The rate of change of width under Ricci ﬂoThe rate of change of width under Ricci ﬂow A part of...

F A C U L T Y O F S C I E N C E U N I V E R S I T Y O F C O P E N H A G E N

Master’s thesisStefan Sommer

The rate of change of width under Ricci flowA part of the proof of the Poincaré conjecture

Speciale for cand.scient graden i matematikInstitut for matematiske fag, Københavns Universitet

Thesis for the Master degree in MathematicsDepartment of Mathematical Sciences, University of Copenhagen

Academic advisor: Jan Philip Solovej

Submitted: October 28, 2008

Abstract

To determine the topology of a manifold, it is useful to investigate if it admits a metricof constant curvature. A strategy for doing this consists of evolving an initial metric onthe manifold by an evolution equation. In this thesis, we present a result concerning theevolution of the notion of width connected to a manifold; we prove that, when the metric ona compact, simply connected 3-dimensional manifold evolves under Ricci flow, there exists acertain upper bound on the rate of change of width.

As a consequence of the result, we show that, under the assumptions above, the evolu-tion cannot continue for infinite time. This particular result was proved by both Perelmanand Colding & Minicozzi II, and was used by Perelman to finish the proof of the Poincareconjecture. We present an exposition of parts of Colding & Minicozzi II’s proof.

The width connected to a manifold is measured as the infimum of maximal slice energiesof sweepouts. A sequence of sweepouts with maximal slice energy converging to the infimumcan be modified to become closer to being harmonic by use of a sweepout tightening map.The map is constructed by a harmonic replacement procedure. This construction is a centralpart of the result.

Using the notion of bubble convergence, a subsequence of the modified sweepouts mustconverge to a collection of harmonic maps. Since we know the rate of change area of suchmaps, the convergence gives information on the rate of change of width. The bubble com-pactness result, which asserts the convergence of a subsequence, constitutes the second majorpart of the thesis.

Resume

For at bestemme en mangfoldigheds topologi, er det nyttigt at undersøge om den kanudstyres med en metrik af konstant krumning. En strategi til at gøre dette bestar i atspecificere hvordan en given startmetrik pa mangfoldigheden kan ændres. I dette specialebeskriver vi et resultat omhandlende ændringen af begrebet bredde, som forbindes meden mangfoldighed. Vi beviser, at der eksisterer en bestemt øvre grænse for ændringsraten,nar metrikken pa en kompakt, enkeltsammenhængende 3-dimensional mangfoldighed ændresunder Ricci flow.

Som en konsekvens af resultatet viser vi, at ændringen ikke kan fortsætte i uendeligtid under antagelserne ovenfor. Dette resultat blev bevist af bade Perelman og Colding &Minicozzi II, og blev brugt af Perelman til at færdiggøre beviset for Poincares formodning.Vi giver en fremstilling af dele af Colding & Minicozzi II’s bevis.

Bredden forbundet til en mangfoldighed bliver malt som et infimum af maksimale skiveen-ergier af sakaldte sweepouts. En følge af sweepouts med maksimal skiveenergi gaende modet sadant infimum kan modificeres til at være tættere pa at være harmonisk. Dette gøresved brug af en sweepout-strammende afbildning, som konstrueres ved en harmonisk substi-tutionsprocedure. Konstruktionen er en central del af resultatet.

Ved brug af begrebet boblekonvergens vises det, at en delfølge af de modificerede sweep-outs ma konvergere mod en familie af harmoniske afbildninger. Da vi kender ændringsratenfor arealet af sadanne afbildninger, far vi herved information om ændringsraten for bredden.Boblekompakthedsresultatet, som sikrer konvergens af en delfølge, udgør den anden hoveddelaf specialet.

i

Til min salig Papa

Contents

Contents vIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Overview 11.1 The Poincare conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hamilton-Perelman’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Perelman’s contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Foundations 52.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Poincare inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Energy and area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Harmonic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Regularity and other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 Change of area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Width and existence time 213.1 Sweepouts and width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Rate of change of width and existence time . . . . . . . . . . . . . . . . . . . . . 243.3 The proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Tightening sweepouts 314.1 Sweepout tightening maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Removing non-constant harmonic slices . . . . . . . . . . . . . . . . . . . . . . . 334.3 Harmonic replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 Comparison maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Continuity of HB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.6 Continuity with scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.7 Continuity of H(·,B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.8 Energy decrease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.9 Construction of the map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Bubble convergence 655.1 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Almost harmonic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Energy preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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Contents

5.4 Varifold convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Combining the results 97

Appendix A Proofs of foundational material 99

Appendix B Differential Geometry 111B.1 The Grassmannian bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111B.2 Tubular neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Appendix C Miscellaneous 115C.2 Convergence in measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116C.3 Generalized difference quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Bibliography 121

Index 125

vi

Introduction

Introduction

In 2003 Grisha Perelman presented a proof of one of the most famous unsolved problems inmathematics; the Poincare conjecture. An important part of his proof was to show that a com-pact, simply connected, 3-dimensional Riemannian manifold under Ricci flow becomes extinctin finite time. The result was proved in the final of the three papers comprising Perelman’sproof of the Poincare conjecture. Perelman had discussed the issue with T. Colding and W.Minicozzi II, who, as a result of this, published a different proof shortly after Perelman’s paper.Their paper, [CM05], was quite brief. To make it accessible to broader audience, Colding andMinicozzi later made an expanded version, [CM07a], available. Though a reader familiar withthe basics of Riemannian geometry will find this paper readable and the strategy of the proofclear, a number of the arguments may still be quite hard to follow. It is the purpose of thisthesis to present Colding and Minicozzi’s proof in an even more accessible way.

Material covered

There are two reasons for choosing Colding and Minicozzi’s proof over Perelman’s. First of all,the availability of the expanded version of Colding and Minicozzi II’s proof in [CM07a] is a goodstarting point for understanding their arguments. In addition, the author of this thesis had theopportunity to meet with Colding when he visited Copenhagen in the summer of 2008. Havingthe chance to meet on first hand was both motivating and helpful in resolving some questionsand thus an additional advantage of choosing Colding and Minicozzi’s approach.

To be able to present more arguments in depth than [CM07a], most, but not all, of thematerial of the paper is covered. The focus is especially on the two main parts of [CM07a];tightening of sweepouts and bubble compactness of almost harmonic, almost conformal maps.For the latter topic, a result on almost harmonic maps from cylinders is left out. Though theresult has a central role in proving the energy preservation property of bubble convergence, thefocus in this thesis will be on the bubble convergence construction as opposed to properties ofharmonic maps.

The actual result concerning the rate of change of width is also covered and applied toRicci flows without surgery. This implies that such flows can only exist in finite time periods.Extending the result to cover Ricci flows with surgery is a matter of understanding the intricaciesof the surgery process and no modification to the rate of change of width result is needed for thisto work. If this is done, it follows that the flow becomes extinct in finite time. In this thesis onlyflows without surgery are covered, and, for this reason, the result presented will be referred toas finite existence time. In addition, neither the fact that simply connectedness of the manifoldimplies positive width nor some additional width related results will be proved.

Before presenting any of the above material, we state the main results of the thesis, outlinethe structure, and give a short introduction to the Poincare conjecture and the role of finiteextinction time in the proof.

Prerequisites

The reader is expected to be familiar with topological, differential, and Riemannian manifolds.The material covered by standard graduate courses on the subjects should suffice. Furthermore,familiarity with Sobolev spaces of maps between Euclidean spaces is assumed. In Chapter 2additional needed background material is covered. The presentation will be brief and focusedonly on the needed results. An index can be found at the end of the thesis, hopefully making it

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Contents

easier to use Chapter 2 as a reference for the background material while reading the rest of thetext.

Writing style

We aim at making the thesis as readable as possible. Though we would like to write out everynon-trivial argument in detail, the amount of material covered makes this impossible withoutthe text being overly long. Some parts of the thesis are quite technical, but we try as much aspossible to motivate the arguments used. Whenever we find it may help the reader, we writereferences above relation symbols in equations. While this sometimes slightly distorts the visuallayout, we expect it will make the arguments easier to follow.

Differences from [CM07a]

The overall structures of the proofs presented follow those of [CM07a]. Indeed, the overallstructure of the thesis is close to that of [CM07a]. The texts differ in the level of detail, inarguments left to the reader in [CM07a] being fully proved here, in numerous added Lemmas,in the use of other articles to reconstruct proofs only sketched in [CM07a], in the Foundationschapter, in added explanations and references. Concluding each chapter we give a few notesregarding the presentation, including the most important of those differences.

Acknowledgments

The author would like to thank his academic adviser Jan Philip Solovej, who suggested thesubject of the thesis and who, throughout the last years of my studies, has been invaluable indiscussing mathematics as well as the ever lasting subject of my future career plans. In addition,I would like to thank Tobias Colding for taking his time to answer questions and offering to advisemy future studies in geometry. Finally, I am grateful to the Department of Mathematics at theUniversity of Washington, Seattle, and John M. Lee in particular, for spawning my interest ingeometry and pushing me through the hardest and most rewarding year of my studies.

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1

Overview

Let M be a compact, 3-dimensional manifold. Later in this thesis, we will define a notion ofwidth of M . The width will be connected to the notion of sweepouts. Assume for a momentthat sweepouts and the space ΩM of sweepouts have been defined. Let [σ] denote the homotopyclass of sweepouts homotopic to σ ∈ ΩM . Suppose the width at time t connected to [σ] isdenoted Wgt([σ]), and let D+

t Wgt([σ]) be the lim sup of forward difference quotients of Wgt([σ])with respect to t. We are then able to state the main result of the thesis:

Theorem 1 (Upper bound on the rate of change of width ([CM05])). Let M be a simplyconnected, compact, differentiable, 3-dimensional manifold and gt a Ricci flow without surgerydefined on an interval I = [0, T ), T ∈ R+∪∞. Then there exist C(M, g0) > 0 and a sweepoutσ ∈ ΩM of M such that, for all t ∈ I,

D+t Wgt([σ]) ≤ −4π +

34(t+ C)

Wgt([σ]) .

The upper bound on the rate of change of width prevents the flow from existing for infinitetime, since the width cannot decrease below zero to become negative. We will use this to showthe following corollary:

Corollary 2 (Finite existence time ([Per03a],[CM05])). Let M be a simply connected, com-pact, differentiable, 3-dimensional manifold and gt a Ricci flow without surgery defined on aninterval I = [0, T ), T ∈ R+ ∪ ∞. Then T <∞.

In Chapter 3 we define sweepouts and width, bound the rate of change of width, and de-duce the implied finite existence time for Ricci flows without surgery. The proof of one resultused in Chapter 3, Theorem 3.3.1, relies on two additional results: the existence of a sweepouttightening map and bubble compactness of almost harmonic, almost conformal maps. We there-fore postpone the proof of the theorem until Chapter 6 after these results have been covered inChapter 4 and 5. Background material needed for the rest of the thesis is given in Chapter 2

Before venturing on to the parts of the thesis leading up to the proof of the above results, webriefly describe the origin of the interest in the rate of change of width and finite existence time.This will lead us to consider the Poincare conjecture and the proof of Hamilton and Perelman.

1.1 The Poincare conjecture

The Poincare conjecture concerns properties for characterizing compact 3-dimensional manifolds.More precisely, it is conjectured that any 3-dimensional compact manifold without boundary

1

1. Overview

and with trivial fundamental group is homeomorphic to the sphere S3. The conjecture turnsout to be true, but finding the answer proved hard enough for it to become one of mathematicsmost famous unsolved problems. It remained so for almost a hundred years after the Frenchmathematician Henry Poincare stated it in 1904.

