).

Now the same computation with U(t) = u+ itv and V (t) = v − itu gives

0 = iR(v, v, u, v)− iR(u, u, u, v)− iR(u, v, v, v) + iR(u, v, u, u).

These two identities imply

0 = R(u, v, v, v)−R(u, v, u, u) = R(v, v, u, v)−R(u, u, u, v),

so that all of the terms arising from the sum with 1 ≤ p, q ≤ 4 vanish. Wehave proved that

`(u,v)(R#) =

∑p,q>4

(R(u, ep, u, eq)R(v, ep, v, eq)

−R(u, ep, v, eq)R(v, ep, u, eq))≥ 0, (13.17)

so by the maximum principle PIC is preserved by Ricci flow.

Chapter 14

The Final Argument

14.1 Proof of the Sphere Theorem

We now have all the ingredients in place to prove the following:

Theorem 14.1 (Differentiable 1/4-Pinched Sphere Theorem). A com-pact, pointwise 1/4-pinched Riemannian manifold of dimension n ≥ 4 is dif-feomorphic to a spherical space form.

Remark 14.2. In fact we prove the stronger result that any compact Rieman-nian manifold (M, g) such that M × R2 has positive curvature on totallyisotropic 2-planes (equivalently, any compact Riemannian manifold with pos-itive complex sectional curvatures) is diffeomorphic to a spherical space-form.This implies the pointwise differentiable 1/4-pinching sphere theorem by Co-rollary 13.13.

Proof. Suppose (M, g0) is a compact Riemannian manifold with positive com-plex sectional curvatures, so that the curvature operator R is in the interiorof CPCSC = CPIC2

for every p ∈ M (cf. Section 13.4.3). Let (M, g(t)) bethe solution of Ricci flow with initial metric g0 on a maximal time interval[0, T ) (the existence of which is guaranteed by the results of Chapter 4). Alsonote that T <∞ since the maximum principle gives a lower bound on scalarcurvature which approaches infinity in finite time (see Section 6.2.1). Thecurvature operators of (M, g0) lie in a compact set K in the interior of thecone CPCSC. By Theorem 13.8 together with Theorem 13.14 (or alternat-ively the results of Section 13.4.2) the cone CPCSC is a closed convex conepreserved by Ricci flow, which is contained in the cone of positive sectionalcurvature and contains the positive curvature operator cone (by Proposi-tion 13.9). Therefore by Theorem 12.2 there exists a pinching family of conesC(s), 0 ≤ s < 1 with C(0) = CPCSC, and by Theorem 12.7 there exists apreserved closed convex set F containing the compact set K, and numbersρ(s) such that F + ρ(s)I ⊂ C(s) for each s > 0. By the maximum principleof Theorem 6.15, the curvature operators of g(t) lie in F for all t ∈ [0, T ).

229

230 14 The Final Argument

Now we perform the blow-up procedure described in Section 8.5. As in-dicated in Theorem 10.21, by the compactness theorem of Chapter 8 to-gether with the injectivity radius lower bound from Chapter 10 (and theregularity estimates from Chapter 7) there exists a limit (M∞, g∞, O∞) ofa sequence of blow-up metrics gi(t) = Qig(ti + Q−1

i t) about points Oi withQi = |R|(Oi, ti) = supM×[0,ti] |R| → ∞, and the limit has |R|(x, t) ≤ 1 for

t ≤ 0. Therefore the curvatures of gi are given by Q−1i times the curvatures

of g, and so lie in the set Q−1i F . On a neighbourhood of O∞, the pullback

metrics Φ∗i gi have scalar curvature bounded away from zero, and so R(g∞)lies in the line R+I by condition (3) of Theorem 12.7.

Schur’s Lemma states that a smooth manifold of dimension at least 3for which R(x) = α(x)I has constant curvature, i.e. α(x) is constant (seeSection 11.3.1). Therefore (M∞, g∞) is a complete Riemannian manifold withconstant positive curvature; hence a spherical space-form by the theoremof Hopf mentioned in the introduction chapter. Since M∞ is compact, theconvergence is in the C∞ sense, rather than only C∞ on compact subsets. Inparticular M is diffeomorphic to a spherical space form. This completes theproof.

14.2 Refined Convergence Result

In this section we discuss more recent work of Brendle [Bre08] in which heproved the following theorem:

Theorem 14.3. Let (M, g0) be a compact Riemannian manifold of dimen-sion n ≥ 3 such that M ×R has positive isotropic curvature. Then M carriesa metric of constant positive curvature, and is diffeomorphic to a sphericalspace form.

As usual, the last part of the statement is given by Hopf’s classification ofspaces with constant positive curvature; the idea of proof is to use Ricci flowto deform the metric g0 to a metric of constant positive curvature.

Remark 14.4. As shown in Section 13.2.1, the cone CPIC1contains the cone

CPIC2= CPCSC, so the assumptions of Theorem 14.3 are weaker than than

in the result proved above. We shall see that the cone CPIC2contains the

cone of 2-positive curvature operators (see Lemma 14.6), so Theorem 14.3also generalises the main results of Chen [Che91] and of Bohm and Wilking[BW08].

We have seen that the cone CPIC1is a closed convex cone in the space Curv

of algebraic curvature operators, which is preserved by the Ricci flow. Thedifficulty is that CPIC1

is no longer contained in the cone of positive sectionalcurvature, so we can no longer apply Theorem 12.2 to find a pinching familyof cones. The work to be done is to find a replacement for this missing step,so that the rest of the argument can be applied as before.

14.2 Refined Convergence Result 231

For n > 3 Brendle [Bre08, Proposition 4] gives a characterization similarto the result of Proposition 13.12:

Proposition 14.5 (Characterisation of CPIC1). Let R ∈ Curv(Rn), for

n ≥ 4. Then R ∈ CPIC1if and only if

R1313 + λ2R1414 +R2323 + λ2R2424 − 2λR1234 ≥ 0

for all orthonormal frames e1, e2, e3, e4 and all λ ∈ [−1, 1].

This implies that PIC on M × R is weaker than 2-positive curvature:1

Lemma 14.6. If R ∈ Curv(Rn), for n ≥ 4, is 2-positive, then R ∈ CPIC1 .

Proof. If R is a 2-positive curvature operator, then for any 4-frame

0 ≤ R(e1 ∧ e3 − λe2 ∧ e4, e1 ∧ e3 − λe2 ∧ e4)

+R(e2 ∧ e3 + λe1 ∧ e4, e2 ∧ e3 + λe1 ∧ e4)

= R1313 + λ2R1414 +R2323 + λ2R2424 − 2λR1234,

where |e1 ∧ e3 − λe2 ∧ e4|2 = |e2 ∧ e3 + λe1 ∧ e4|2 = 1 + λ2 and

〈e1 ∧ e3 − λe2 ∧ e4, e2 ∧ e3 + λe1 ∧ e4〉 = 0.

Therefore by Proposition 14.5, R ∈ CPIC1.

Another reason why the result of Theorem 14.3 seems very satisfying isbecause of its meaning in the three-dimensional case:

Lemma 14.7. On a 3-manifold, the cones

CPIC1(R3) = R ∈ Curv(R3) : Ric(R) ≥ 0

CPIC2(R3) = R ∈ Curv(R3) : R ≥ 0.

That is, the result of Theorem 14.3 for n = 3 recovers Hamilton’s result[Ham82b] for manifolds with positive Ricci curvature, while R ∈ CPIC2

amounts to the assumption of positive sectional curvatures.

Proof. The cone CPIC2is easy to identify: From Proposition 13.9, CPCSC is

contained in the cone of positive sectional curvatures, and contains the coneof positive curvature operators. However in three dimensions these coincide,since the curvature tensor, when n = 3, can be rewritten as a symmetricbilinear form Λ given by (6.5). So if the sectional curvatures are all positive,Λ(v, v) is also positive since it is ‖v‖2 times the sectional curvature of the2-plane normal to v.

1 A self-adjoint bilinear form is 2-positive if it has positive trace on 2-dimensionalsubspaces (equivalently, the sum of the smallest two eigenvalues is positive).


Now consider the cone CPIC1 . First, we show that this is contained in the

cone of positive Ricci curvature: Let R ∈ CPIC1(R3) and v ∈ R3 with ‖v‖ = 1.

Choose an orthonormal basis e1, e2, e3 = v for R3×0, and let e4 = (0, 1).Then by equation (13.2) we have

0 < KC(e1 + ie2, e3 + ie4) = R1313 + R2424 + R1414 + R2323 − 2R1234

= R1313 +R2323

= Rj3j3 = Ric(R)33 = Ric(R)(v, v).

Since v is arbitrary, Ric(R) > 0. For the converse, we must suppose thatRic(R) > 0, and show that all complex sectional curvatures of totally isotropic2-planes in R4

C are positive. Let Π be any such 2-plane, and let X be a vectorin Π orthogonal to e4. As X is isotropic, we can write X = e1 + ie2 for someorthonormal pair e1, e2 in R3 × 0. Moreover Π is totally isotropic, so Π isspanned by e1 + ie2 and (cos θe3 + sin θe4) + i(− sin θe3 + cos θe4) for someθ. Now we can compute the complex sectional curvature directly:

KC(Π) = K(R)C(e1 + ie2, (cos θe3 + sin θe4) + i(− sin θe3 + cos θe4))

= K(R)C(e1 + ie2, (cos θ − i sin θ)(e3 + ie4))

= K(R)C(e1 + ie2, e3 + ie4)

= R1313 +R2323 = Ric(R)33 > 0.

14.2.1 A Preserved Set Between CPIC1and CPIC2

. The main in-gredient in the proof of Theorem 14.3 is the construction of a new preservedset E. Given R ∈ Curv(Rn), define S ∈ Curv(Rn+2) by

S(v1, v2, v3, v4) = R(v1, v2, v3, v4) + 〈x1 ∧ x2, x3 ∧ x4〉

for all vj = (vj , xj) ∈ Rn+2 ' Rn × R2. Then E is defined as follows:2

E = R ∈ Curv(Rn) : S ∈ CPIC(Rn+2).

This is a closed, convex O(n)-invariant set.

Proposition 14.8 (Characterisation of E). Let R ∈ Curv(Rn), for n ≥ 4.Then R ∈ E if and only if

R1313 + λ2R1414 + µ2R2323 + λ2µ2R2424 − 2λµR1234 + (1− λ2)(1− µ2) ≥ 0

for all orthonormal frames e1, e2, e3, e4 and all λ, µ ∈ [−1, 1].

We refer the reader to [Bre08, Proposition 7] for a proof. From Proposi-

tions 13.12, 14.5 and 14.8 we can relate E to the cones CPIC1and CPIC2

:

2 Note that S is the curvature operator of M × S2, so E corresponds to PIC onM × S2.


Corollary 14.9. The set E lies between CPIC1 and CPIC2 in the sense that

CPIC2⊂ E ⊂ CPIC1

. Furthermore we have⋃c>0

(cE) = CPIC1and

⋂c>0

(cE) = CPIC2.

In particular, if K is any compact subset in the interior of CPIC1 , then K ⊂λE for some λ > 0.

Proof. If R ∈ CPIC2 then

R1313 + λ2R1414 + µ2R2323 + λ2µ2R2424 − 2λµR1234 ≥ 0,

so for any c > 0,

R1313 + λ2R1414 + µ2R2323 + λ2µ2R2424 − 2λµR1234

+ c(1− λ2)(1− µ2) ≥ 0,

and R ∈ cE for any c > 0.If R ∈ cE, then choosing µ = 1 in the characterization of Proposition 14.8

givesR1313 + λ2R1414 +R2323 + λ2R2424 − 2λR1234 ≥ 0

for every 4-frame and every |λ| ≤ 1, so R ∈ CPIC1.

Now we show⋃c>0(cE) = CPIC1

. The forward inclusion is clear, so weprove the reverse: Suppose R is not in cE for any c > 0. That is, for any cthere exists a frame Oc and numbers λc, µc ∈ [−1, 1] such that

ROc1313 +λ2

cROc1414 +µ2

cROc2323 +λ2

cµ2cR

Oc2424−2λcµcR

Oc1234 + c(1−λ2

c)(1−µ2c) < 0.

By compactness we can find a subsequence cj →∞ such that Ocj → O, λcj →λ, and µcj → µ. If (1 − λ2)(1 − µ2) > 0 then we have a contradiction sincethe last term approaches infinity, so we can assume without loss of generalitythat µ = 1. But then RO

1313 + λ2RO1414 + RO

2323 + λ2RO2424 − 2λRO

1234 ≤ 0,

contradicting the fact that R is in the interior of CPIC1. This proves that⋃

c>0 cE = CPIC1 .The characterization of the intersection is clear from Propositions 14.8 and

13.12.

The first step in proving that E is preserved by the Ricci flow is thefollowing lemma:

Lemma 14.10. Let R ∈ Curv(Rn), and let S be the induced operator inCurv(Rn+2). Then

S#(va, vb, vc, vd) = R#(va, vb, vc, vd)

for all vectors vi = (vi, xi) ∈ Rn × R2.


Proof. Let e1, . . . , en be a basis for Rn. Suppose also that e1, . . . , en+2 isan orthonormal basis of Rn × R2 such that ek = (ek, (0, 0)) for k = 1, . . . , n.Note that

n+2∑p,q=1

SapcqSbpdq =

n∑p,q=1

SapcqSbpdq +

n+2∑p,q=n+1

SapcqSbpdq

=

n∑p,q=1

RapcqRbpdq + 〈xa, xb〉〈xc, xd〉 .

By interchanging vc and vd we obtain

n+2∑p,q=1

SapdqSbpcq =

n∑p,q=1

RapdqRbpcq + 〈xa, xb〉〈xd, xc〉 .

So by subtracting the first equation from this we get S#abcd = R#

abcd.

Proposition 14.11. Let e1, e2, e3, e4 be an orthonormal 4-frame, let λ, µ ∈[−1, 1] and suppose that the curvature operator R ∈ E. If

R1313 + λ2R1414 + µ2R2323 + λ2µ2R2424 − 2λµR1234

+ (1− λ2)(1− µ2) = 0, (14.1)

then we have

Q1313 + λ2Q1414 + µ2Q2323 + λ2µ2Q2424 − 2λµQ1234 ≥ 0.

Proof. Define the orthonormal 4-frame e1, e2, e3, e4 for Rn × R2 by

e1 = (e1, 0, 0) e2 = (µe2, 0,√

1− µ2)

e3 = (e3, 0, 0) e4 = (λe4,√

1− λ2, 0).

By hypothesis S has nonnegative isotropic curvature, and so (14.1) impliesthat

S1313 + S1414 + S2323 + S2424 − 2S1234 = 0.

From the argument of Section 13.3 (in particular Claim 13.16), or alternat-ively that of Section 13.5 (in particular equation (13.17)), we also have

S#1313 + S#

1414 + S#2323 + S#

2424 + 2S#1342 + 2S#

1423 ≥ 0.

Using Lemma 14.10 we obtain

R#1313 + λ2R#

1414 +µ2R#2323 + λ2µ2R#

2424 + 2λµR#1342 + 2λµR#

1423 ≥ 0. (14.2)

Moreover,


R21313 + λ2R2

1414 + µ2R22323 + λ2µ2R2

2424 + 2λµR21342 + 2λµR2

1423

= (R13pq − λµR24pq)2 + (λR14pq + µR23pq)

2 ≥ 0. (14.3)

The result now follows by adding (14.2) and (14.3), together with the factthat Q = Q(R) satisfies the first Bianchi identity by Remark 11.12.

Now by the same argument used in Section 13.3, we see that Proposi-tion 14.11 implies:

Proposition 14.12. The set E is preserved by the Ricci flow.

14.2.2 A Pinching Set Argument. We are now in a position to givethe outline of the proof of Theorem 14.3, of which the main ingredient isProposition 14.12. The picture looks very nice: We already know that thecone CPIC2

has a pinching family linking it to the constant positive curvature

ray. Now we have the region between CPIC1 and CPIC2 filled out by a family

of preserved sets, each of which is asymptotic to the cone CPIC2near infinity.

It seems highly plausible that with this information we could find a pinchingset to finish the proof.

One might hope to modify the argument in the proof of Theorem 12.2 toproduce a suitable pinching set. The fact that the sets λE are not cones doesnot pose any great difficulty here, however there is a more serious difficulty:The vector field Q does not point strictly into them. For example these setsall contain the radial line through the curvature operator of Sn−1 × R, andthe vector Q at such points is also radial, so is tangent to the boundary of E.Thus our first step is to modify E, using an idea from [BW08, Proposition3.2], to fix this problem. We use the notation of Section 11.4.

Lemma 14.13. Let G be a closed O(n)-invariant preserved convex set, suchthat

1. G \ 0 is in the open half-space of positive scalar curvature;2. G is contained in the cone of non-negative Ricci curvature;3. G contains the cone of positive curvature operators.

Suppose that

b ∈(

0,

√2n(n− 2) + 4− 2

n(n− 2)

), and a = b+

n− 2

2b2.

Then the set la,b(G) is strictly preserved, in the sense that Q(R) is in theinterior of the tangent cone of la,b(G) at any non-zero boundary point R.

Proof. By Lemma 11.31 it suffices to show that the vector field Xa,b = Da,b+Q points strictly into G, and since we know Q points into G it remains toshow Da,b points strictly into G. Using Theorem 11.32, Da,b(R) is given by


Da,b(R) =((n− 2)b2 − 2(a− b)

)Ric∧ Ric

+ 2aRic ∧ Ric + 2b2 Ric 2 ∧ id

+tr(Ric

2)

n+ 2n(n− 1)a

(nb2(1− 2b)− 2(a− b)(1− 2b+ nb2)

)I.

By our choice of a and b, the first term vanishes; by assumption (2) thesecond term is non-negative; and the third is manifestly non-negative. Thelast term is non-negative, since the term in the bracket can be rewritten asb2(2 − 4b − (n − 2)nb2) by the choice of a and b, i.e. 2(a − b) = (n − 2)b2.By examining the roots of the quadratic equation 2 − 4b − n(n − 2)b2 onesee that the desired quantity is strictly positive for the chosen range of b.Therefore Da,b(R) ≥ 0 for R ∈ G. Furthermore, the inequality is strict: Inthis range the last term is strictly positive unless σ = 0, and in that case wehave Ric = Scal

n I, so the first term is strictly positive (since Scal > 0 unlessR = 0 by assumption (1)).

Finally, by assumption (3), we have S/ε ∈ G for any non-negativecurvature operator S and any ε ∈ (0, 1). So by convexity (1 − ε)R + S =(1 − ε)R + ε(S/ε) ∈ G. As G is closed, R + S ∈ G. But then S ∈ G − R ⊂⋃h>0

G−Rh = TRG. Thus the non-negative curvature operators are in TRG,

and the positive curvature operators are in the interior of TRG, so in partic-ular Da,b(R) points strictly into G at R.

Corollary 14.14. For a and b as given in Lemma 14.13, the sets la,b(E) and

la,b(CPIC2) are strictly preserved.

Proof. We verify the assumptions of Lemma 14.13 in these cases: The sets Econtain CPIC2

, so assumption (3) of the lemma follows from Proposition 13.9.Assumptions (1) and (2) hold for n = 3 by Lemma 14.7. For n ≥ 4, Proposi-tion 14.5 with λ = 0 implies R1313 +R2323 ≥ 0 for all orthonormal frames andall R ∈ CPIC1

. For any k, Rickk = 12(n−2)

∑i 6=j, i,j 6=k(Rikik +Rjkjk) ≥ 0, and

Scal = 12(n−2)

∑k

∑i6=j i,j 6=k(Rikik +Rjkjk) ≥ 0. If Scal = 0 then each of the

non-negative summands must be zero, so R1313 +R1212 = 0, and subtractinggives R2323 − R1212 = 0. Since this is true for all frames, we have R1212 = 0for all frames, and hence R = 0.