The proof of the conjecture was given in 2002 and 2003 by Grisha Perelman in three articles;[Per02], [Per03b], and [Per03a]. He continued a program laid out by Richard Hamilton for solvingthe conjecture using an evolution equation for the geometry of Riemannian manifolds called theRicci flow. For this reason, the proof is often attributed to both Hamilton and Perelman.

Before discussing the proof, it is worth considering the equivalent question in dimensionsnot equal to 3. It is well-kown that compact 2-manifolds are completely characterized by theirfundamental group, implying that a simply connected compact 2-manifold is homeomorphic toS2. In the nineteen sixties it was proved that for 5-dimensions and above, the correspondingquestion also has a positive answer. The 4-dimensional case was solved in 1982.

We note that a more detailed overview of Hamilton and Perelman’s proof can be found in theintroduction of [MT07]. The reader may also find interest in the description of the conjecture,the circumstances regarding the publishing of Perelman’s papers, and background on Perelmanhimself in [NG06].

1.2 Hamilton-Perelman’s proof

Though the Poincare conjecture is a purely topological question, Hamilton and Perelman usedgeometrical methods in their proof. To understand the mechanics of this, we must recall thatcertain connections between topology and geometry exist. Given a topological manifold M ofdimension n there exists a standard method of equipping it with differential and geometric struc-ture, making it a Riemannian manifold (M, g). The geometric structure gives rise to notionsof curvature. One of these is the sectional curvature and it is a classic theorem that, if M iscompact and has constant positive sectional curvature, the universal covering of M is homeo-morphic to Sn. In particular, if M is also simply connected, M itself is homeomorphic to Sn.Hence, certain geometries restrict the underlying topology.

This observation leads to the following overall strategy for proving the conjecture. Given acompact, simply connected, 3-dimensional manifold M , find a differential structure and a metricg0 making (M, g0) a Riemannian manifold. Then deform g0 creating a family of metrics gt,and hope that for some t, (M, gt) has constant positive sectional curvature. If this turns out tobe the case, the theorem above would imply that M is homeomorphic to S3 and hence completethe proof.

1.3 The Ricci flow

The part missing for the above strategy to work is determining how to deform the metric.While we may not be able to find a deformation which actually generates a metric of constantcurvature, we will nevertheless have this idea in mind. We will therefore look for some way ofsmoothing out irregularities in the geometry; areas of large curvature should tend to curve lessand vice versa. In 1982 Hamilton found a way of doing this by defining the Ricci flow. TheRicci flow states that a family of metrics gt should evolve by the Ricci flow equation

∂tgt = −2 Ricgt ,

2

Perelman’s contributions

where Ricgt is the Ricci curvature of the manifold at time t. One quick observation is that metricsof constant curvature are, up to rescaling, fixed points of the flow. One would expect suchbehavior of a deformation moving the geometry towards constant curvature. In 2 dimensions,the Ricci flow - or more precisely, a solution to the Ricci flow - observes exactly the behaviorof smoothening irregularities. The same is to some extent the case in 3 dimensions. Using theRicci flow, Hamilton proved that if the Ricci curvature of g0 is positive, M does admit a metricof constant positive curvature. But the analysis is more complicated in the general case. Oneproblem is that singularities in the flow might occur. Hamilton proved short time existence ofthe flow, but could produce manifolds on which long time existence was impossible due to theexistence of singularities. Though some special cases of singularities with nice structure could bedealt with by a process called surgery, Hamilton could not rule out the existence of singularitieswhich could not be handled through surgery.

1.4 Perelman’s contributions

The obstacles that Hamilton was not able to overcome were ultimately dealt with by Perelman.Perelman ruled out the existence of singularities of all but two kinds. He differed between thetwo by considering on which subsets of the manifold the curvature tends to infinity. For thefirst kind this happens only on a proper subset and the curvature is bounded on the rest of themanifold. Such singularities are called ε-necks, due to the fact that the geometry of the neck closeto the singularity time is close to the geometry of S2 × I for an interval I. Hamilton knew howto perform surgery on ε-necks, but Perelman improved the surgery procedure. Loosely speaking,surgery consists of excising the neck part of the manifold obtaining two components with holes,which are then patched up by gluing 3-dimensional balls to the hole of each component. Afterthis is done, the flow can continue on each of the obtained components and due to the nicetopological structure of the neck, one can keep track of the topological changes to the manifoldcaused by the surgery. Perelman was able to construct the metric on the patches so that hecould rule out the singular times accumulating; he was able to show that in any finite timeperiod, only a finite number of singularities can arise.

The second kind of singularities is different in that one actually seeks for them to occur. Inthis case, the curvature at all points of the manifold tends to infinity. Perelman could showdirectly what the underlying topology will be in this case and he could therefore stop the flowhaving no need for additional information. Perelman called this extinction time.

To review, one starts with a manifold, compact and simply connected and any metric g0.The metric is now changed using a Ricci flow until a singularity occurs. If it is of the first kind,the neck is excised and the flow is restarted for each of the obtained components. If, for eachcomponent, at some point a singularity of the second kind occurs - the component becomingextinct - one has full knowledge of the topology of that component. If all components becomeextinct and since we know which topological changes happen during the surgery process, we canreconstruct the original topology from the topology of the components. Indeed, Perelman couldconclude that the topology will be that of S3.

For this strategy to work, we need to ensure that all components become extinct in finitetime. Perelman proved this in his last paper, and the result, except for the parts related tosurgery, is the focus of this thesis.

3

2

Foundations

In this chapter foundational material necessary in the rest of the thesis is presented. The purposeis to introduce notation, to present precise definitions for the notions we will use, and to provesome results not present in the books to which we refer. We will assume that the reader has agood understanding of differential and Riemannian geometry and is familiar with Sobolev spacesof maps between Euclidean spaces. From this starting point, we will define the notions we willlater use. It is worth noting, that the rest of the thesis will use the results from this chapter ina tool-like fashion; we will refer to specific results, but not use technical details or deep insightin the theory behind the results. Therefore, it should be possible to understand the rest of thethesis on the basis of reading this chapter. We advice the reader wanting more details to consultthe references given throughout the text.

We will introduce notation as we describe each part of the material. Proofs of the statedresults are located in Appendix A. Such proofs can consist of just a reference or be given in fulldetail. Results may not be stated or proved in the most general setting possible; we only focuson developing what is necessary for the rest of the thesis.

A brief note on the use of constants: We will rarely have constants in the sense of fixed realnumbers. Rather, we will have reals depending on objects, which will be unchanged throughmost of the thesis. E.g., the lower bound on the energy of a harmonic map will depend on thedomain and target manifolds of the map. Since the domain Σ will be S2 and the target manifoldM may be thought of as fixed, the lower bound can be regarded as constant. In such cases, wewill use the notation ε(Σ,M) when defining ε to indicate the dependence of Σ and M .

2.1 Manifolds

Throughout the thesis, the word manifold will be used for smooth, connected manifolds withoutboundary. A surface is a 2-dimensional manifold without boundary. We will often use Σ todenote a surface and M to denote a manifold. If we wish to specify the dimension of M to ben, we write Mn.

The notation related to differential geometry mostly follows the notation in [Lee03]. Inshort, we use the Einstein summation convention where appropriate. Thus viei denotes the sum∑

i viei. Vectors will have lower indices. The tangent bundle is denoted TM , the differential or

pushforward of F : M → N , dF , and its restriction to TpM , dpF . The bundle of k-covariant,l-contravariant tensors is denoted Tk

l M , and T kl M the corresponding space of tensor fields. IfM ⊂ M is immersed, we let NM be the normal bundle and N M the smooth sections of NM .We write cl (U) for the closure of a set U and int (U) for the interior of U .

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2. Foundations

When endowing a manifold M with a metric g, we will write (M, g) and say that M isRiemannian. When it is clear from the context that a manifold or surface must be Riemannian,we will not always explicitly mention it. Metrics will always be smooth. The distance mapassociated with the metric is denoted d(M,g) or dM . Often Σ will be the domain manifold withrespect to some map and M the target. In such situations, the metric on the domain manifoldwill be denoted h and the metric on the target g. We will make use of the Embedding Theoremof Nash to isometrically embed (M, g) in Rm for some m > 0, confer [Nas56, Theorem 3]. Notethat the embedding will change if we change the metric.

By a geodesic ball Br(x), x ∈M , r > 0 of (M, g) we mean a set V such that expx |Br(0) is adiffeomorphism onto its image V .

A metric h on a surface Σ is called conformal if, for each point x ∈ Σ, there exists a chartcontaining x in which hij = ρ2δij with ρ : M → R being a positive function. Such a chartis called conformal. A map f ∈ C1(Σ,M) to a manifold (M, g) is said to be conformal if themetric h on Σ is conformal and for each conformal chart on Σ, g(∂x1f, ∂x1f) = g(∂x2f, ∂x2f)and g(∂x1f, ∂x2f) = 0. A conformal chart is a conformal map. Later we will introduce mapswith non-continuous partial derivatives. Such maps are said to be conformal almost everywhere,if the two equalities are satisfied almost everywhere.

2.1.1 The sphere S2

We will let the compact surface S2 ⊆ R3 have the induced metric. We let S2+ denote the northern

hemisphere (x1, x2, x3) |x3 > 0 and S2− the southern hemisphere (x1, x2, x3) |x3 < 0. In

addition, the set (x1, x2, x3) |x3 = 0 is called the equator and we will let p+ and p− denotethe north pole (0, 0, 1) and south pole (0, 0,−1) respectively.

The stereographic projection Πp− : S2 \ p+ → R2 is the inverse of the map

(x, y) 7→ 1x2 + y2 + 1

(2x, 2y, x2 + y2 − 1

). (2.1.1)

Let p ∈ S2 and p is antipodal point. We let Πp,R : S2 \ p → R2 be a composition Π R, whereR ∈ O(3) is an isometry S2 → S2 taking p to p−. We will never be concerned with which R wechoose, thus we abuse notation and write just Πp meaning Πp,R for some relevant R.

Lemma 2.1.1. The metric on S2 is conformal. In addition, for any p ∈ S2, the stereographicprojection Πp is conformal.

Let Dλ : R2 → R2, λ > 0 be the dilation z 7→ λz. A dilation Dp,λ : S2 \ p → S2 \ p is acomposition (Πp)−1 Dλ Πp. We extend such maps to maps S2 → S2 by defining Dp,λ(p) = p.It is easily checked that maps so defined are diffeomorphisms. Thus we can regard dilations asmaps S2 → S2.

Since dilations of R2 are conformal, we get the following corollary of Lemma 2.1.1.

Corollary 2.1.2. For any λ > 0 and p ∈ S2, the dilation Dp,λ is conformal.

The conformality of the stereographic projection will allow us to pull back maps S2 → Mto R2 without changing characteristics such as the energy which we will define shortly. In fact,this will be so important that we will introduce a way of specifying radius and center of ballsunder such a pullback.

We let a projected ball B be a pair (U,Λ), such that U ⊂ S2 and Λ : S2 → R2 a compositionof a finite number of dilations S2 → S2 and a stereographic projection, such that, for some r > 0

6

The sphere S2

and x ∈ R2, the ball Br(x) ⊂ R2 is the image of U under Λ. That is, there exist λ1, . . . , λk,λi > 0 and p, p1, . . . , pk, p, pi ∈ S2 such that Λ = Πp Dpk,λk · · · Dp1,λ1 and Br(x) = Λ(U).We often write just B for the set U ⊂ S2 letting the map be implicit. When we need to specify rand x above, we write (U,Λ, r, x) or Br(x). A closed projected ball is the closure of B for someprojected ball B.