Now things look much better: Consider the nested family of preserved setsA(s) for 0 < s <∞ defined by

A(s) =

1−ss la(s),b(s)(E) for 0 < s < 1

la(1),b(1)(CPIC2) for s = 1

C(s− 1)⋂la(1),b(1)(CPIC2

) for s > 1

where C(s) is the pinching family of preserved cones constructed by applying

Theorem 12.2 with C(0) = CPIC2 , and we choose


a(s) = b(s) +n− 2

2b(s)2 and b(s) =

s

2

√2n(n− 2) + 4− 2

n(n− 2)

for 0 < s ≤ 1. By Lemma 14.13, A(s) is strictly preserved for 0 < s ≤ 1, andA(s) is also strictly preserved for s > 1 since it is an intersection of two strictlypreserved sets. Since C(s) approaches the ray of constant positive curvature

operators as s→∞, so does A(s). We also have that A(s) approaches CPIC1

as s → 0 by Proposition 14.9, since 1−ss E does and la(s),b(s) approaches the

identity transformation.It is now straightforward to modify the argument of Theorem 12.7 to find

a pinching set: The Theorem below is only a slight modification, and thestructure of the proof is the same.

Theorem 14.15. Let A(s)s∈[0,∞) be a continuous nested family of closedconvex O(n)-invariant preserved sets in Curv of maximal dimension, con-tained in the half-space of curvature operators with positive scalar curvature.Assume that for each s > 0 and for each λ ∈ (0, 1) we have λA(s) ⊆ A(s)and A(s) =

⋂λ>0 (λA(s)) is a strictly preserved cone, in the sense that for

all R ∈ ∂A(s) \ 0, Q(R) is in the interior of the tangent cone to A(s) atR. Assume further that the family of cones A(s) is continuous. Then ifK is any compact set contained in the interior of A(0), there exists a closedconvex O(n)-invariant set F ⊂ A(0) with the following properties:

1. Q(R) ∈ TRF for every R ∈ ∂F ;2. K ⊂ F ;3. For each s > 0 there exists ρ(s) such that F + ρ(s)I ⊂ A(s).

Proof. The family A(s) is continuous in s, and A(s) approaches CPIC1 ass → 0, so given any compact K in the interior of A(0) we have K ⊂ A(s0)for some s0 > 0. We will construct a set F of the form

F = A(s0) ∩∞⋂i=1

(A(si)− 2ihI

),

where h > 0 and (si) is an increasing sequence approaching infinity.Such a set is manifestly convex. This construction also gives condition (3)

of the theorem automatically. Also, since each set A(s) is O(n)-invariant, sois the set F . We first choose s0 > 0 and h ≥ 1 such that

K ⊂ A(s0) ∩ scal(R) ≤ h.

Lemma 14.16. For any s ≥ s0 there exists N(s) ≥ 1 (non-decreasing in s)such that if s ∈ [s0, s], T ∈ ∂A(s) with scal(T ) ≥ N(s), then Q(S) is in thetangent cone to A(s) at T for every S with |S − T | ≤ 2|I|scal(T )/N(s).

Proof of Lemma. If the Lemma fails, then there exist sequences si ∈ [s0, s],Ti ∈ ∂A(si) and Si ∈ Curv with scal(Ti) ≥ i and |Si−Ti| ≤ 2scal(Ti)/i, suchthat Q(Si) is not in the tangent cone to A(si) at Ti.


Passing to a subsequence we can assume that si → s ∈ [s0, s]. For each N ∈N, and for each i ≥ N we have Ti/scal(Ti) ∈ A(si)/scal(Ti)∩scal(R) = 1 ⊆A(si)/N ∩scal(R) = 1, which is compact. Therefore there is a subsequenceconverging in A(s)/N ∩ scal(R) = 1. Taking a diagonal subsequence givesTi/scal(Ti)→ T ∈

⋂N≥1 (A(s)/N) ∩ scal(R) = 1 = A(s) ∩ scal(R) = 1.

Since Q(Si) is not in the tangent cone of A(si) at Ti, there existsa linear function ì with ‖ì‖ = 1 such that ì(Ti) = supA(si) ì andì(Q(Si)) > 0. Passing to a subsequence we have ì → `, and `(T ) =supA(s) `, so T ∈ ∂A(s) and ` ∈ NT A(s). Also we have ì(Q(Si/scal(Ti))) =

ì(Q(Si))/(scal(Ti))2 ≥ 0, and |Si/scal(Ti)− Ti/scal(Ti)| ≤ 2/i → 0. So

Si/scal(Ti) → T , and by continuity of Q, Q(Si/scal(Ti)2) → Q(T ) while

`(Q(T )) = limi→∞ ì(Q(Si/scal(Ti))) ≥ 0. But this is a contradiction to theassumption that A(s) is strictly preserved, since it implies that `(Q(T )) < 0for every ` ∈ NT A(s).

Lemma 14.17. There exists a non-increasing function δ : [s0,∞) → R+

such that whenever s ∈ [s0, s] and R + cI ∈ A(s) with scal(R) ≤ cN(s) forc ≥ 1, then R+ 2cI ∈ A(s+ δ(s)).

Proof of Lemma. Otherwise there exist sequences si ∈ [s0, s], ci ≥ 1 and Ri ∈Curv with Ri+ciI ∈ A(si) and scal(Ri) ≤ ciN(s), but Ri+2ciI /∈ A(si+1/i).If there exists a subsequence with ci bounded, then we can find a subsequencewith ci → c, si → s, Ri → R with R + cI ∈ A(s), scal(R) ≤ cN(s), butR + 2cI /∈ A(s′) for all s′ > s. But this is impossible, since R + cI ∈ A(s)implies that R+ 2cI is in the interior of A(s), so R+ 2cI ∈ A(s+ δ) for someδ > 0 by the continuity of the family A(s).

The remaining possibility is that ci →∞. In this case we have Ri/ci+ I ∈A(si)/ci ⊂ A(s)/k for i ≥ k, and scal(Ri/ci) ≤ N(s). So for a subsequencewe have Ri/ci → R with R+ I ∈ A(s) for some s ∈ [s0, s]. However, we haveRi + 2ciI /∈ A(s + 1/i), so R + 2I /∈ A(s + δ) for all δ > 0. But this is acontradiction since R + 2I is in the interior of A(s), and hence is containedin A(s+ δ) for some δ > 0, by continuity of the family A(s).

We now construct the sequence si inductively by taking si+1 = si + δ(si)for each i ≥ 1. Note that since δ is a non-increasing positive function,limi→∞ s(i) =∞. Let

Fj = A(s0) ∩j⋂i=1

(A(si)− 2ihI

).

We prove that for each j,

Fj+1 ∩ scal(R) ≤ 2jN(sj)h = Fj ∩ scal(R) ≤ 2jN(sj)h. (14.4)

To see this we must show that the set on the right is contained in that onthe left. If R ∈ Fj ∩ scal(R) ≤ 2jN(sj)h then R + 2jhI ∈ A(sj) and


scal(R) ≤ 2jN(sj)h. Therefore R+ 2j+1hI ∈ C(sj+1) by Lemma 14.17 (withc = 2jh).

It follows that F ∩scal(R) ≤ 2jh = Fj ∩scal(R) ≤ 2jh, so F is closed.Also, we have K ⊂ A(s0) ∩ scal(R) ≤ h = F ∩ scal(R) ≤ h ⊂ F .

It remains to show thatQ(R) is in TRF for every R ∈ ∂F . To prove this it isenough to prove, by the inclusion (14.4), that Q(R) is in TR(A(sj)− 2jhI) =TR+2jhIA(sj) for every R ∈ ∂(A(sj) − 2jhI) with scal(R) ≥ 2j−1N(sj)h.Let T = R + 2jhI and S = R, so that scal(T ) ≥ N(sj) and |S − T | ≤2|I|scal(T )/N(sj). By Lemma 14.16 Q(S) is in the tangent cone to C(sj) atT , hence Q(R) ∈ TR+2jhIA(sj) as required.

Remark 14.18. Having constructed the pinching set, the remainder of theproof of Theorem 14.3 is exactly the same as that given in Section 14.1.

Appendix A

Gateaux and Frechet Differentiability

Following [LP03] there are two basic notions of differentiability for functionsf : X → Y between Banach spaces X and Y .

Definition A.1. A function f is said to be Gateaux differentiable at x ifthere exists a bounded linear1 operator Tx ∈ B(X,Y ) such that ∀ v ∈ X,

limt→0

f(x+ tv)− f(x)

t= Txv.

The operator Tx is called the Gateaux derivative of f at x.

If for some fixed v the limit

δvf(x) :=d

dt

∣∣∣t=0

f(x+ tv) = limt→0

f(x+ tv)− f(x)

t

exists, we say f has a directional derivative at x in the direction v. Hencef is Gateaux differentiable at x if and only if all the directional derivativesδvf(x) exist and form a bounded linear operator Df(x) : v 7→ δvf(x).

If the limit (in the sense of the Gateaux derivative) exists uniformly in von the unit sphere of X, we say f is Frechet differentiable at x and Tx is theFrechet derivative of f at x. Equivalently, if we set y = tv then t→ 0 if andonly if y → 0. Thus f is Frechet differentiable at x if for all y,

f(x+ y)− f(x)− Tx(y) = o(‖y‖)

and we call Tx = Df(x) the derivative of f at x.Note that the distinction between the two notion of differentiability is

made by how the limit is taken. The importance being that the limit in theFrechet case only depends on the norm of y.2

1 Some authors drop the requirement for linearity here.2 In terms of ε-δ notation the differences can expressed as follows. Gateaux: ∀ε > 0and ∀v 6= 0, ∃ δ = δ(ε, v) > 0 such that, ‖f(x + tv) − f(x) − tTv‖ ≤ ε|t| whenever

241

242 A Gateaux and Frechet Differentiability

A.1 Properties of the Gateaux Derivative

If the Gateaux derivative exists it unique, since the limit in the definition isunique if it exists.

A function which is Frechet differentiable at a point is continuous there,but this is not the case for Gateaux differentiable functions (even in thefinite dimensional case). For example, the function f : R2 → R defined byf(0, 0) = 0 and f(x, y) = x4y/(x6 + y3) for x2 + y2 > 0 has 0 as its Gateauxderivative at the origin, but fails to be continuous there. This also providesan example of a function which is Gateaux differentiable but not Frechetdifferentiable. Another example is the following: If X is a Banach space, andϕ ∈ X ′ a discontinuous linear functional, then the function f(x) = ‖x‖ϕ(x)is Gateaux differentiable at x = 0 with derivative 0, but f(x) is not Frechetdifferentiable since ϕ does not have limit zero at x = 0.

Proposition A.2 (Mean Value Formula). If f is Gateaux differentiablethen

‖f(y)− f(x)‖ ≤ ‖x− y‖ sup0≤θ≤1

‖Df(θx+ (1− θ)y)‖.

Proof. Choose u∗ ∈ X such that ‖u∗‖ = 1 and ‖f(y) − f(x)‖ = 〈u∗, f(y) −f(x)〉. By applying the mean value theorem to h(t) = 〈u∗, f(x + t(y − x))〉we find that | 〈u∗, f(y)〉− 〈u∗, f(x)〉 | = ‖h(1)−h(0)‖ ≤ sup0≤t≤1 ‖h′(t)‖ and

h′(t) =

⟨u∗,

d

dtf(x+ t(y − x))

⟩=

⟨u∗, lim

s→0

f(x+ (t+ s)(y − x))− f(x+ t(y − x))

s

⟩= 〈u∗, Df(x+ t(y − x)) (y − x)〉 .

So |h′(t)| ≤ ‖Df(x+ t(y − x)) (y − x)‖ ≤ ‖Df(x+ t(y − x))‖ ‖y − x‖.

If the Gateaux derivative exist and is continuous in the following sense,then the two notions coincide.

Proposition A.3. If f is Gateaux differentiable on an open neighbourhoodU of x and Df(x) is continuous,3 then f is also Frechet differentiable at x.

Proof. Fix v and let g(t) = f(x + tv) − f(x) − tDf(x)v, so g(0) = 0. Bycontinuity of the Gateaux derivative with the mean value theorem we findthat

|t| < δ. Frechet: ∀ε > 0, ∃ δ = δ(ε) > 0 such that ‖f(x + v) − f(x) − Tv‖ ≤ ε‖v‖whenever ‖v‖ < δ.3 In the sense that Df : U → B(X,Y ) is continuous at x so that limx→x ‖Df(x) −Df(x)‖ = 0. In words, the derivative depends continuous on the point x.

A.1 Properties of the Gateaux Derivative 243

‖f(x+ tv)− f(x)− tDf(x)v‖ = ‖g(1)‖≤ ‖v‖ sup

0≤t≤1‖Df(x+ tv)−Df(x)‖

= o(‖v‖)

The notion of Gateaux differentiability and Frechet differentiability alsocoincide if f is Lipschitz and dim(X) <∞, that is:

Proposition A.4. Suppose f : X → Y is a Lipschitz function from a finite-dimensional Banach space X to a (possibly infinite-dimensional) Banachspace Y . If f is Gateaux differentiable at some point x, then it is also Frechetdifferentiable at that point.

Proof. As the unit sphere SX of X is compact, it is totally bounded. So givenε > 0 there exists a finite set F = F (ε) ⊂ X such that SX =

⋃uj∈F Bε(uj).

Thus for all u ∈ SX there is an index j such that ‖u− uj‖ < ε.By hypothesis choose δ > 0 such that

‖f(x+ tuj)− f(x)− tDf(x)uj‖ < ε |t|

for |t| < δ and any index j. It follows that for any u ∈ SX ,

‖f(x+ tu)− f(x)− tDf(x)u‖ ≤ ‖f(x+ tu)− f(x+ tuj)‖+ ‖f(x+ tuj)− f(x)− tDf(x)uj‖+ ‖tDf(x)(uj − u)‖

≤ (C + ‖Df(x)‖+ 1)ε |t|

for |t| < δ, where C is the Lipschitz constant of f . Hence δ is independent ofu and so f is also Frechet differentiable at x.

In the infinite dimensional case the story is very different. Broadly speakingin such a situation there are reasonably satisfactory results on the existenceof Gateaux derivatives of Lipschitz functions, while results on existence ofFrechet derivatives are rare and usually very hard to prove. On the otherhand, in many applications it is important to have Frechet derivatives of f ,since they provide genuine local linear approximation to f , unlike the muchweaker Gateaux derivatives.

Appendix B

Cones, Convex Sets and SupportFunctions

The geometric concept of tangency is one of the most important tools inanalysis. Tangent lines to curves and tangent planes to surfaces are definedclassically in terms of differentiation. In convex analysis, the opposite ap-proach is exploited. A generalised tangency is defined geometrically in termsof separation; it is expressed by supporting hyperplanes and half-spaces. Herewe look at convex sets (particularly when they are defined by a set of lin-ear inequalities) and the characterisation of their tangent and normal cones.This will be needed in the proof of the maximum principle for vector bundlesdiscussed in Section 6.4.

B.1 Convex Sets

Let E be a (finite-dimensional) inner product space, and E∗ its dual space.A subset A ⊂ E is convex set if for every v, w ∈ A, θv + (1 − θ)w ∈ Afor all θ ∈ [0, 1]. A set Γ ⊂ E is a cone with vertex u ∈ E if for everyv ∈ Γ we have u + θ(v − u) ∈ Γ for all θ ≥ 0. A half-space is a set ofthe form x ∈ E : `(x) ≤ c where ` is a non-trivial linear function onE, i.e. ` ∈ E∗ \ 0. In such a case we normalise so that ` is an element ofS∗ = ω ∈ E∗ : ‖ω‖ = 1.

A supporting half-space to a closed convex set A is a half-space whichcontains A and has points of A arbitrarily close to its boundary. A supportinghyperplane to A is a hyperplane which is the boundary of a supporting half-space to A. That is, supporting hyperplanes to A take the form x : `(x) = cwhere ` ∈ E∗ \ 0 and c = sup`(v) : v ∈ A.

B.2 Support Functions

If A is a closed convex set in E, the support function of A is a functions = sA : E∗ → R ∪ ∞ defined by

s(`) = sup`(x) : x ∈ A

245

246 B Cones, Convex Sets and Support Functions

for each ` ∈ E∗ \0. Here s is a homogeneous degree one convex function onE∗. For each ` with s(`) <∞, the half-space x : `(x) ≤ s(`) is the uniquesupporting half-space of A which is parallel to x : `(x) ≤ 0.

Theorem B.1. The convex set A is the intersection of its supporting half-spaces:

A =⋂`∈S∗x ∈ E : `(x) ≤ s(`).

Proof. Firstly, the set A is contained in this intersection since it is containedin each of the half-spaces. To prove the reverse inclusion it suffices to showfor any y /∈ A there exists ` ∈ S∗ such that `(y) > s(`).

Let x be the closest point to y in A, and define ` ∈ E∗ by `(z) = 〈z, y−x〉.Suppose `(w) > `(x) for some w ∈ A. Then x + t(w − x) ∈ A for 0 ≤ t ≤ 1,and

d

dt‖y − (x+ t(w − x))‖2

∣∣∣t=0

= −2〈y − x,w − x〉 = −2(`(w)− `(x)) < 0,

contradicting the fact that x is the closed point to y in A. Therefore `(z) ≤`(x) for all z ∈ A, so s(`) = supA ` = `(x) < `(y). The same holds for˜= `/‖`‖ ∈ S∗.

B.3 The Distance From a Convex Set

For a closed convex set A in E, the function dA : E → R given by

dA(x) = inf‖x− y‖ : y ∈ A

is Lipschitz continuous, with Lipschitz constant 1, and is strictly positive onE \A. We call this the distance to A. It has the following characterisation interms of the support function of A.

Theorem B.2. For any y /∈ A,

dA(y) = sup`(y)− s(`) : ` ∈ S∗.

Proof. Let x be the closest point to y in A. So for any ` ∈ S∗ we have

`(y)− s(`) = `(y)− supA` ≤ `(y)− `(x) = `(y − x)

≤ ‖`‖ ‖y − x‖ = ‖y − x‖ = dA(y),

while the particular choice of `( · ) = 〈y−x, · 〉/‖y−x‖ gives equality through-out.

B.5 Convex Sets Defined by Inequalities 247

B.4 Tangent and Normal Cones

A convex set may have non-smooth boundary, so there will not in generalbe a well-defined normal vector or tangent plane. Nevertheless we can makesense of a set of normal vectors, as follows:

Definition B.3. Let A be a closed bounded convex set in E, and let x ∈ ∂A.The normal cone to A at x is defined by

NxA = ` ∈ E∗ : `(x) = s(`).

In other words, NxA is the set of linear functions which achieve their max-imum over A at the point x (so that the corresponding supporting half-spaceshave x in their boundary). The set NxA is a convex cone in E∗ with vertexat the origin.

Complementary to this is the following definition:

Definition B.4. The tangent cone TxA to A at x is the set

TxA =⋂

`∈NxA

z ∈ E : `(z) ≤ 0.

That is, x+TxA is the intersection of the supporting half-spaces of A with xon their boundary. It follows that A−x ⊂ TxA. Indeed TxA may alternativelybe characterised as the closure of

⋃ 1h (A − x) : h > 0. The tangent cone

TxA is a closed convex cone in E with vertex at the origin (in fact it is thesmallest such cone containing A− x).

B.5 Convex Sets Defined by Inequalities

In many cases the convex set A of interest is explicitly presented as an inter-section of half-spaces, in the form

A =⋂`∈B

x ∈ E : `(x) ≤ φ(`) (B.1)

where B is a given closed subset of E∗ \ 0 and φ : B → R is given. If Bdoes not intersect every ray from the origin, this definition will involve onlya subset of the supporting half-spaces of A. In this situation we have thefollowing characterisation of the support function of A:

Theorem B.5. Let E be of dimension n, and suppose A is defined by(B.1). For any ` ∈ E∗ with s(`) < ∞ there exist `1, . . . , `n+1 ∈ B andλ1, . . . , λn+1 ≥ 0 such that

` =

n+1∑i=1

λi`i and s(`) =

n+1∑i=1

λiφ(`i).