1

cos(d(p-,p))

sin(d(p-,p))

d(p-,p)

p

p_(p)

Figure 2.1: Change of radii under stereographic projection.

As sketched in Figure 2.1, for any p ∈ S2,

‖Πp−(p)‖R2 =sin(d(p, p−))

1 + cos(d(p, p−)). (2.1.2)

For some r ∈ (0, π), let U be the geodesic ball Br(0). Then (U,Πp−) is a projected ball with

Λ(U) =

(x, y) ∈ R2 | ‖(x, y)‖ < sin(r)1 + cos(r)

.

By the following proposition, the converse is also true.

Proposition 2.1.3. The set of projected balls and the set of geodesic balls in S2 are equal.

As a result of the proposition, the notion of projected balls has the sole purpose of giving usa different way of specifying radii and center of balls. E.g., in the light of (2.1.2), if the projectedball Br(x) is equal to the geodesic ball Br(x), we in general do not have Bcr(x) = Bcr(x) forc > 0. Varying the radius of a projected ball while keeping the center in R2 fixed will be used somuch that we will introduce the notation cB to denote the projected ball Bcr(x) when B = Br(x).In addition, if B = Bα is a family of projected balls, we will let cB be the family cBα. Fora projected ball B, we will sometimes be interested in the radius of the corresponding geodesicball. We will call this radius the geodesic radius of B.

In addition to the conformality of Λ when (U,Λ) is a projected ball, we will use the resultof the following lemma. The availability of this result is the reason for defining Λ to be thesomewhat complicated composition of known maps, instead of just requiring it to be someconformal map S2 → R2. By a half space in R2 we mean a set

(x, y) ∈ R2 | kxx+ kyy < α

for

kx, ky, α ∈ R and one of kx or ky non-zero.

7

2. Foundations

Lemma 2.1.4. Let B1 = (U1,Λ1) and B2 = (U2,Λ2) be projected balls. Then Λ2(U1) and Λ1(U2)are balls, complement of closures of balls in R2, or half spaces in R2.

Projected balls will always be written in calligraphic font; B, whereas geodesic balls andballs in Euclidean spaces are written in normal font; B. Collections of balls will be written insans serif font; B = B1, . . . ,Bk.

2.1.2 Families of metrics

Though we will in general work with a fixed smooth manifold M , the metric on M will not befixed. Usually, gt will be a family of metrics on M parametrized by t on some interval I. Wesay that gt is a smooth family if the map I ×M → T2M , (t, p) 7→ gt(p) is smooth. Whendealing with quantities depending on the metric, e.g. the scalar curvature S, we add the metricas subscript to indicate the dependence, e.g. Sgt when gt is a family of metrics.

2.1.3 Curvature and Ricci flow

Let (M, g) be a Riemannian manifold and let R ∈ T 31(M) denote the curvature endomorphism

R(X,Y, Z) = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z .

The choice of sign on R follows [Lee97] but is not important. In coordinates, we write R =R lijk dx

i ⊗ dxj ⊗ dxk ⊗ ∂l with R lijk defined by R(∂i, ∂j)∂k = R l

ijk ∂l. The Ricci curvaturetensor Ric ∈ T 2(M) is the trace of R on the first and last entries, confer [Lee97, Page 124]. Incoordinates,

Ric = Rijdxi ⊗ dxj

where Rij = R kkij .

Definition 2.1.5 (Ricci flow). A Ricci flow is a smooth family of metrics gt, t ∈ I on Mwhich satisfies the PDE

dtgt = −2 Ricgt .

Note that a Ricci flow is a solution to the PDE, while the PDE itself is called the Ricci flow,or Hamilton’s Ricci flow, giving credit to Richard Hamilton, who introduced the flow in [Ham82].Only a small number of results concerning the Ricci flow will be used in this thesis. Short timeexistence for example, is essential to the usefulness of the Ricci flow but is not covered here.

A property we will need though is a lower bound on the evolution of the scalar curvatureof a Ricci flow. Recall that when taking trace of the Ricci curvature tensor we get the scalarcurvature S, confer [Lee97, Page 124]. In coordinates,

S = trace Ric = gijRij .

Note that the metric is used to convert the covariant 2-tensor Ric into a(

11

)-tensor of which we

can take the trace.

Lemma 2.1.6. Suppose Mn is a compact manifold and gt, t ∈ [0, T ) a Ricci flow of M .Then, for all t ∈ [0, T ),

Sgt ≥1

1minp∈M Sg0 (p) −

2n t

.

8

Sobolev spaces

When M has dimension 3, we often write the result of Lemma 2.1.6 as the simpler expression

Sgt ≥ −3

2(t+ C),

with C(M, g0) > 0.

2.2 Sobolev spaces

We here aim at extending the usual notion of W 1,2 orH1 Sobolev spaces to maps (N,h)→ (M, g)between manifolds. Confer [Eva98, Chapter 5] for an introduction to Sobolev spaces in the realcase.

Let (N,h) be an oriented manifold. A map u : N → Rm is locally p-integrable or inLploc(N,R

m) if, for each i ∈ 1, . . . ,m and every chart (ϕ,U), uiϕ−1 is measurable and∫K |u

iϕ−1|p

√det(hαβ) <∞ for every compact K ⊂ ϕ(U). Similarly, we say that u ∈ Lp(N,Rm) if, for

each i ∈ 1, . . . ,m and every chart (ϕ,U), ui ϕ−1 is measurable,∫ϕ(U) |u

i ϕ−1|p√

det(hαβ) <∞, and

∫N |u

i|p < ∞. Note that if u ∈ L2(N,Rm) also∫N ‖u‖

2Rm < ∞. The set L∞(N,Rm) is

defined likewise.The integral over N above is defined locally as in [Spi65, Chapter 5] with respect to the

volume element on N . We always write∫N f meaning

∫N fdVh. Whenever we integrate on a

manifold, we will implicitly assume it being oriented.Assuming u, v ∈ L2

loc(N,Rm), if, for every chart, the partial derivatives ∂xj (ui ϕ−1) existin the weak sense (confer [Eva98, Section 5.2.1]) and are in L2(ϕ(U),R), we define the innerproduct

〈du, dv〉 =∑i

⟨dui, dvi

⟩h∗M

=∑i

hαβ∂xα(ui ϕ−1)∂xβ (vi ϕ−1) .

Note that the inner product is defined only for almost every point of N . We let ‖du‖2 = 〈du, du〉and say that u ∈ W 1,2

loc (N,Rm) if∫N ‖du‖

2 < ∞. If in addition u ∈ L2(N,Rm), we say thatu ∈ W 1,2(N,Rm). The entity

∫N ‖du‖

2 is closely related to the notion of energy which we willdefine shortly. Therefore, both W 1,2 and W 1,2

loc maps have finite energy, but W 1,2loc maps may only

locally be L2.1 Often N will be compact and the two spaces equal. We endow W 1,2(N,Rm)with the inner product

〈u, v〉2W 1,2(N,Rm) =∫N〈u, v〉Rm +

∫N〈du, dv〉 .

It can be checked that the inner product makes W 1,2(N,Rm) a Hilbert space. We defineW 1,2

0 (N,Rm) to be the closure of C∞c (N,Rm) in the W 1,2(N,Rm) topology, confer [Eva98,Page 245].

If M ⊂ Rm is isometrically embedded and u ∈ W 1,2loc (N,Rm) we say that u ∈ W 1,2

loc (N,M) ifu(x) ∈ M for almost every x ∈ N . In the same way we define W 1,2(N,M). It is important tonote that neither W 1,2

loc (N,M) nor W 1,2(N,M) are vector spaces, but W 1,2(N,M) neverthelessis a metric space using the norm inherited from W 1,2(N,Rm). Note also that W 1,2

loc (N,M)equals W 1,2(N,M) if N has finite volume and M is compact. If M is a closed subset of Rm,

1We note that some authors require that W 1,2loc maps only locally have finite energy. In addition, in some texts

the notation W 1,2 is used for the set we call W 1,2loc . We need to differ between W 1,2 and W 1,2

loc , and we will worksolely with maps which globally have finite energy. In lack of standard notation, we use the symbols W 1,2 andW 1,2loc as defined above.

9

2. Foundations

the set W 1,2(N,M) is closed in W 1,2(N,Rm); L2 convergence of a sequence implies pointwiseconvergence of a subsequence almost everywhere. If M is closed, the pointwise limits of thissubsequence must be in M . For a surface Σ and M compact, W 1,2(Σ,M) is the completion ofC∞(Σ,M) but in general this is not the case, confer [Hel02, Page 33].

Note that for u ∈ W 1,2(N,M), the induced norm ‖u‖W 1,2(N,Rm) of u is dependent on theembedding of M . It is interesting though, that, at least when M is compact, [Hel02, Lemma1.4.3] shows that the definition of W 1,2

loc (N,M) is independent of the embedding. By modifyingthe argument of the lemma slightly, it can be seen that when M is compact, u ∈W 1,2

loc (N, (M, g))if and only if u ∈W 1,2

loc (N, (M, g)), where g is any metric on M . In addition, if also N is compact,the sets W 1,2(N, (M, g)) and W 1,2(N, (M, g)) are equal with the same topology.

If U is an open subset of N , it is itself a manifold and W 1,2loc (U,M) is defined. If U is closed, we

define W 1,2loc (U,M) to be W 1,2

loc (int (U) ,M) with the convention that W 1,2loc (int (U) ,M) is empty

if the interior of U is empty. The same applies to W 1,2(U,M).We let u, v ∈W 1,2

loc (N,Rm) and consider the inner product 〈du, dv〉 for a moment. If we, forp ∈ N choose orthonormal coordinates, we have

〈du, dv〉 (p) =∑i,α

∂xα(ui ϕ−1)(p)∂xα(vi ϕ−1)(p) = tr (dpuTdpv) ,

where we on the right hand side considered dpu and dpv as the Jacobian matrices of partialderivatives. Thus, still in such coordinates,

‖du‖ = tr (dpuTdpu)1/2 = ‖dup‖F .

Here ‖ · ‖F is the submultiplicative Frobenius or Hilbert-Schimidt matrix norm, confer [HJ90,Page 291]. This allows us to use facts from linear algebra, e.g. that there exists cF > 0 suchthat

cF ‖du‖ ≤ ‖du‖∞ ≤ ‖du‖ , (2.2.1)

where ‖du‖∞(p) = supv∈TpN,‖v‖=1 ‖du(v)‖. In addition, we will need that

‖GF‖F ≤ ‖G‖∞‖F‖F (2.2.2)

for matrices G,F .

2.2.1 Weak and strong topology

Let (uj), ui ∈W 1,2(N,Rm). We say that (uj) converges or converges strongly to u ∈W 1,2(N,Rm)

and write ujW 1,2(N,Rm)−→ u, if ‖uj − u‖W 1,2(N,Rm) → u. When u, ui ∈ W 1,2(N,M) this will be

equivalent to writing ujW 1,2(N,M)−→ u.

If, for all continuous, linear functionals f : W 1,2(N,Rm) → R, f(uj) → f(u), we say that

(uj) converges to u weakly and write ujW 1,2(N,Rm)

u. By the Riesz representation theorem([LL01, Theorem 2.14]), this is equivalent to requiring

⟨uj , v

⟩W 1,2(N,Rm)

→⟨uj , v

⟩W 1,2(N,Rm)

forall v ∈W 1,2(N,Rm).

Lemma 2.2.1. Suppose (uj), uj ∈ W 1,2(N,M), u ∈ W 1,2(N,M) and V ⊂ N is open. If

ujW 1,2(N,M)−→ u then uj |V

W 1,2(V,M)−→ u|V . If ujW 1,2(N,Rm)

u then uj |VW 1,2(V,Rm)

u|V .