248 B Cones, Convex Sets and Support Functions

It follows that the support function s of A on all of E∗ can be recovered fromthe given function φ on B.

Proof. Firstly, for ` ∈ B note that if `(x) = φ(`) for some x ∈ A, then`(x) = sup`(y) : y ∈ A = s(`). Now define

B = R+` ∈ B : ∃x ∈ A with `(x) = φ(`).

That is, B consists of positive scalar multiples of those ` in B for whichequality holds in equation (B.1). Note that B is closed. Also let

φ(ϑ) =

cφ(`) if ϑ = c` where c ≥ 0, ` ∈ B+∞ otherwise

Thus we have that φ(ϑ) = s(ϑ) for ϑ ∈ B. From (B.1) and by the construction

of φ we have

A =⋂ϑ∈E∗

x ∈ E : ϑ(x) ≤ φ(ϑ).

In which case we see that

s(`) = sup`(x) : x ∈ A

= sup`(x) : x ∈ E, ϑ(x) ≤ φ(ϑ),∀ϑ ∈ E∗

= sup`∗(`) : `∗ ≤ φ, `∗ ∈ (E∗)∗

since (E∗)∗ = E. That is, the epigraph of s is the convex hull of the epigraph

of φ (cf. [Roc70, Corollary 12.1.1]). Now we observe by the Caratheodorytheorem [Roc70, Corollary 17.1.3] that

s(`) = inf

n+1∑i=1

λiφ(ì) : ì ∈ B, λi ≥ 0,

n+1∑i=1

λiì = `

.

The infimum is attained since B is closed. The result follows since each ì ∈ Bis a non-negative multiple of some element ¯

i of B with φ(¯i) = s(¯

i).

From this theorem we obtain a useful result for the normal cone:

Theorem B.6. Let E be of dimension n, and suppose A is defined by (B.1).Then for any x ∈ ∂A, NxA is the convex cone generated by B ∩ NxA. Thatis, for any ` ∈ NxA there exist k ≤ n + 1 and `1, . . . , `k ∈ B ∩ NxA andλ1, . . . λk ≥ 0 such that ` =

∑ki=1 λiì.

Proof. Let ` ∈ NxA. By Theorem B.5 there exist `1, . . . , `n+1 and λi ≥ 0such that s(`) =

∑n+1i=1 λiφ(ì) and ` =

∑n+1i=1 λiì. Since ` ∈ NxA we have

`(x) = s(`) =

n+1∑i=1

λis(ì) ≥n+1∑i=1

λiì(x) = `(x),

B.5 Convex Sets Defined by Inequalities 249

so that equality holds throughout, and s(ì) = ì(x) (hence ì ∈ NxA) foreach i with λi > 0.

This in turn gives a useful characterisation of the tangent cone:

Theorem B.7. Let E be of dimension n, and suppose A is defined by (B.1).Then for any x ∈ ∂A,

TxA =⋂

`∈B: `(x)=φ(`)

z ∈ E : `(z) ≤ 0

and the interior of TxA is given by the intersection of the corresponding openhalf-spaces.

Proof. Any point z in TxA satisfies `(z) ≤ 0 for every ` ∈ E \ 0 with`(x) = s(`). In particular, if ` ∈ B and `(x) = φ(x), then φ(x) = s(`) and`(z) ≤ 0. Conversely, if `(z) ≤ 0 for all ` ∈ B with `(x) = φ(`) (equivalently,for all ` ∈ B ∩ NxA) and ϑ is any element of NxA, then by Theorem B.6

there exist ì ∈ B∩NxA and λi > 0 for i = 1, . . . , k such that ϑ =∑ki=1 λiì,

and so ϑ(z) =∑ki=1 λiì(z) ≤ 0. Since this is true for all ϑ ∈ NxA, z is in

TxA.

Appendix C

Canonically Identifying Tensor Spaceswith Lie Algebras

In studying the algebraic decomposition of the curvature tensor, one needs tomake several natural identification between tensor spaces and Lie algebras.By doing so, one is able to use the Lie algebra structure in conjunction withthe tensor space construction to elucidate the structure of the quadratic termsin the curvature evolution equation.

C.1 Lie Algebras

A Lie algebra consists of a finite-dimensional vector space V over a field Fwith a bilinear Lie bracket [, ] : (X,Y ) 7→ [X,Y ] that satisfies the properties:

1. [X,X] = 02. [X, [Y, Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0

for all vectors X,Y and Z.Any algebra A over a field F can be made into a Lie algebra by defining

the bracket[X,Y ] := X · Y − Y ·X.

A special case of this arises when A = End(V ) is the algebra of operatorendomorphisms of a vector space V . In which case the corresponding Liealgebra is called the general Lie algebra gl(V ). Concretely, setting V = Rngives the general linear Lie algebra gl(n,R) of all n × n real matrices withbracket [X,Y ] := XY − Y X. Furthermore, the special linear Lie algebrasl(n,R) is the set of real matrices of trace 0; it is a subalgebra of gl(n,R).The special orthogonal Lie algebra so(n,R) = X ∈ sl(n,R) : XT = −X isthe set of skew-symmetric matrices.

251

252 C Canonically Identifying Tensor Spaces with Lie Algebras

C.2 Tensor Spaces as Lie Algebras

Suppose U = (U, 〈·, ·〉) is a real N -dimensional inner product space withorthonormal basis (eα)Nα=1. Let Eαβ be the matrix of zero’s with a 1 in the(α, β)-th entry. The matrix product then satisfies EαβEλη = δβλEαη.

The tensor space U ⊗ U is equipped with an inner product

〈x⊗ y, u⊗ v〉 = 〈x, u〉〈y, v〉 .

The set (eα ⊗ eβ)Nα,β=1 forms an orthonormal basis. We identify U ⊗ U 'gl(N,R) by defining the linear transformation

x⊗ y : z 7→ 〈y, z〉x (C.1)

for any x⊗ y ∈ U ⊗U . The map simply identifies y with its dual. Under thisidentification, the inner product on gl(N,R) is given by the trace norm:

〈A,B〉 = trATB,

for any A,B ∈ gl(N,R). To see why, observe that eα ⊗ eβ ' Eαβ and so

trETαβEλη = trEβαEλη = tr δαλEβη = δαλδβη = 〈eα ⊗ eβ , eλ ⊗ eη〉 .

C.3 The Space of Second Exterior Powers as a LieAlgebra

Consider the n-dimensional real inner product space V = (V, 〈·, ·〉) with or-thonormal basis (ei)

ni=1. As usual, let (ei) be the corresponding dual basis for

V ∗. Define∧2

V = V ⊗ V/I to be the quotient algebra of the tensor spaceV ⊗ V by the ideal I generated from x⊗ x for x ∈ V . In which case

x ∧ y = x⊗ y (mod I),

for any x, y ∈ V . The space∧2

V is called the second exterior power of Vand elements x∧ y are referred to as bivectors.1 The canonical inner producton∧2

V is given by

〈x ∧ y, u ∧ v〉 = 〈x, u〉〈y, v〉 − 〈x, v〉〈y, u〉 . (C.2)

With respect to this, the set (ei ∧ ej)i<j forms an orthonormal basis for the

n(n − 1)/2-dimensional vector space∧2

V . We identify∧2

V ' so(n) by

1 The geometric interpretation of x ∧ y is that of an oriented area element in theplane spanned by x and y. The object x∧ y is referred to as a bivector as it is a two-dimensional analog to a one-dimensional vector. Whereas a vector is often utilisedto represent a one-dimensional directed quantity (often visualised geometrically asa directed line-segment), a bivector is used to represent a two-dimensional directedquantity (often visualised as an oriented plane-segment).

C.3 The Space of Second Exterior Powers as a Lie Algebra 253

mapping ei ∧ ej to the linear map L(ei ∧ ej) of rank 2 which is a rotationwith angle π/2 in the (i, j)-th plane. This is equivalent defining the lineartransformation

x ∧ y : z 7→ 〈y, z〉x− 〈x, z〉 y. (C.3)

Under this identification, the inner product on so(n) is given by the tracenorm

〈A,B〉 =1

2trATB = −1

2trAB

where A,B ∈ so(n). To see this, note that

(ei ∧ ej)T · (ek ∧ e`) = (Eji − Eij)(Ek` − E`k)

= δikEj` − δi`Ejk + δj`Eik − δjkEi`

and so tr (ei ∧ ej)T · (ek ∧ e`) = 2(δikδj` − δi`δjk) = 2 〈ei ∧ ej , ek ∧ e`〉.

Example C.1. When n = 3 and V = R3 we observe that

e2 ∧ e3 7−→ Rx = E23 − E32 =

0

0 1−1 0

e1 ∧ e3 7−→ Ry = E13 − E31 =

0 0 10 0 0−1 0 0

e1 ∧ e2 7−→ Rz = E12 − E21 =

0 1−1 0

0

where Rx, Ry, Rz are π/2-rotations about the x, y and z axis. Whence anyX ∈ so(3) can be written as

X =

0 c b−c 0 a−b −a 0

= aRx + bRy + cRz,

since XT = −X and trX = 0 by definition. Furthermore, if Y = uRx+vRy+wRz then the inner product 〈X,Y 〉 = au+ bv+ cw = (a, b, c) · (u, v, w) is theusual Euclidean inner product.

C.3.1 The space∧2

V ∗ as a Lie Algebra. As done in the above

passage,∧2

V ∗ = V ∗⊗V ∗/I is the quotient algebra of V ∗⊗V ∗ by the idealI = 〈x⊗ x |x ∈ V ∗〉. The canonical inner product given by (C.2), except nowapplied to dual vectors. The wedge ∧ is an antisymmetric bilinear productwith the additional property that

254 C Canonically Identifying Tensor Spaces with Lie Algebras

(ei ∧ ej)(ek, e`) = det

(ei(ek) ei(e`)ej(ek) ej(e`)

)= δikδj` − δi`δjk.