Lemma 2.2.2. The unit ballu ∈W 1,2(N,Rm) | ‖u‖ ≤ 1

is weakly compact. If M is a closed

subset of Rm and N ⊂ Rn, the unit ballu ∈W 1,2(N,M) | ‖u‖ ≤ 1

is weakly compact.

10

Additional properties

2.2.2 Additional properties

Let V be an open or closed subset of some manifold N . We will introduce the shorthandC0 ∩W 1,2

loc (V,M) for the set W 1,2loc (V,M) ∩ C0(cl (V ) ,M). Note that the maps are required to

be continuous on cl (V ). We define the set C0 ∩W 1,2 likewise and equip it with the norm

‖u‖C0∩W 1,2(V,M) = ‖u‖W 1,2(V,M) + ‖u‖C0(cl(V ),Rm) .

We will construct a number of C0∩W 1,2loc (R2,M) maps by cutting and pasting already existing

maps together. The following lemma justifies this procedure.

Lemma 2.2.3 (Gluing Lemma). Let B ⊂ R2 be a ball, and u ∈ C0 ∩W 1,2loc (B,M), v ∈ C0 ∩

W 1,2loc (R2 \ B,M) be maps such that u|∂B = v|∂B. Then there exists a unique map w ∈ C0 ∩

W 1,2loc (R2,M) such that w|B = u and w|R2\B = v.

Lemma 2.2.4. Let BR(x) ⊂ R2 be a ball, and suppose u ∈W 1,2(BR(x),M). Let u : BR(x)→Mbe a representative of u. Then

u|∂Br(x) ∈W 1,2(∂Br(x),M) (2.2.3)

for almost all r ∈ (0, R). In particular, if in addition u ∈ C0(cl (BR(x)) ,M) then

u|∂Br(x) ∈ C0 ∩W 1,2(∂Br(x),M)

for almost all r ∈ (0, R).

Theorem 2.2.5 (Fundamental Theorem of Calculus). Let u ∈ W 1,2((a, b),Rm). Then u has acontinuous representative u which, for all x, y ∈ (a, b), satisfies

u(x)− u(y) =∫ x

y∂tu(t)dt .

The following definition will be used for sequences of maps converging away from a finite setof “singular” points. This will be important when we turn to bubble convergence later.

Definition 2.2.6. Let M ⊂ Rm be embedded, (uj), uj ∈W 1,2(N,M) a sequence of maps, andS ⊂ N a finite set. Then (uj) is said to converge strongly except on S to a map u ∈W 1,2(N,M)

if ujW 1,2(N,Rm)

u and ujW 1,2(K,M)−→ u for every compact set K ⊂ Σ \ S.

We will also need a notion of convergence of sets.

Definition 2.2.7. Let S ⊆ N and (V j), V j ⊆ N a sequence of open or closed sets. Then (V j) issaid to exhaust N \S in the limit, if, for all j, V j ⊂ V j+1 and if, for any compact set K ⊂ N \Sthere exists JK such that, for all j ≥ JK , K ⊂ V j .

Lemma 2.2.8. Suppose u ∈ L2(N,M) and S ⊂ N . Then, if (V j) is a sequence of sets exhaust-ing N \ S in the limit,

limj→∞

∫V j‖u‖2 =

∫N\S‖u‖2 .

In particular, if u ∈W 1,2(N,M), then

limj→∞

‖u|V j‖W 1,2(V j ,M) = ‖u|N\S‖W 1,2(N\S,M) .

11

2. Foundations

Note that if S has measure zero, the equalities above reduce to limj→∞∫V j ‖u‖ =

∫N ‖u‖

and limj→∞ ‖u|V j‖W 1,2(V j ,M) = ‖u‖W 1,2(N,M).

2.3 Poincare inequalities

Theorem 2.3.1 (Poincare’s inequality). Assume U ⊂ Rn is open and bounded. Let u ∈W 1,2

0 (U,Rm), and r > 0 and x ∈ Rn be such that U ⊂ Br(x). Then∫U‖u‖2 ≤ r2

2

∫U‖du‖2 .

Here we will use Theorem 2.3.1 to bound the energy of a map on the boundary of a ballvanishing at a point. The bound presented is not the tightest possible.

Corollary 2.3.2. Let B = Br(x) ⊂ R2 be a ball. Then, for all u ∈ C0∩W 1,2(∂B,M) vanishingat a point p ∈ ∂B, we have∫

∂B‖u‖2 ≤ π2r2

2

∫∂B‖du‖2 = π2r2E(u) .

2.4 Energy and area

We define the energy of a W 1,2loc (N,Rm) map u by

E(u) =12

∫N‖du‖2 .

This is a natural generalization of the usual energy of a curve. The map E : W 1,2loc (N,Rm)→ R

itself is called the energy functional. Often we wish to measure the energy of u on some subsetV of N . We then write E(u|V ) meaning 1

2

∫V ‖du‖

2. If V is a collection of such sets, we use theshorthand E(u|V) meaning E(u|∪V ∈VV ).

For x ∈ N such that the differential map dxu is defined, the inner products on N and Rm

define an adjoint map dxuT by

〈dxu(X), Y 〉 = h(X, dxuT (Y ))

for all X ∈ TxN and Y ∈ TxRm. For such x we define the Jacobian determinant Ju : N → R by

Ju =√

det(duTdu) .

In local coordinates at N ,

Ju =

√det((∑

i hαγ∂xγu

i∂xβui)βα

)=

√det((hαβ

)βα

(∑i ∂xαu

i∂xβui)βα

),

where (aβα)βα denotes the matrix with entries aβα. In normal coordinates, the expression reduces to√det(duTdu) considering du as the Jacobian matrix of partial derivatives and duT its transpose.

Note that Ju(x) = 0 if and only if rank dxu < dimN .

12

Energy and area

The area of u ∈W 1,2loc (Σ,Rm) for a surface Σ is given by

Area(u) =∫

ΣJu .

If u is an embedding, this corresponds to the usual notion of the area of u(Σ). If f is a realvalued map defined on u(Σ), we will define∫

uf =

∫Σ

(f u)Ju .

This again extends the usual notion of integration on a parametrized surface.

Lemma 2.4.1. Let Σ be a surface and u ∈ W 1,2loc (Σ,Rm). Then Area(u) ≤ E(u). If the metric

on Σ is conformal, Area(u) = E(u) if and only if u is conformal almost everywhere.

Lemma 2.4.2 (Convexity of Energy). For any u, v ∈W 1,2loc (N,Rm) and t ∈ [0, 1],

E(tu+ (1− t)v) ≤ tE(u) + (1− t)E(v) .

Corollary 2.4.3 (Triangle inequality for energy). Let u, v ∈W 1,2loc (N,Rm). Then

E(v + u) ≤ 2E(v) + 2E(u) .

Lemma 2.4.4 (Weak lower semicontinuity of energy). If ujW 1,2(N,Rm)

u then

E(u) ≤ lim inf j →∞E(uj) .

Lemma 2.4.5. Let Σ be a surface. Suppose (uj), uj ∈ C1(Σ,Rm) and ujC1(Σ,Rm)→ u, u ∈

C1(Σ,Rm). If Area(Σ) is bounded, then ujW 1,2(Σ,Rm)−→ u.

Lemma 2.4.6. If (uj), uj ∈ W 1,2(N,M) converges strongly to u ∈ W 1,2(N,M) except on Sthen, for any precompact set K such that cl (K) ⊆ N \ S,

limj→∞

E(uj − u|K)→ 0

andlimj→∞

E(uj |K)→ E(u|K) .

Let Σ be a surface with metric h and (ϕ,U) a chart. If u ∈W 1,2loc (Σ,Rm) then

∫UJu =

∫ϕ(U)

√det((hαβ

)βα

(∑i ∂xαu

i∂xβui)βα

)√det(hαβ)dx1 ∧ dx2

=∫ϕ(U)

√det((∑

i ∂xαui∂xβu

i)βα

)dx1 ∧ dx2

(2.4.1)

13

2. Foundations

showing that the area of u|U is not dependent on the metric on Σ. Suppose now h is conformaland (ϕ,U) a conformal chart such that hαβ = ρ2δαβ and hαβ = ρ−2δαβ with ρ : Σ→ R positive.Then

12

∫U‖du‖2 =

12

∫ϕ(u)

∑i

hαβ∂xαui∂xβu

i√

det(hαβ)dx1 ∧ dx2

=12

∫ϕ(u)

∑i,α

∂xαui∂xαu

idx1 ∧ dx2(2.4.2)

implying that the energy is dependent only on the conformal structure of the surface, not onthe metric itself. Since a conformal diffeomorphism between surfaces corresponds to changingthe metric without changing the conformal structure, we get the following result.

Lemma 2.4.7. Suppose c : Σ1 → Σ2 is a diffeomorphism between the surfaces Σ1 and Σ2, andf ∈ W 1,2

loc (Σ2,Rm). Then Area(f) = Area(f c). If, in addition, c and the metrics on Σ1 andΣ2 are conformal then E(f) = E(f c).

Let B = (U,Λ) be a projected ball, and u ∈ W 1,2(U,M). Since Λ is conformal, byLemma 2.4.7,

E(u) = E(u Λ−1) . (2.4.3)

Often, we will use theorems concerning the energy of maps defined on subsets of R2. By (2.4.3)and since u Λ−1 ∈ W 1,2(Λ(U),M), Λ(U) ⊂ R2, the results of such theorems transfer to mapsdefined on projected balls.

Definition 2.4.8. LetN andM be a manifolds, U ⊂ N open and bounded, and u ∈W 1,2(U,M).We say that v is equal to u on the boundary or coincide on the boundary, if u− v ∈W 1,2

0 (U,M).Note that if u and v are continuous, this is equivalent to u|∂U = v|∂U and, if ∂U is Lipschitz, itis equivalent to u and v having equal trace.

A map u ∈ W 1,2(U,M) is said to be energy minimizing, if E(u) ≤ E(v) for all v equal to uon the boundary.

Theorem 2.4.9 ([Mor66]). Suppose U ⊂ Rm is open and bounded. If u ∈ W 1,2(U,M) thenthere exists an energy minimizing map v ∈W 1,2(U,M) equal to u on the boundary.

Again the conformal equivalence between S2 \p and R2 using the stereographic projection,allows us to use Theorem 2.4.9 on subsets of S2 as well.

2.5 Harmonic maps

Let M ⊂ Rm be embedded and Mδ a δ-tubular neighborhood of M with projection r : Mδ →M ,confer Appendix B.2. Let u ∈W 1,2

loc (N,M). Then u is said to be weakly harmonic if it is a criticalpoint of the energy functional in the sense that for all maps v ∈W 1,2

0 (N,Rm) with ‖v(x)‖ < δ(x)for almost every x ∈ N ,

limt→0

E(r(u+ tv))− E(u)t

= 0 .

In particular, energy minimizing maps are weakly harmonic. If u ∈ C∞(N,M) we say that u isharmonic if it satisfies the Euler-Lagrange equations of the energy functional:

∆hui + hαβΓiM,jk(u(·)) ∂u

j

∂xα∂uk

∂xβ= 0 ,

14

Regularity and other properties

where ∆h is the Laplacian

∆hf =1√

dethαβ∂xα

(√dethαβh

αβ∂xβf).

If u is harmonic and in W 1,2loc (N,M) then it is also weakly harmonic, confer [Jos05, Lemma 8.1.1].

On the other hand, if u is smooth and weakly harmonic, it is harmonic, confer [Hel02, Lemma1.4.10]. See [Eva98, Section 2.2] for an introduction to harmonic maps Rn → R, and [Jos05,Chapter 8] for more details on harmonic maps between manifolds.