Any ϕ ∈∧2

V ∗ may be written as

ϕ =1

2

∑i,j

ϕijei ∧ ej =

∑i<j

ϕijei ∧ ej (C.4)

where ϕij := ϕ(ei, ej). Moreover, the pairing of bivectors with its dual is givenby (ei ∧ ej)(ek ∧ e`) = (ei ∧ ej)(ek, e`) in order to preserve orthonormality.

Remark C.2. A quick consistency check confirms the summation conventionused in (C.4) allows the coefficients ϕij that appear in the sum to agree withthe component ϕ(ei, ej). Indeed, we observe that(1

2

∑i,j

ϕijei ∧ ej

)(ek, e`) =

1

2ϕij(δikδj` − δi`δjk) =

1

2(ϕk` − ϕ`k) = ϕk`

which is equal to ϕ(ek, e`) by definition. Furthermore one also find that⟨ϕ, ek ∧ e`

⟩= 1

2

∑i,j ϕij

⟨ei ∧ ej , ek ∧ e`

⟩= ϕk`. Thus the convention is con-

sistent.

We identify∧2

V ∗ with the Lie algebra so(n) by sending ei∧ej 7→ Eij−Ejias before. This equips

∧2V ∗ with a Lie algebra structure. In particular the

bracket

[ei ∧ ej , ek ∧ e`] = (ei ∧ ej) · (ek ∧ e`)− (ek ∧ e`) · (ei ∧ ej)= (Eij − Eji)(Ek` − E`k)− (Ek` − E`k)(Eij − Eji)= EijEk` − EijE`k − EjiEk` + EjiE`k

− EkÈij + EkÈji + E`kEij − E`kEji= δiè

j ∧ ek + δjkei ∧ e` − δikej ∧ e` − δjèi ∧ ek

In which case, given any φ, ψ ∈∧2

V ∗ one computes

[φ, ψ] =1

4φijψk`[e

i ∧ ej , ek ∧ e`]

=1

4φijψk`(δiè

j ∧ ek + δjkei ∧ e` − δikej ∧ e` − δjèi ∧ ek)

=1

4

(φpjψkpe

j ∧ ek + φipψpèi ∧ e` − φpjψpèj ∧ e` − φipψkpei ∧ ek

)=

1

2

∑i,j

(φipψpj − ψipφpj)ei ∧ ej

Therefore we (naturally) define the components of the bracket, with respectto the basis (ei ∧ ej)i<j , by

C.3 The Space of Second Exterior Powers as a Lie Algebra 255

[φ, ψ]ij := φipψpj − ψipφpj (C.5)

for any φ, ψ ∈∧2

V ∗.

C.3.1.1 Structure Constants. Now suppose (ϕα) is an orthonormal basis

for∧2

V ∗. The structure constants cαβγ for the bracket (C.5), with respect tothe basis (ϕα), are defined by

[ϕα, ϕβ ] = cαβγ ϕγ .

As (ϕα) are orthonormal, the structure constants can be directly computedfrom

cαβγ =⟨[ϕα, ϕβ ], ϕγ

⟩.

It is easy to check that the tri-linear form⟨[ϕα, ϕβ ], ϕγ

⟩is fully antisymmet-

ric, thus the structure constants cαβγ are anti-symmetric in all three compon-

ents. Moreover, if (σα) orthonormal basis for∧2

V dual to (ϕα), then thecorresponding structure constants cγαβ are given by

[σα, σβ ] = cγαβσγ .

From the identification of∧2

V with∧2

V ∗ we also have cγαβ = cαβγ .

References

[AM94] Uwe Abresch and Wolfgang T. Meyer, Pinching below 1/4, injectivityradius, and conjugate radius, Journal of Differential Geometry 40 (1994),no. 3, 643–691.

[AM97] , Injectivity radius estimates and sphere theorems, Comparisongeometry, 1997, pp. 1–47.

[AN07] Ben Andrews and Huy T. Nguyen, Four-manifolds with 1/4-pinched flagcurvature, 2007. preprint.

[AS53] W. Ambrose and I. M. Singer, A theorem on holonomy, Transactions ofthe American Mathematical Society 75 (1953), no. 3, 428–443.

[Ban87] Shigetoshi Bando, Real analyticity of solutions of Hamilton’s equation,Mathematishe Zeitschrift 195 (1987), no. 1, 93–97.

[Ber00] Marcel Berger, Riemannian geometry during the second half of the twenti-eth century, Reprint of the 1998 original, University Lecture Series, vol. 17,AMS, 2000.

[Ber03] , A Panoramic View of Riemannian Geometry, Springer-VerlagBerlin Heidelberg, 2003.

[Ber60a] , Les varietes Riemanniennes 1/4-pincees, Annali della ScuolaNormale Superiore di Pisa, Classe di Scienze 3e serie 14 (1960), no. 2,161–170.

[Ber60b] , Sur quelques varietes riemanniennes suffisamment pincees, Bul-letin de la Societe Mathematique de France 88 (1960), 57–71.

[Bes08] Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, 2008. Reprint of the 1987 edition.

[BL00] Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear func-tional analysis, Volume 1, American Mathematical Society ColloquiumPublications, vol. 48, AMS, 2000.

[Bre08] Simon Brendle, A general convergence result for the Ricci flow in higherdimensions, Duke Mathematical Journal 145 (2008), no. 3, 585–601.arXiv:0706.1218v4 [math.DG].

[BS09a] Simon Brendle and Richard Schoen, Manifolds with 1/4-pinched curvatureare space forms, Journal of the American Mathematical Society 22 (2009),no. 1, 287–307. arXiv:0705.0766v3 [math.DG].

[BS09b] , Sphere theorems in geometry, to appear in Surveys in DifferentialGeometry, 2009. arXiv:0904.2604v1 [math.DG].

[BW08] Christoph Bohm and Burkhard Wilking, Manifolds with positivecurvature operators are space forms, Annals of Mathematics 167 (2008),no. 3, 1079–1097. arXiv:math/0606187v1 [math.DG].

257

http://www.msri.org/communications/books/Book30/contents.html

http://www.msri.org/communications/books/Book30/contents.html

http://arxiv.org/abs/0706.1218v4



http://arxiv.org/abs/math/0606187v1

258 REFERENCES

[Car10] Mauro Carfora, Renormalization Group and the Ricci flow, 2010.arXiv:1001.3595v1 [hep-th].

[CCCY03] H. D. Cao, B. Chow, S. C. Chu, and S. T. Yau (eds.), Collected papers onRicci flow, Series in Geometry and Topology, vol. 37, International Press,2003.

[CCG+07] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther,James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni,The Ricci flow: Techniques and applications — Part I: Geometric aspects,Mathematical Surveys and Monographs, vol. 135, AMS, 2007.

[CCG+08] , The Ricci flow: Techniques and applications — Part II: Analyticaspects, Mathematical Surveys and Monographs, vol. 144, AMS, 2008.

[CE08] Jeff Cheeger and David G. Ebin, Comparison theorems in Riemanniangeometry, Revised reprint of the 1975 original, AMS Chelsea Publishing,2008.

[CGT82] Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagationspeed, kernel estimates for functions of the Laplace operator, and thegeometry of complete Riemannian manifolds, Journal of Differential Geo-metry 17 (1982), no. 1, 15–53.

[Che70] Jeff Cheeger, Finiteness theorems for Riemannian manifolds, AmericanJournal of Mathematics 92 (1970), 61–74.

[Che91] Haiwen Chen, Pointwise 1/4-pinched 4-manifolds, Annals of Global Ana-lysis and Geometry 9 (1991), no. 2, 161–176.

[CK04] Bennett Chow and Dan Knopf, The Ricci flow: An Introduction, Math-ematical Surveys and Monographs, vol. 110, AMS, 2004.

[CLN06] Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, GraduateStudies in Mathematics, vol. 77, AMS, 2006.

[Con08] Lawrence Conlon, Differentiable manifolds, 2nd ed., Modern BirkhauserClassics, Birkhauser Boston, 2008.

[CTZ08] Bing-Long Chen, Siu-Hung Tang, and Xi-Ping Zhu, Complete classific-ation of compact four-manifolds with positive isotropic curvature, 2008.arXiv:0810.1999v1 [math.DG].

[CZ06] Bing-Long Chen and Xi-Ping Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature, Journal of Differential Geo-metry 74 (2006), no. 2, 177–264. arXiv:math/0504478v3 [math.DG].

[Dac04] Bernard Dacorogna, Introduction to the calculus of variations, Translatedfrom the 1992 French original, Imperial College Press, London, 2004.

[dC92] Manfredo Perdigao do Carmo, Riemannian geometry, Translated fromthe second Portuguese edition by Francis Flaherty, Mathematics: Theoryand Applications, Birkhauser, 1992.

[DeT83] Dennis M. DeTurck, Deforming metrics in the direction of their Riccitensors, Journal of Differential Geometry 18 (1983), no. 1, 157–162.

[ES64] James Eells Jr and J. H. Sampson, Harmonic mappings of Riemannianmanifolds, American Journal of Mathematics 86 (1964), no. 1, 109–160.

[Esc86] Jost-Hinrich Eschenburg, Local convexity and nonnegative curvature—Gromov’s proof of the sphere theorem, Inventiones Mathematicae 84(1986), no. 3, 507–522.

[Eva98] Lawrence C. Evans, Partial differential equations, Graduate Studies inMathematics, vol. 19, AMS, 1998.

[EW76] J. Eells and J. C. Wood, Restrictions on harmonic maps of surfaces,Topology 15 (1976), no. 3, 263–266.

[Fri85] Daniel Harry Friedan, Nonlinear models in 2 + ε dimensions, Annals ofPhysics 163 (1985), no. 2, 318–419.

[GKR74] Karsten Grove, Hermann Karcher, and Ernst A. Ruh, Jacobi fields andFinsler metrics on compact Lie groups with an application to differenti-able pinching problems, Mathematische Annalen 211 (1974), 7–21.

http://arxiv.org/abs/1001.3595

http://au.arxiv.org/abs/0810.1999v1


REFERENCES 259

[Gro66] Detlef Gromoll, Differenzierbare Strukturen und Metriken positiverKrummung auf Spharen, Mathematische Annalen 164 (1966), 353–371.