From the definition of harmonic maps we immediately get the following lemma stating thatbeing harmonic is a local property:

Lemma 2.5.1. If u : N → M is a map such that, for each x ∈ N , there exists an open set Ucontaining x and u|U is harmonic then u is harmonic.

We will use the following fact and the connected constant εL extensively.

Lemma 2.5.2 (Convexity of Energy for low energy maps). There exists a constant εL(M, g) > 0such that if B ⊂ R2 is a ball, u, v ∈ W 1,2(B,M) with u − v ∈ W 1,2

0 (B,M), v weakly harmonicand E(v) ≤ εL then

12E(u)− 1

2E(v) ≥ 1

4E(u− v) = E(

12

(u− v)) .

Corollary 2.5.3. Let B ⊂ R2 be a ball and u, v ∈ W 1,2(B,M) with u − v ∈ W 1,20 (B,M). If v

is weakly harmonic, E(u), E(v) ≤ εL and E(v) = E(u) then u = v.

Corollary 2.5.4 (Uniqueness of low energy harmonic maps). Let B ⊂ R2 be a ball and u, v ∈W 1,2(B,M) with u−v ∈W 1,2

0 (B,M). If both v and u are weakly harmonic and E(u), E(v) ≤ εLthen u = v.

As a consequence of Corollary 2.5.4 and Theorem 2.4.9, any weakly harmonic map withenergy less than εL is minimizing.

Proposition 2.5.5 (Unique continuation). Let u, v : N → M be harmonic maps. If u and vagree on an open subset of N , then u = v.

2.6 Regularity and other properties

Weakly harmonic maps are not in general smooth or even continuous. In some special cases,we do though have regularity results, confer [Hel02, Section 1.5] for a survey. From now on wewill work solely with harmonic maps from surfaces to a compact manifold. It turns out that inprecisely this case the question of regularity has a nice answer.

Theorem 2.6.1 ([Hel02]). Let Σ be a surface and M compact. If u ∈ W 1,2loc (Σ,M) is weakly

harmonic then u is smooth and thus harmonic.

Because of Theorem 2.6.1, the terms weakly harmonic and harmonic will have the samemeaning when applied to maps from surfaces to compact manifolds.

The following result extends the continuity of harmonic maps to the boundary of a ball.

15

2. Foundations

Theorem 2.6.2 ([Qin93]). Let B ⊂ R2 be a ball, and u : cl (B) → M a map with u|B ∈W 1,2(B,M) and u|B harmonic. If M is compact and u|∂B continuous then u is continuous oncl (B).

In addition, if U is a set of maps cl (B) → M with u|B ∈ W 1,2(B,M) for u ∈ U such thatthe sets

u|∂B |u ∈ U and u|B |u ∈ U

are equicontinuous2 and, furthermore, given ε > 0 and p ∈ ∂B there exists R > 0 such that, forall u ∈ U ,

E(u|B∩BR(p)) < ε

then the set U is equicontinuous.

The following results on harmonic maps from surfaces proved in [SU81] and summarized in[Par96] will be important.

Theorem 2.6.3 ([SU81]). Let Σ be a compact surface with conformal metric and M a manifold.Then there exists a constant εSU (Σ, h,M, g) > 0, such that

(H1) (Local Compactness) if Br(x) ⊂ Σ is a geodesic ball and (uj), uj : Br(x) → M a se-quence of harmonic maps with E(uj) ≤ εSU for all j then there exist a harmonic map

u : Br/2(x)→M and a subsequence (ujk) such that ujk |Br/2(x)

C1(Br/2(x),M)→ u,

(H2) (Removable Singularities) if U ⊂ Σ is open, p ∈ U , u : U \ p → M a harmonic map,and E(u) <∞ then u extends to a harmonic map U →M .

Suppose u is an immersion. The surface u(Σ) ⊂M then gives rice to the second fundamentalform A : T 1 Σ × T 1 Σ → N u(Σ), confer [Lee97, Page 133]. When e1, e2 is an orthonormalbasis for du(TpΣ), p ∈ Σ, the norm of A is given by

‖A‖2 =∑i,j

g(A(ei, ej), A(ei, ej))

and the mean curvature vector is

H =∑i

A(ei, ei) .

The surface u(Σ) is said to be minimal if H = 0 for all p ∈ Σ. This is equivalent to u beingharmonic and conformal, confer [Jos05, Section 3.6]. For maps from S2 we have the followingresult:

Proposition 2.6.4. Let u : S2 →M be harmonic. Then u is conformal.

Thus, if u : S2 →M is a harmonic immersion, u(Σ) is a minimal surface.

2See [Roy88, Page 177] for the definition of equicontinuity.

16

Change of area

2.7 Change of area

We now turn to investigating how the area of a map varies when the metric g on the targetmanifold is changed. Since an isometric embedding (M, g) ⊂ Rm is dependent on the metric, itbecomes convenient to have an expression for the Jacobian determinant independent of such anembedding:

Ju =

√det((hαγgij∂xγu

i∂xβuj)βα

)=

√det((hαγg(du(∂xγ ), du(∂xβ ))

)βα

).

Lemma 2.7.1. Let Σ be a surface, M a compact, differentiable manifold, I a compact interval,and gt, t ∈ I a smooth family of metrics on M . Then there exists a non-negative, continuousmap ψ : I × I → R with ψ(t, t) = 0, t ∈ I such that, for any s, t ∈ I and f ∈W 1,2

loc (Σ,M),

|Areagt(f)−Areags(f)| ≤ ψ(t, s) Areagt(f) .

Remark 2.7.2. If fα, fα ∈ W 1,2loc (Σ,M) is a family of maps of which we have some control

of the area, Lemma 2.7.1 implies that t 7→ Areagt(fα) is an equicontinuous family of maps.E.g., if E(f j)→ C for j →∞ and some C, then given t ∈ I and ε > 0 we can choose δ > 0 suchthat ψ(t, s) < ε/(C + 1) whenever |s− t| < δ. Then for all but finitely many j ∈ N,∣∣Areagt(f

j)−Areags(fj)∣∣ ≤ ψ(t, s) Areagt(f

j) ≤ ψ(t, s)Egt(fj) < ψ(t, s)(C + 1) < ε

proving the equicontinuity.

Lemma 2.7.3. Let M be a compact, differentiable manifold and gt, t ∈ I a smooth family ofmetrics on M . Let Σ be a surface and f ∈W 1,2

loc (Σ,M). Then Areagt(f) is a C1 function of t.

The above lemma can be strengthened to get smoothness of t 7→ Areagt(f), but we will notneed that.

We now consider how the area varies when M is 3-dimensional and gt a Ricci flow on M .Note that, in the lemma below, when nf is undefined Jf = 0 so that the integral is well-defined.

Lemma 2.7.4. Suppose M3 is a compact manifold, Σ a surface, and gtt∈I a Ricci flow of M .Let f ∈W 1,2

loc (Σ,M). Then

dt Areagt f = −∫

Σ(Sgt −Ricgt(nf , nf )) Jf,gt ,

where nf is a unit normal to f(Σ).

Corollary 2.7.5. Suppose M3 is a compact manifold, Σ a surface, and gtt∈I a Ricci flow ofM with I compact. Then Areagt(f) is a C2 function of t for each f ∈ W 1,2

loc (Σ,M) and thereexists a constant C(M, gt) ≥ 0 such that for all f ∈W 1,2

loc (Σ,M) and t ∈ I,

|dt2 Areagt f | ≤ C Areagt(f) .

We will need to apply Lemma 2.7.4 to minimal surfaces. The upper bound on the change ofarea we thus obtain will be central to proving the main rate of change of width result. To beginwith, we will show the upper bound for immersed, minimal surfaces. Afterwards, we will usethe theory of branched immersions to deal with non-immersed minimal surfaces.

17

2. Foundations

Let f : Σ → M be an immersion, and f(Σ) minimal. We write Kgt(ei, ej) for the sectionalcurvature associated with the plane span (ei, ej) with respect to gt, and Kf(Σ)(x) for the intrinsicGauss curvature of f(Σ) at f(x). When e1, e2 is an orthonormal basis for df(TxΣ) at time tthe Gauss Equation ([Lee97, Theorem 8.4]) asserts

Kf(Σ) = Kgt(e1, e2) + gt(Agt(e1, e1), Agt(e2, e2))− gt(Agt(e1, e2), Agt(e1, e2))H=0= Kgt(e1, e2)− gt(Agt(e1, e1), Agt(e1, e1))− gt(Agt(e1, e2), Agt(e1, e2))H=0= Kgt(e1, e2)− 1

2‖Agt‖2gt .

(2.7.1)

Letting e3 = nf be normalized with respect to gt, we have (confer [Lee97, Page 147])

Sgt =∑i

Ricgt(ei, ei) =∑i,j

gt(Rgt(ej , ei)ei, ej) = 2∑i<j

Kgt(ei, ej) (2.7.2)

and hence

Sgt −Ricgt(nf , nf ) = Sgt −Kgt(e1, nf )−Kgt(e2, nf )(2.7.2)

=12

Sgt +Kgt(e1, e2)

(2.7.1)=

12

Sgt +Kf(Σ) +12‖Agt‖2gt .

Thus, when f(Σ) is a minimal immersed surface, Lemma 2.7.4 gives


Σ

(12

Sgt +Kf(Σ) +12‖Agt‖2gt

)Jf,gt . (2.7.3)

We now turn to the case where f is not immersed. We refer to [ET88] for further elaboration ofthe discussion presented here. Since f is not immersed, there are points p ∈ Σ with rank df < 2.Such points are called critical and the Gauss map, which connects a normal vector to the surfacef(Σ), is not defined for such points. A critical point is called smooth, if the Gauss map has asmooth extension to p. If f is smooth and conformal and all its critical points are smooth then fis said to be a conformal, smoothly branched immersion. The usual Gauss-Bonnet Theorem (seee.g. [Lee97, Theorem 9.7]) extends to cover such maps. The version presented here is weakerthan, but follows immediately from, [ET88, Theorem 4].

Theorem 2.7.6. Let f : Σ→M be a conformal smoothly branched immersion from the compactsurface Σ to the manifold M . Then ∫

ΣKf(Σ)Jf ≥ 2πχ(Σ) , (2.7.4)

where χ(Σ) is the Euler characteristic of Σ.

If f is non-constant, conformal and harmonic then, by [ET88, Theorem 3], f is a conformalsmoothly branched immersion. Thus (2.7.4) holds for such maps. Recall that χ(S2) = 2 (see[Lee97, Page 169]). By Proposition 2.6.4, when f : S2 → M is harmonic it is conformal. Thus,if it is also non-constant, ∫

ΣKf(Σ)Jf ≥ 4π . (2.7.5)

Since the set of smooth critical points of such f is finite (see [ET88, Page 628]) and a conformal,harmonic map is minimal away from its critical points ([ET88, Page 626]), (2.7.3) holds in thiscase too. Discarding the term regarding the second fundamental form, we have proved thefollowing result.

18

Change of area

Proposition 2.7.7. Let f : S2 →M be non-constant and harmonic. Then

dt Areagt f ≤ −4π − 12

∫Σ

Sgt Jf,gt ,

Notes

The content of this chapter is a collection of known material from various sources. In particular,the definition of Sobolev spaces between manifolds is based on [Hel02], [Jos05], [Eva98], and[Spi65] as we where unable to find a single source giving a formal and detailed exposition of theentire subject. The notion of projected balls is based on the notion of balls in S2 of [CM07a].Such a ball is defined to be the preimage under a stereographic projection of a ball in R2. Wecompose the stereographic projection with dilations which later on will let us make explicit thedilation invariance of the notion of almost harmonic maps. With a few exceptions, all proofsconsisting of more than a reference are written by the author of this text. This includes therest of the Appendices. The exceptions are the change of area results under Ricci flow fromLemma 2.7.4 and onwards and Lemma C.1.6. The presentation of these results is a detailedversion of the exposition given in [CM07a, Chapter 1].

19

3

Width and existence time

This chapter is devoted to the two main results of the thesis; the upper bound on the rate ofchange of width and finite existence time. We will prove the results except for some intermediateresults. For one of these, we will need the results of the following two chapters. We will thereforestate the result here, but defer the proof to Chapter 6. In addition, we omit proving that simplyconnectedness implies positive width and that two different characterizations of the width areequivalent. As previously discussed, we will only prove finite existence time for Ricci flowswithout surgery. We will, though, indicate how the proof can be extended to hold when surgeryoccurs.

We start by discussing sweepouts, width, and positiveness of width. Following this we showhow the rate of change of width implies finite existence time. The proof of the rate of change ofwidth result with the exceptions described above will be given at the end of the chapter.

3.1 Sweepouts and width

The definition of width we will introduce relies on sweepouts. In this section we will introduceboth concepts. We will furthermore discuss why simply connectedness implies positive width.Throughout the section, we let (M, g) be a compact manifold isometrically embedded in Rm.

Definition 3.1.1. We let ΩM be the subset of the set of maps S2 × [0, 1]→M such that eachσ ∈ ΩM has the following properties:

(1) for each s ∈ [0, 1], σ(·, s) ∈ C0 ∩W 1,2(S2,M) ,

(2) the map s 7→ σ(·, s) ∈ C0([0, 1], C0 ∩W 1,2(S2,M)) ,

(3) σ(S2, 0) = p0 and σ(S2, 1) = p1 for some p0, p1 ∈M .

The set [0, 1] is denoted the parameter space.

As noted in Section 2.2, the compactness of S2 and M implies that W 1,2(S2,M) except forthe norm is independent of the metric g. Thus, the same applies to ΩM . Note also that Property(1) and (2) imply that any σ ∈ ΩM is also in C0(S2×[0, 1],M). To see this let (pk, sk) ∈ S2×[0, 1]be a sequence converging to (p, s) ∈ S2 × [0, 1]. Given ε > 0 use Property (1) to choose K ∈ Nsuch that, for all k ≥ K,

‖σ(p, s)− σ(pk, s)‖Rm < ε/2 .

21

3. Width and existence time

Then use the uniform continuity of Property (2) to enlarge K ∈ N such that, for all k ≥ K andany p ∈ S2,

‖σ(p, s)− σ(p, sk)‖Rm < ε/2 .

By the triangle inequality,

‖σ(p, s)− σ(pk, sk)‖Rm ≤ ‖σ(p, s)− σ(pk, s)‖Rm + ‖σ(pk, s)− σ(pk, sk)‖Rm < ε

proving the continuity.We call a map σ ∈ ΩM a sweepout of M , and, for each s ∈ [0, 1], the map σ(·, s) is called

a slice. It is useful to go down one dimension to illustrate the cause of the name and to getintuition for what happens in the higher dimensional case. If we consider a compact surfaceembedded in R3 and define the sweepout σ instead to be a map S1× [0, 1], Figure 3.1 shows theloops σ(·, s) sweeping over the surface as s changes. Similarly, if M is 3-dimensional, one canthink of the images σ(·, s) of S2 as sweeping over M as s varies.

(·,s)

(·,0)

(·,1)

Figure 3.1: Sweeping out a compact hypersurface

If we aim at getting topological information about M the notion of sweepouts is somewhattoo strong; one could take the sweepout ((x, y, z), s) 7→ s(s− 1)(x, y, z) ∈ R3 as an example of anon-trivial sweepout mapping into a homotopically trivial (non-compact) manifold. We remedythis by considering homotopy classes of sweepouts.

Definition 3.1.2 (Homotopy in ΩM ). Two maps σ0, σ1 ∈ ΩM are homotopic if there exists ahomotopy from σ0 to σ1; that is, a continuous map H : (S2 × [0, 1]) × [0, 1] → M such thatH(·, 0) = σ0 and H(·, 1) = σ1.

Let σ ∈ ΩM . Then [σ] denotes the homotopy class of σ; the set of all σ ∈ ΩM such that σis homotopic to σ. A homotopy class [σ] is said to be trivial if [σ] = [c] for some constant mapc ∈ ΩM . Thus, [σ] is trivial if it is homotopic to a constant map.

Note that [σ] is trivial if σ is homotopic to a map c with c(·, s) being constant for each s,since such a c is homotopic to a constant map. This would not in general be the case if theparameter space was e.g. S1 instead of [0, 1].

Definition 3.1.3 (Width). The width W ([σ]) of a homotopy class [σ] is given by

W ([σ]) = infσ∈[σ]

maxs∈[0,1]

E(σ(·, s)) .

22

Sweepouts and width

The width of a homotopy class is clearly non-negative and it is dependent on the metricbecause the energy is. If we need to specify which metric is used, we write Wg([σ]). For our use,positivity of the width will be essential. We must therefore find conditions to ensure this. Thefollowing result leads us somewhat along the way. In essence, it states that when the energyof a sweepout is sufficiently small, the image of the sweepout in M is too small to envelophomotopically non-trivial parts.

Proposition 3.1.4. There exists εmin(M, g) > 0 such that if maxs∈[0,1]E(σ(·, s)) < ε then [σ]is trivial.

Proof. See [CM07a, Page 3 and 12].

Therefore, if M has a sweepout belonging to a non-trivial homotopy class, the width ofthe homotopy class will be positive for any metric on M . Suppose for a moment that M is3-dimensional. Using an algebraic topology argument, confer [CM05, Page 2] or the argumentof [MM88, Lemma 3], one can show that if π3(M) is non-trivial, a non-trivial homotopy classexists. By [Hat, Proposition 3.7], π3(M) is non-trivial if M is simply connected. Thus, we getthe following result:

Proposition 3.1.5 (Positivity of width). Let M be a simply connected, compact, differentiable,3-dimensional manifold. Then there exists σ ∈ ΩM such that Wg([σ]) is positive for any metricg on M .

Proof. Combining Proposition 3.1.4 with [MM88, Lemma 3] and [Hat, Proposition 3.7] gives theresult.

We will need some additional properties of the width. Since Area(u) ≤ E(u), for anyu ∈W 1,2(S2,M), we have

W ([σ]) ≥ infσ∈[σ]

maxs∈[0,1]

Area(σ(·, s)) .

In fact, right hand side in the inequality provides an equivalent definition of the width of asweepout:

Proposition 3.1.6. For any sweepout σ ∈ ΩM ,

W ([σ]) = infσ∈[σ]

maxs∈[0,1]

Area(σ(·, s)) . (3.1.1)

Proof. See [CM07a, Proposition 1.5].

The width measured with respect to a smooth family of metrics is continuous:

Lemma 3.1.7. Let gt, t ∈ I be a smooth one-parameter family of metrics on M with I acompact interval. Let σ ∈ ΩM be a sweepout of M . Then Wgt([σ]) is continuous in t.

Proof. We will use the area characterization (3.1.1) of the width.Fix t ∈ [0, 1] and let ε > 0. Let σt ∈ [σ] and t′1 ∈ [0, 1] be such that σt(·, t′1) is the slice of

maximal area with respect to the metric gt and Areagt(σt(·, t′1)) < Wgt([σ])+ε/2. Let ψ : I×I →R be the map of Lemma 2.7.1. Choose δ > 0 such that ψ(t, s), ψ(s, t) < ε/(2 Areagt(σt(·, t′1)) for

23


all s ∈ I with |s−t| < δ. Let s′1 be such that Areags(σt(·, s′1)) is maximal. Then, by Lemma 2.7.1for such s,

Areags(σt(·, s′1)) ≤ ψ(t, s) Areagt(σt(·, s′1)) + Areagt(σt(·, s′1))≤ ψ(t, s) Areagt(σt(·, t′1)) + Areagt(σt(·, t′1))< ε/2 +Wgt([σ]) + ε/2 .

Thus, Wgs([σ]) < ε+Wgt([σ]).On the other hand, suppose for contradiction that |s − t| < δ and Wgt([σ]) ≥ ε + Wgs([σ]).

Then let σs ∈ [σ] and s′2 be such that Areags(σs(·, s′2)) < Wgs([σ]) + ε/2 and Areags(σs(·, s′2))maximal. Furthermore, let t′2 be such that Areagt(σs(·, t′2)) is maximal. By Lemma 2.7.1,

Areagt(σs(·, t′2)) ≤ ψ(s, t) Areags(σs(·, t′2)) + Areags(σs(·, t′2))≤ ψ(s, t) Areags(σs(·, s′2)) + Areags(σs(·, s′2))< ψ(s, t) (Wgs([σ]) + ε/2) +Wgs([σ]) + ε/2 .

(3.1.2)

By assumption Wgs([σ]) + ε/2 < Wgt([σ]). Hence

Areagt(σs(·, t′2))(3.1.2)< ψ(s, t)Wgt([σ]) +Wgs([σ]) + ε/2≤ ψ(s, t) Areagt(σt(·, t′1)) +Wgs([σ]) + ε/2< Wgs([σ]) + ε .

That is, Wgt([σ]) < ε+Wgs([σ]) which contradicts the assumption. Thus we conclude

|Wgs([σ])−Wgt([σ])| < ε

whenever |s− t| < δ.

Though the width is continuous, we have no information regarding differentiability. Toremedy this, when measuring the rate of change of the width, we use the weaker notion of thelim sup of forward difference quotients. For any function f : R→ R we define

D+x f(x) = lim sup

h→0+

f(x+ h)− f(x)h

.

Note that if f is differentiable then D+x f(x) = f ′(x). In Appendix C.3 we prove some basic

results regarding forward difference quotients.

3.2 Rate of change of width and existence time

We are now ready to relate the rate of change of the width under Ricci flow to existence timeof the flow; assuming Theorem 1 is established, we derive Corollary 2.

Recall from Chapter 1 that exactly two kinds of singularities can cause the Ricci flow tostop; the ones which can be handled through surgery and the extinction singularity. In theresult below, we will not distinguish between the different kinds. We prove that within finitetime some undefined singularity will occur. After the proof we will briefly discuss what isnecessary to extend the proof to cover Ricci flows with surgery. If we assume this is done, theresult would imply that in finite time a singularity which cannot be handled through surgerywill occur. That is, the manifold becomes extinct.

24

The proof of Theorem 1

Proof of Corollary 2. Let σ ∈ ΩM be the sweepout ofM whose existence is assured by Theorem 1.By Theorem 1 and Lemma C.3.1,

D+t (Wgt([σ])(t+ C)−3/4) ≤ D+

t Wgt([σ])(t+ C)−3/4 +Wgt([σ])(−3

4(t+ C)−7/4

)≤(−4π +

34(t+ C)

Wgt([σ]))

(t+ C)−3/4 +Wgt([σ])(−3

4(t+ C)−7/4

)= −4π(t+ C)−3/4 .

Thus D+t (Wgt([σ])(t+C)−3/4) is strictly negative for all t. By Corollary C.3.4, using the conti-

nuity from Lemma 3.1.7, for any t,

Wgt([σ])(t+ C)−3/4 −Wg0([σ])(0 + C)−3/4 ≤∫ t

0D+t (Wgt([σ])(t+ C)−3/4)dt

≤∫ t

0−4π(t+ C)−3/4dt = −16π

((t+ C)1/4 − (0 + C)1/4

).