[Gro99] David Gross, Renormalization groups, Quantum fields and strings: Acourse for mathematicians, 1999.

[GSW88] Michael B. Green, John H. Schwarz, and Edward Witten, SuperstringTheory 1: Introduction, Cambridge Monographs on Mathematical Phys-ics, Cambridge University Press, 1988.

[GT83] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equa-tions of second order, 2nd ed., Grundlehren der mathematischen Wis-senschaften, vol. 224, Springer-Verlag Berlin Heidelberg, 1983.

[GVZ08] Karsten Grove, Luigi Verdiani, and Wolfgang Ziller, A positively curvedmanifold homeomorphic to T1S4, 2008. arXiv:0809.2304 [math.DG].

[GW09] Roe Goodman and Nolan R. Wallach, Symmetry, Representations, andInvariants, Graduate Texts in Mathematics, vol. 255, Springer, 2009.

[GW88] R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds,Pacific Journal of Mathematics 131 (1988), no. 1, 119–141.

[Ham74] Richard S. Hamilton, Harmonic maps of manifolds with boundary, Lec-ture Notes in Mathematics, vol. 471, Springer-Verlag Berlin Heidelberg,1974.

[Ham82a] , The inverse function theorem of Nash and Moser, Bulletin (newseries) of the American Mathematical Society 7 (1982), no. 1, 65–222.

[Ham82b] , Three-manifolds with positive Ricci curvature, Journal of Differ-ential Geometry 17 (1982), no. 2, 255–306.

[Ham86] , Four-manifolds with positive curvature operator, Journal of Dif-ferential Geometry 24 (1986), no. 2, 153–179.

[Ham93] , The Harnack estimate for the Ricci flow, Journal of DifferentialGeometry 37 (1993), no. 1, 225–243.

[Ham95a] , A compactness property for solutions of the Ricci flow, AmericanJournal of Mathematics 117 (1995), no. 3, 545–572.

[Ham95b] , The formation of singularities in the Ricci flow, Surveys in dif-ferential geometry, 1995, pp. 7–136. Proceedings of the Conference onGeometry and Topology held at Harvard University, Cambridge, Mas-sachusetts, April 23–25, 1993.

[Ham97] , Four-manifolds with positive isotropic curvature, Communica-tions in Analysis and Geometry 5 (1997), no. 1, 1–92.

[Har67] Philip Hartman, On homotopic harmonic maps, Canadian Journal ofMathematics 19 (1967), 673–687.

[Hel62] Sigurdur Helgason, Differential geometry and symmetric spaces, Pure andApplied Mathematics, vol. 12, Academic Press, 1962.

[HK78] Ernst Heintze and Hermann Karcher, A general comparison theorem withapplications to volume estimates for submanifolds, Annales Scientifiquesde l’Ecole Normale Superieure. Quatrieme Serie 11 (1978), no. 4, 451–470.

[Hop01] Heinz Hopf, Collected Papers: Gesammelte Abhandlungen (Beno Eck-mann, ed.), Springer-Verlag Berlin Heidelberg, 2001.

[Hop25] , Ueber Zusammenhange zwischen Topologie und Metrik vonMannigfaltigkeiten, PhD thesis, Friedrich-Wilhelms-Universitat zu Berlin(1925).

[Hop26] , Zum Clifford-Kleinschen Raumproblem, Mathematische Annalen95 (1926), no. 1, 313–339.

[Hui85] Gerhard Huisken, Ricci deformation of the metric on a Riemannian man-ifold, Journal of Differential Geometry 21 (1985), no. 1, 47–62.

[Jos08] Jurgen Jost, Riemannian geometry and geometric analysis, 5th ed., Uni-versitext, Springer-Verlag Berlin Heidelberg, 2008.


http://edoc.hu-berlin.de/docviews/abstract.php?lang=ger&id=26999

260 REFERENCES

[Jos83] , Harmonic mappings between Riemannian manifolds, Proceedingsof the Centre for Mathematics and its Applications, vol. 4, AustralianNational University, Canberra, 1983.

[Kar70] Hermann Karcher, A short proof of Berger’s curvature tensor estimates,Proceedings of the American Mathematical Society 26 (1970), no. 4, 642–644.

[Kaz81] Jerry L. Kazdan, Another proof of Bianchi’s identity in Riemannian geo-metry, Proceedings of the American Mathematical Society 81 (1981),no. 2, 341–342.

[KL08] Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geometry &Topology 12 (2008), no. 5, 2587–2855. arXiv:math/0605667 [math.DG].

[Kli59] Wilhelm Klingenberg, Contributions to Riemannian geometry in thelarge, The Annals of Mathematics 69 (1959), no. 3, 654–666.

[Kli61] , Uber Riemannsche Mannigfaltigkeiten mit positiver Krummung,Commentarii Mathematici Helvetici 35 (1961), no. 1, 47–54.

[KM63] Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres I,The Annals of Mathematics 34 (1963), no. 3, 504–537.

[KN96] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differentialgeometry. Vol. I, Wiley Classics Library, John Wiley & Sons Inc., 1996.Reprint of the 1963 original, A Wiley-Interscience Publication.

[Lee02] John M. Lee, Introduction to smooth manifolds, Graduate Texts in Math-ematics, vol. 218, Springer, 2002.

[Lee97] , Riemannian manifolds: An introduction to curvature, GraduateTexts in Mathematics, vol. 176, Springer, 1997.

[LP03] Joram Lindenstrauss and David Preiss, On Frechet differentiability ofLipschitz maps between Banach spaces, Annals of Mathematics 157(2003), no. 1, 257–288. arXiv:math/0402160v1 [math.FA].

[LP87] John M. Lee and Thomas H. Parker, The Yamabe problem, Bulletin (NewSeries) of the American Matmatical Society 17 (1987), no. 1, 37–91.

[Mar86] Christophe Margerin, Pointwise pinched manifolds are space forms, Geo-metric measure theory and the calculus of variations, 1986, pp. 307–328.

[Mar94a] , Une caracterisation optimale de la structure differentielle stand-ard de la sphere en terme de courbure pour (presque) toutes les dimen-sions. I. Les enonces, Comptes Rendus de l’Academie des Sciences. SerieI. Mathematique 319 (1994), no. 6, 605–607.

[Mar94b] , Une caracterisation optimale de la structure differentielle stand-ard de la sphere en terme de courbure pour (presque) toutes les dimen-sions. II. Le schema de la preuve, Comptes Rendus de l’Academie desSciences. Serie I. Mathematique 319 (1994), no. 7, 713–716.

[Mil56] John W. Milnor, On manifolds homeomorphic to the 7-sphere, The Annalsof Mathematics 64 (1956), no. 2, 399–405.

[MM88] Mario J. Micallef and John Douglas Moore, Minimal two-spheres andthe topology of manifolds with positive curvature on totally isotropic two-planes, Annals of Mathematics 127 (1988), no. 1, 199–227.

[Mok88] Ngaiming Mok, The uniformization theorem for compact Kahler mani-folds of nonnegative holomorphic bisectional curvature, Journal of Differ-ential Geometry 27 (1988), no. 2, 179–214.

[Mos60] Jurgen Moser, A new proof of De Giorgi’s theorem concerning the regu-larity problem for elliptic differential equations, Communications on Pureand Applied Mathematics 13 (1960), 457–468.

[MT07] John W. Morgan and Gang Tian, Ricci flow and the Poincare conjecture,Clay Mathematics Monographs, vol. 3, American Mathematical Society,2007. arXiv:math/0607607v2 [math.DG].

[Ngu08] Huy T. Nguyen, Invariant curvature cones and the Ricci flow, PhD thesis,The Australian National University (2008).

http://arxiv.org/abs/math/0605667

http://arxiv.org/abs/math/0402160


REFERENCES 261

[Ngu10] , Isotropic Curvature and the Ricci Flow, InternationalMathematics Research Notices 2010 (2010), no. 3, 536–558.doi:10.1093/imrn/rnp147.

[Ni04] Lei Ni, The entropy formula for linear heat equation, Journal of GeometricAnalysis 14 (2004), no. 1, 87–100.

[Nis86] Seiki Nishikawa, On deformation of Riemannian metrics and manifoldswith positive curvature operator, Curvature and Topology of RiemannianManifolds, 1986, pp. 202–211.

[Nom56] Nomizu, Katsumi, Lie groups and differential geometry, Publications ofthe Mathematical Society of Japan, vol. 2, The Mathematical Society ofJapan, 1956.

[NW07] Lei Ni and Baoqiang Wu, Complete manifolds with nonnegative curvatureoperator, Proceedings of the American Mathematical Society 135 (2007),no. 9, 3021–3028.

[Oku66] Okubo, Tanjiro, On the differential geometry of frame bundles, Annali diMatematica Pura ed Applicata, Serie Quarta 72 (1966), no. 1, 29–44.

[OSW06] T. Oliynyk, V. Suneeta, and E. Woolgar, A gradient flow for world-sheet nonlinear sigma models, Nuclear Physics B 739 (2006), 441–458.arXiv:hep-th/0510239v2.

[Per02] Grisha Perel’man, The entropy formula for the Ricci flow and its geomet-ric applications, 2002. arXiv:math/0211159v1 [math.DG].

[Pet06] Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Math-ematics, vol. 171, Springer, 2006.

[Pet87] Stefan Peters, Convergence of Riemannian manifolds, Compositio Math-ematica 62 (1987), no. 1, 3–16.

[Pol98] Joseph Polchinski, String theory 1: An introduction to the bosonic string,Cambridge Monographs on Mathematical Physics, Cambridge UniversityPress, 1998.

[PRS08] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Vanishing and fi-niteness results in geometric analysis, A generalization of the Bochnertechnique, Progress in Mathematics, vol. 266, Birkhauser, 2008.

[PW08] Peter Petersen and Frederick Wilhelm, An exotic sphere with positivesectional curvature, 2008. arXiv:0805.0812 [math.DG].