Hence

Wgt([σ]) ≤ (t+ C)3/4(−16π

((t+ C)1/4 − C1/4

)+Wg0([σ])C−3/4

)= −16π(t+ C) + (t+ C)3/4

(16πC1/4 +Wg0([σ])C−3/4

)showing that for sufficiently large t, Wgt([σ]) is negative. Since the width with respect to anymetric is non-negative, the Ricci flow cannot exist at such t. Thus T <∞.

If instead gt is a Ricci flow with surgery, we may, as discussed in Chapter 1, encountersingular times, at which the topology of M is changed through surgery. If we wish to extend theabove results to hold in such cases, we first need to check that the obtained components satisfythe assumptions of Theorem 1. Since the components will also be simply connected, this willindeed be the case. Then we would need to know how the width of each component is related tothe width of M just before the singular time; if the width of each component is not significantlygreater than the width of M then, because of the scarcity of singular times, we would still beable to apply the argument of Corollary 2 using Theorem 1 at non-singular times. It turns outthat this is exactly the case; for each component there exists a map from M just before thesingular time to the component which is close to distance non-increasing. Using this map, onecan show that the area of the sweepouts used to measure the width cannot increase much. Thus,with some extra work, the finite existence time will apply to Ricci flows with surgery as well.

3.3 The proof of Theorem 1

The proof of Theorem 1 relies on properties of the Ricci flow covered in Section 2.7 together withTheorem 3.3.1 below. Theorem 3.3.1 is the consequence of two additional results; the existenceof a sweepout tightening map and bubble compactness of almost harmonic, almost conformalmaps. The proofs of those will be the subject of the remainder of the thesis. We will thereforedefer proving the theorem until Chapter 6 when the needed results have been covered.

Theorem 3.3.1 asserts the existence of a sequence of sweepouts with maximal slice energyconverging to the width, and, in addition, that certain slices of sweepouts in the sequence become

25


closer to finite collections of harmonic maps. When the width is positive, those maps will benon-constant. We can therefore apply Proposition 2.7.7 to get a bound on the rate of change ofarea of the harmonic maps and relate the bound to the area of slices of the sweepouts. This willbe done in the lemma following the statement of Theorem 3.3.1. Finally, we will use the lemmato bound the rate of change of width and prove Theorem 1.

In the statement of Theorem 3.3.1, when f ∈W 1,2(S2,M), we let nf (p) denote a unit normalto df(TpS2) whenever Jf (p) 6= 0.

Theorem 3.3.1. Let M be a compact manifold with metric g and let σ ∈ ΩM be a sweepout.Then there exists a sequence (γj), γj ∈ [σ] satisfying

(1) maxs∈[0,1]Eg(γj(·, s))j→Wg([σ]) ,

(2) for any ε > 0 there exist J ∈ N and δ > 0 so that if j ≥ J and s ∈ [0, 1] with

Areag(γj(·, s)) > Wg([σ])− δ

then there exists a finite collection of harmonic maps u1, . . . , uk, ui : S2 →M with∣∣∣∣∣Areag(γj(·, s))−∑i

Areag(ui)

∣∣∣∣∣ < ε (3.3.1)

and ∣∣∣∣∣∫γj(·,s),g

(Sg −Ricg(nγj(·,s), nγj(·,s))

)−∑i

∫ui,g

(Sg −Ricg(nui , nui))

∣∣∣∣∣ < ε . (3.3.2)

In the lemma below, we will get information on how the area of slices of sweepouts in thesequence varies in small time intervals. Here positive width becomes essential. It enables us toensure that the harmonic maps of the above theorem are non-constant. We assume that M issimply connected in Theorem 1 in order to apply this lemma.

Lemma 3.3.2. Let M be a compact manifold and σ ∈ ΩM a sweepout. Let gt, t ∈ I be aRicci flow. Fix a time t0 ∈ I and suppose Wgt0

([σ]) > 0. Use Theorem 3.3.1 with the metric gt0to get a sequence (γj) of sweepouts satisfying the properties to the theorem. Then there existsC(M, gt, [σ], t0) > 0 with the property that, for any ε > 0, there exist J ∈ N and H > 0 suchthat, for any j ≥ J , h ∈ [0, H) and s ∈ [0, 1],

Areagt0+h(γj(·, s))− max

s∈[0,1]Areagt0 (γj(·, s))

≤(−4π + Cε− S0(t0)

2maxs∈[0,1]

Areagt0 (γj(·, s)))h+ Ch2 ,

(3.3.3)

where S0(t) is a non-positive lower bound for the scalar curvature at time t.

Proof. Let ε > 0 and shrink it so that it is less than Wgt0([σ])/2. Since (γj) has the properties

of Theorem 3.3.1 we can let J ∈ N and δ > 0 be given by Property (2) of the theorem. Sincemaxs∈[0,1]Egt0 (γj(·, s))→Wgt0

([σ]) by Property (1), we can assume Egt0 (γj(·, s)) ≤Wgt0([σ])+1

26


and hence also Areagt0 (γj(·, s)) ≤Wgt0([σ]) + 1. We shrink δ so that it is less than Wgt0

([σ])/2.Let j ≥ J and s ∈ [0, 1] and suppose

Areagt0 (γj(·, s) > Wgt0([σ])− δ . (3.3.4)

Then, by Property (2) of Theorem 3.3.1, there exists a finite collection of harmonic mapsu1, . . . , uk, ui : S2 → M satisfying (3.3.1) and (3.3.2). The inequality (3.3.1), Wgt0

([σ])being positive, and the requirements on δ and ε imply that

∑i Areagt0 (ui) > 0. Thus, at least

one of the maps ui must be non-trivial. We discard the trivial maps. By Lemma 2.7.4 and(3.3.2),

dt Areagt(γj(·, s))|t0 < ε+

∑i

dt Areagt(ui)|t0 .

Using Proposition 2.7.7 and the non-triviality of ui, we get

dt Areagt(γj(·, s))|t0 ≤ ε+

∑i

(−4π − 1

2

∫ui

Sgt0

)≤ ε− 4π − 1

2

∑i

Areagt0 (ui) minx∈M

Sgt0 (x) .

From Lemma 2.1.6 we have a lower bound S0(t) for minx∈M Sgt(x) dependent on g0. We canassume S0(t) is non-positive. Combined with (3.3.1), we get

dt Areagt(γj(·, s))|t0 ≤ C1ε− 4π − 1

2Areagt0 (γj(·, s))S0(t0) ,

where C1 = −S0(t0)/2 + 1. Since S0(t) ≤ 0,

dt Areagt(γj(·, s))|t0 ≤ C1ε− 4π − 1

2maxs∈[0,1]

Areagt0 (γj(·, s))S0(t0) . (3.3.5)

By Corollary 2.7.5 and Remark 2.7.2, using the upper bound on the areas, Areagt0+h(γj(·, s))

is a C2 function of t and there exist H,C2(M, gt, t0, H) > 0 independent of s and j such thatwhen h ∈ [0, H),∣∣∣dh2 Areagt0+h

(γj(·, s))∣∣∣ Corollary 2.7.5

≤ C2 Areagt0+h(γj(·, s))

Remark 2.7.2≤ C2(Areagt0 (γj(·, s)) + 1) ≤ C3

with C3 = C2(Wgt0([σ]) + 2). Hence, by Taylor expanding (see [Sol01, Page 62,63]),

Areagt0+h(γj(·, s)) ≤ Areagt0 (γj(·, s)) + dh Areagt0+h

(γj(·, s))|0h+C3

2h2

≤ maxs∈[0,1]

Areagt0 (γj(·, s)) + dt Areagt(γj(·, s))|t0h+

C3

2h2 .

(3.3.6)

Combining (3.3.5) and (3.3.6), we get


s∈[0,1]Areagt0 (γj(·, s))

≤(C1ε− 4π − S0(t0)

2maxs∈[0,1]

Areagt0 (γj(·, s)))h+

C3

2h2 .

Thus, when (3.3.4) holds, the result follows by setting C = max(C1, C3/2).

27


IfAreagt0 (γj(·, s) ≤Wgt0

([σ])− δ (3.3.7)

then


s∈[0,1]Areagt0 (γj(·, s)) ≤ Areagt0+h

(γj(·, s))−Wgt0([σ])

(3.3.7)

≤ Areagt0+h(γj(·, s))−Areagt0 (γj(·, s)− δ .

By the equicontinuity of Remark 2.7.2 and the upper bound on the areas, we can shrink H suchthat still H > 0 and, for all j ≥ J , h ∈ [0, H), and s ∈ [0, 1],

Areagt0+h(γj(·, s))−Areagt0 (γj(·, s)) < δ/2 .

HenceAreagt0+h

(γj(·, s))− maxs∈[0,1]

Areagt0 (γj(·, s)) < −δ/2 .

Then (3.3.3) follows by possibly shrinking H even further.

Proof of Theorem 1. By the assumptions on M and Proposition 3.1.5, there exists a sweepoutσ ∈ ΩM such that Wgt([σ]) > 0 for all t. Let C(M, g0) > 0 be such that − 3

2(t+C) is a negativelower bound for Sgt by Lemma 2.1.6. We wish to show that, for any t0 ∈ I,

D+t Wgt0

([σ]) ≤ −4π +3

4(t0 + C)Wgt0

([σ]) (3.3.8)

which follows if, for any t0 ∈ I, there exists Ct0 > 0 so that given ε > 0 there exists H > 0, with

Wgt0+h([σ])−Wgt0

([σ])h

≤ −4π + Ct0ε+3

4(t0 + C)Wgt0

([σ]) + Ct0h (3.3.9)

for all h ∈ [0, H). Fix a t0 ∈ I and let (γj) be the sequence of sweepouts from Theorem 3.3.1.By positivity of the width, Lemma 3.3.2 applies. Let Ct0 be given by the lemma and supposeε > 0. Let h > 0 and for any j ∈ N, let sj ∈ [0, 1] be such that Areagt0+h

γj(·, sj) is maximal.By Proposition 3.1.6,

Wgt0+h([σ]) = inf

σ∈[σ]maxs∈[0,1]

Areagt0+h(σ(·, s))

so thatWgt0+h

([σ]) ≤ Areagt0+h(γj(·, sj)) . (3.3.10)

Since, by Property (1) of Theorem 3.3.1, maxs∈[0,1]Egt0 (γj(·, s))→Wgt0([σ]), we have, for all j,

Wgt0([σ]) = inf

σ∈[σ]maxs∈[0,1]

Areagt0 (σ(·, s))

≤ maxs∈[0,1]

Areagt0 (γj(·, s)) ≤ maxs∈[0,1]

Egt0 (γj(·, s))→Wgt0([σ])

so thatmaxs∈[0,1]

Areagt0 γj(·, s)→Wgt0

([σ]) . (3.3.11)

28


Lemma 3.3.2 asserts the existence of H > 0 such that, when h ∈ [0, H) and sufficiently large j,(3.3.3) and (3.3.10) together with Lemma 2.1.6 imply that

Wgt0+h([σ])− max

s∈[0,1]Areagt0 γ

j(·, s)

≤(−4π + Ct0ε+

34(t0 + C)

maxs∈[0,1]

Areagt0 γj(·, s)

)h+ Ct0h

2 .

Then letting j →∞, (3.3.11) gives

Wgt0+h([σ])−Wgt0

([σ]) ≤(−4π + Ct0ε+

34(t0 + C)

Wgt0([σ])

)h+ Ct0h

2 .

Since this holds for any ε > 0, we get (3.3.9). Then, since the argument holds for any t0 ∈ I,(3.3.8) follows concluding the proof.