[Rau51] Harry Ernest Rauch, A contribution to differential geometry in the large,The Annals of Mathematics 54 (1951), no. 1, 38–55.

[Roc70] Ralph Tyrell Rockafellar, Convex analysis, Princeton MathematicalSeries, vol. 28, Princeton University Press, 1970.

[Ruh71] Ernst A. Ruh, Curvature and differentiable structure on spheres, Com-mentarii Mathematici Helvetici 46 (1971), 127–136.

[Ruh73] , Krummung und differenzierbare Struktur auf Spharen, II, Math-ematische Annalen 205 (1973), 113–129.

[Ruh82] , Riemannian manifolds with bounded curvature ratios, Journal ofDifferential Geometry 17 (1982), no. 4, 643–653.

[Sch07] Richard Schoen, Manifolds with pointwise 1/4-pinched curvature, PacificNorthwest Geometry Seminar, 2007 Fall Meeting, University of Oregon,2007.

[Sch86] Friedrich Schur, Ueber den Zusammenhang der Raume constantenRiemann’schen Krummungsmaasses mit den projectiven Raumen, Math-ematische Annalen 27 (1886), no. 4, 537–567.

[Shi89] Wan-Xiong Shi, Deforming the metric on complete Riemannian mani-folds, Journal of Differential Geometry 30 (1989), 223–301.

[SSK71] K. Shiohama, M. Sugimoto, and H. Karcher, On the differentiable pinch-ing problem, Mathematische Annalen 195 (1971), 1–16.

http://arxiv.org/abs/hep-th/0510239v2



http://www.math.washington.edu/~lee/PNGS/2007-fall/schoen-problems.pdf

http://www.math.washington.edu/~lee/PNGS/2007-fall/schoen-problems.pdf

262 REFERENCES

[Str08] Michael Struwe, Variational methods, 4th ed., Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge, vol. 34, Springer-Verlag Berlin Heidel-berg, 2008.

[Tao06] Terence Tao, Perel’man’s proof of the Poincare conjecture: A nonlinearPDE perspective, 2006. arXiv:math/0610903v1 [math.DG].

[Tao08] , Research Blog on MATH 285G: Perelmans proof of the Poincareconjecture, Lecture Notes, University of California, Los Angeles, 2008.http://wordpress.com/tag/285g-poincare-conjecture/.

[Top05] Peter Topping, Diameter control under Ricci flow, Communications inAnalysis and Geometry 13 (2005), no. 5, 1039–1055.

[Top06] , Lectures on the Ricci flow, London Mathematical Society LectureNote Series, vol. 325, Cambridge University Press, 2006.

[Top58] V. A. Toponogov, Riemannian spaces having their curvature bounded be-low by a positive number, Doklady Akademii Nauk SSSR 120 (1958),no. 4, 719–721.

[Vaz84] J. L. Vazquez, A strong maximum principle for some quasilinear el-liptic equations, Applied Mathematics and Optimization. An InternationalJournal with Applications to Stochastics 12 (1984), no. 3, 191–202.

[Wal07] Nolan Wallach, The decomposition of the space of curvature operators,2007. http://math.ucsd.edu/~nwallach/curvature.pdf.

[Won61] Wong, Yung-Chow, Recurrent tensors on a linearly connected differen-tiable manifold, Transactions of the American Mathematical Society 99(1961), no. 2, 325–341.


http://wordpress.com/tag/285g-poincare-conjecture/

http://math.ucsd.edu/~nwallach/curvature.pdf

Index

adjunct matrix, 59algebraic curvature operator, 173

decomposition, 181Arzela-Ascoli theorem, 135

Berger, xvilemma, 23

β-function, 164Bianchi identities

contracted second, 22diffeomorphism invariance, 73generalisation, 91

bivector, 252blow-up of singularities, 137

local noncollapsing, 162theorem, 163

bounded geometry, 132

Cartan’s formula, 32Cheeger-Gromov convergence, 128Christoffel symbols, 13C∞-convergence

Cheeger-Gromov, 128on compact sets, 127

-operator, 177compact convergence, see uniform

convergencecompactness theorem

for metrics, 133for Ricci flow, 134

proof, 136cone, 210, 245

pinching family, 195pinching set, 202

conjugate heat operator, 152connection

compatible, 16

definition, 12dual and product bundles, 16second covariant derivative, 15subbundles, 28symmetric, 17tenor fields, 13

contractions, 7metric, 11

convex in the fibre, 103convex set, 245

defined by inequalities, 247distance from, 246

coupling constant, 165curvature operator, see algebraic

curvature operatorcurvature tensor

decomposition, 65definition, 18dual and tensor product bundles, 18evolution equation, 96Riemannian, 20spatial, 91temporal, 91tensor bundles, 19time-derivative, 89

δ-pinching, xviiderivation, 2

tangent bundle, 3DeTurck’s Trick, 75

motivation, 77divergence theorem, 32

Eells and Sampson, 46Einstein metrics, 50Einstein-Hilbert functional, 141

variation, 141

263

264 Index

ellipticityBianchi identities, 72

equicontinuous, 135exotic spheres, xviii

Petersen and Wilhelm, xix

F-functional, 142monotonicity, 148variation, 142

fibre bundledefinition, 2section, 3

Fm-functional, 144Frechet derivative, 241frame bundle, 4, 83

orthonormal, 5, 84time-dependent, 85

frameslocal, 3

Gateaux derivative, 56, 241examples, 242

geodesics, 10global existence, 35short-time existence, 10

gradient flowdefinition, 139Dirichlet energy, 140gradient vector field, 139

harmonic mapconvergence, 46DeTurck’s Trick, 78Dirichlet and Neumann problems, 47energy, 39energy density bounds, 42examples, 38geodesics, 36gradient flow, 39Hartman’s lemma, 44homotopy class, 38invariance property, 79Laplacian, 40uniqueness, 47

heat flow method, 41Hessian, 15Hopf

conjecture, xviconstant curvature metrics, xv

injectivity radius, 132definition, 161local noncollapsing, 161

integration by parts, 32

invariant subbundle, 169invariant subset, 169invariant subspace, 169isotropic

subspace, 208vector, 209

Klingenberg, xvi, 138lemma, 161

Kulkarni-Nomizu product, 63, 179

Laplaciandefinition, 17of curvature, 52, 95

Levi-Civita connection, 17time derivative, 55

Lie algebra, 251linearisation, 71

Ricci tensor, 72local noncollapsing, 157

implies injectivity bounds, 161theorem, 160

locally compact, 126Log Sobolev inequality, 154long-time existence, 120Lotka-Volterra equations, 109

maximum principlescalar, see scalar maximum principlesymmetric 2-tensors, 101, 107vector bundle, see vector bundle

maximum principlemetric tensors, 9

on dual bundles, 11on tensor product bundles, 11product, 10

Moser iteration, 42µ-functional, 153

normal cone, 103, 247null-eigenvector assumption, 101, 108

parabolic rescaling, 51, 137parabolicity, 70parallel subbundle, 28parallel transport, 105, 107

definition, 27invariance under, 103

pinching problem, xvipinching sets, 202pointed Riemannian manifold, 128pointwise δ-pinching, xviiipointwise bounded, 135positive background measure, 144

Index 265

positive complex sectional curvaturedefinition, 209

positive isotropic curvature, xvii, 215conjecture, xxiidefinition, 209

preserved cone, 187preserved set, 187P (temporal curvature tensor), 91pullback bundle, 24

connection, 25restriction, 24

pushforward, 25

quadratic curvature tensor, 52intepretation, 174properties, 176

Rauch, xviregularisation, 164regularity

global shi estimates, 117Rellich compactness theorem, 154renormalisation, 164

flow, 165group equations, 165

restriction, 24Ricci curvature

definition, 21Ricci flow, xix

coupled, 146coupled modified, 145diffeomorphism invariance, 51evolution of curvature, 60exact solutions, 50gradient flow, 144higher dimensions, xxintroduction, 49

Ricci tensortraceless, 64

Ricci-DeTurck flow, 77Riemannian curvature, 20

symmetry properties, 20Riemannian metric

canonical, 141definition, 10series expansion, 29space of, 141

S-tensor, 88scalar maximum principle, 99

comparison principle, 98doubling time estimates, 101, 122, 138scalar curvature lower bounds, 100shi estimates, 119

Schur’s lemma, 182second derivative test, 97second exterior power space, 252

as a Lie algebra, 253section, 3sectional curvature

definition, 22sections

as a module over C∞(M), 3#-operator, 177shi estimates, see regularityσ-models, 165singular solution, 120singularity, 120space forms, xvspace-time

canonical connection, 90spatial tangent bundle, 28, 40, 87

canonical connection, 88vector fields, 88

sphere theoremBrendle and Schoen, xxiChen, xxclassical proof, xviidifferentiable, xixHuisken, 66Rauch-Klingenberg-Berger, xviitopological, xvi

support function, 103, 245supporting

half-space, 245hyperplane, 245

symbol, 69elliptic, 70principal, 70total, 70

tangent cone, 103, 247tensor, 5∗-convention, 118contraction, 6fields, 8products, 6pullback , 25tensor test, 9test for tensorality, 9

torsion tensor, 92totally isotropic subspace, 209trace, 6trilinear form, 1782-positive curvature operator, xx

Uhlenbeck’s trickabstract bundle, 81

266 Index

evolution of curvature, 96orthonormal frame, 82

uniform convergence, 126compact subsets, 126, 135

variation formulascurvature, 56, 90metric, 54

vector bundledefinition, 2dual, 8maximum principle, 103metrics, 11subbundle, 4tensor, 8

tensor product, 8time-dependent metrics, 87

vector field points into the set, 103

W-functional, 149compatible, 153gradient flow, 152monotonicity, 152symmetry properties, 149variation, 150

wedge, 252wedge product, 179Weyl tensor, 183

definition, 64

The Ricci Flow in Riemannian Geometry - MSIandrews/book.pdf · The Ricci Flow in Riemannian...

Documents

Transcript of The Ricci Flow in Riemannian Geometry - MSIandrews/book.pdf · The Ricci Flow in Riemannian...