Notes

The chapter is primarily a detailed exposition of results of [CM07a, Section 1]. The continuityof the width, the part of Appendix C concerning forward difference quotients, and the discussionon the different kinds of singularities and extending the result to Ricci flow with surgery areadded.

29

4

Tightening sweepouts

In this chapter we prove the existence of a sweepout tightening map. Such a map sends sweepoutsto sweepouts with the image sweepouts in a sense being closer to harmonic. This property willallow us to use the map on each element of a certain sequence of sweepouts. The obtainedsequence of image sweepouts can then be used with the bubble convergence result of the nextchapter leading to the proof of Theorem 3.3.1.

The map is constructed by a careful application of a harmonic replacement procedure. Mostof the chapter concerns properties of this procedure. The actual construction of the map takesplace at the end of the chapter.

We note that the notion of a sweepout tightening map is close to what in [CM07a] is called anenergy decreasing map. We have chosen the different name because, in our view, it captures theidea behind the construction and because the map on each slice is only energy non-increasing.

4.1 Sweepout tightening maps

Let us go down one dimension to gain some intuition for how a tightening map can be con-structed. As in Section 3.1 let σ : S1 × [0, 1] be a sweepout of a compact surface embedded inR3. In Figure 4.1 we see the loop σ(·, s) for a s ∈ [0, 1]. Since harmonic maps S1 → M areprecisely geodesics, if we wish to make the loop closer to being harmonic, it is natural to cutout parts of the loop being far from geodetic and replace them with geodesics with the sameendpoints. Since geodesics on small intervals are distance minimizing, being far from geodesiccan be measured as being longer than geodesics. Since the length of a curve is roughly equivalentto the energy, a good strategy would be to replace parts of the loop having high energy. Suchparts can be images σ(I, s) with I being small, closed subsets of S1. Back in three dimensions,the parts of the loop considered above naturally correspond to images of closed balls on S2.Thus, applying the same strategy, we aim at replacing σ(·, s) at high energy balls with maps ofminimal energy. When the balls are sufficiently small, such maps are precisely harmonic.

If a slice is “far” from being harmonic, the tightening procedure is expected to make asignificant decrease in the overall energy of the slice, due to the replacement with maps ofminimal energy. On the contrary, if the slice is “close” to being harmonic, the energy decreasewill be slight. In this case, we expect no balls to exist on which further replacement wouldchange the slice significantly. This property is expressed in Property (4) below, which formalizesour notion of the tightening map producing sweepouts closer to being harmonic.

We note that the required properties of a sweepout tightening map, in particular Property(4), are not only chosen as to formalize the above intuitive idea, but also to be sufficiently weak

31

4. Tightening sweepouts

(·,s)

replacement

(I,s)(high energy)

Figure 4.1: Tightening a slice of a sweepout of a compact hypersurface.

to allow the existence of a map. E.g., we introduce the continuous map ψ : [0,∞) → [0,∞).It would be natural always to let this map be just the identity, but then the proof of existencepresented below would not be adequate. This is also the reason for introducing the set ΩM .

Definition 4.1.1 (Sweepout tigtening maps). Let (M, g) be a manifold isometrically embeddedin Rm. Define

ΩM = σ ∈ ΩM |W ([σ]) > 0 and for each s ∈ [0, 1], σ(·, s) is either constant or not harmonic .

A sweepout tightening map is a triple (Ψ, ψ, εT ), where Ψ : ΩM → ΩM , ψ ∈ C0([0,∞), [0,∞))and εT > 0, such that, for all σ ∈ ΩM ,

(1) Ψ(σ) ∈ [σ] ,

(2) for each s ∈ [0, 1] , E(Ψ(σ)(·, s)) ≤ E(σ(·, s)) ,

(3) ψ(0) = 0 ,

(4) there exists r ∈ (0, 1] such that, for each s ∈ [0, 1] such that E(σ(·, s)) ≥W ([σ])/2 and foreach finite collection of disjoint projected balls B with E(Ψ(σ)(·, s)|B) ≤ εT , we have, foreach energy minimizing map v : ∪B∈BrB →M with v|∪B∈B∂rB = Ψ(σ)(·, s)|∪B∈B∂rB, that

E(v −Ψ(σ)(·, s)|∪B∈BrB) ≤ ψ(E(σ(·, s))− E(Ψ(σ)(·, s))) .

The following theorem is the main result of the chapter.

Theorem 4.1.2. Let (M, g) be a compact manifold isometrically embedded in Rm. Then thereexist Ψ, ψ, ε such that (Ψ, ψ, ε) is a sweepout tightening map.

We begin by proving showing that the requirement that each slice of the sweepouts in thedomain ΩM must be either constant or non-harmonic is not an obstacle to the use of the mapon general sweepouts. We then continue with a detailed discussion of harmonic replacement; wefirst prove that harmonic replacement is possible and then turn to continuity properties. Thelast step before the proof of Theorem 4.1.2 will be investigating what happens to the energywhen iterating harmonic replacement.

32

Removing non-constant harmonic slices

Throughout the chapter, we will let (M, g) be a fixed compact manifold. As usual we canassume that M is isometrically embedded in Rm. We will let εH > 0 be a constant less thanthe constant εL of Lemma 2.5.2, less than εSU of the Local Compactness Property (H1) ofTheorem 2.6.3, strictly less than εmin/2 with εmin the lower width bound of Proposition 3.1.4,and less than 1. In addition, B = (U,Λ) will be a projected ball and B ⊂ R2 the image Λ(U).By definition, B is then a ball.

4.2 Removing non-constant harmonic slices

We will prove that, given a sweepout σ ∈ ΩM , there exists a sweepout σ which has no non-constant harmonic slices and is close in energy and homotopic to σ. Thus, if W ([σ]) > 0, σ willbe in the domain ΩM of a sweepout tightening map. We will need a result on the density ofcertain sweepouts proved in [CM07a]. After stating the result, we turn to constructing σ.

Lemma 4.2.1 (Density of C2 sweepouts). For any σ ∈ ΩM and ε > 0 there exists σ ∈ [σ] suchthat

maxs∈[0,1]

‖σ(·, s)− σ(·, s)‖W 1,2(S2,Rm) ≤ ε ,

for each s ∈ [0, 1], σ ∈ C2(S2,Rm), and the map s 7→ σ(·, s) is C0([0, 1], C2(S2,Rm)).

Proof. See [CM07a, Lemma D.1].

Corollary 4.2.2. For any σ ∈ ΩM and ε > 0 there exists σ ∈ [σ] such that

maxs∈[0,1]

E(σ(·, s)) < maxs∈[0,1]

E(σ(·, s)) + ε (4.2.1)

and for no s ∈ [0, 1], the slice σ(·, s) is non-constant and harmonic.

Proof. Given σ and ε > 0 use Lemma 4.2.1 to get a sweepout σ ∈ [σ] with the properties of thelemma and maxs∈[0,1] ‖σ(·, s)− σ(·, s)‖W 1,2(S2,Rm) sufficiently small so that

maxs∈[0,1]

E(σ(·, s)) < maxs∈[0,1]

E(σ(·, s)) + ε/2 .

Using a stereographic projection, we can consider σ as a map R2 →M as well. The strategy isnow to construct a map Φ : R2 → R2, so that each slice of σ = σ Φ has the desired property.The regularity of σ allows us to define

C = sups∈[0,1]

supp∈cl(B1)(0)

‖dσ(p, s)‖2 .

Now chose R ∈ (0,min((ε/(5Cπ))1/2 , 1)] and define

Φ(r, θ) =

(2r, θ) r < R/2 ,(R, θ) R/2 ≤ r < R ,(r, θ) R ≤ r

effectively squeezing the area between BR/2(0) and BR(0) into a circle of radius R. Since Φ ishomotopic to the identity through the homotopy

H(r, θ, t) =

((1 + t)r, θ) r < R/(1 + t) ,

(R, θ) R/(1 + t) ≤ r < R ,(r, θ) R ≤ r ,

33


σ is homotopic to σ, so that σ ∈ [σ]. Since ‖dΦ‖ ≤ 2, the choice of R implies that for anys ∈ [0, 1],

2 |E(σ(·, s))− E(σ(·, s))| ≤∫BR(0)

‖dσ(·, s)‖2 +∫BR(0)

‖dσ(·, s)‖2

≤∫BR(0)

4‖(dσ)(Φ(·), s)‖2 +∫BR(0)

‖dσ(·, s)‖2

≤ πR2(4C) + πR2C ≤ 5π( ε

5Cπ

)C = ε

ensuring that (4.2.1) holds.Assume σ(·, s) is harmonic for some s ∈ [0, 1]. Then, regarding σ as a map S2 → M , by

Proposition 2.6.4, σ is conformal. Since ∂rσ = 0 on BR(0) \ BR/2(0), the conformality impliesthat also ∂θσ = 0 is zero on the set, so that σ|BR(0)\BR/2(0) = 0. Then by Proposition 2.5.5, σ isconstant on all of R2 as required.

4.3 Harmonic replacement

Though we will only use harmonic replacement on slices of sweepouts, the procedure works forgeneral maps u ∈ C0 ∩W 1,2(S2,M) as well. Hence we will turn our attention to such mapsinstead of sweepouts for a while.

Theorem 4.3.1 (Existence of replacement). Let u ∈ C0 ∩W 1,2(B,M) with E(u) ≤ εH . Thenthere exists a unique energy minimizing map w ∈ C0 ∩W 1,2(B,M) equal to u on the boundary.

Proof. Let u ∈ C0 ∩W 1,2(B,M) be the map u Λ−1. The existence of an energy minimizingmap w ∈W 1,2(B,M) is the result of Theorem 2.4.9. Since

E(w) ≤ E(u) ≤ εH ≤ εL ,

w is unique by Corollary 2.5.4. By Theorem 2.6.2, w ∈ C0(cl (B) ,M). By conformal invarianceof the energy, w = w Λ is energy minimizing. The uniqueness of w follows from the uniquenessof w.

We will let HB denote the map from the C0 ∩ W 1,2(B,M) maps with energy less thanεH to C0 ∩ W 1,2(B,M) defined by HB(u) = w, where w is the energy minimizing map ofTheorem 4.3.1. To ease notation, we let MB be the set

u ∈ C0 ∩W 1,2(B,M) |E(u) ≤ εH

so

that HB :MB →MB. We let MB have the norm

‖ · ‖B = ‖ · ‖C0(cl(B),Rm) + (2E(·))1/2 = ‖ · ‖C0(cl(B),Rm) + ‖ · ‖W 1,2(B,Rm) − ‖ · ‖W 0,2(B,Rm) .

The reason for dropping the ‖ · ‖W 0,2(B,Rm) term is the fact that the norm on MB is nowconformally invariant. Hence, for any u ∈MB,

‖u‖B = ‖u Λ−1‖B .

Therefore, in terms of the norm, we are completely free to regard u as a map defined on B ⊂ R2.We use this identification in some of the following proofs. This will especially be important

34

Comparison maps

when scaling the radius of the ball. Some results, in general those not concerning the abovenorm, will not be independent of which domain we choose. Such results will be proved for mapsdefined on balls in R2.

Since

‖ · ‖B ≤ ‖ · ‖C0(cl(B),Rm) + ‖ · ‖W 1,2(B,Rm) ≤ ‖ · ‖B + ‖ · ‖C0(cl(B),Rm) Area(S2)1/2 ,

MB has the subspace topology of C0 ∩W 1,2(S2,M). Thus, from a topological point of view, wehave not changed anything in defining this simpler norm.

4.4 Comparison maps

A basic tool used in the rest of the chapter is comparison maps. The use of these also highlightsone of the reasons why we perform replacement with energy minimizing maps instead of justharmonic maps.

Suppose u ∈MB and w = HB(u). Then of course

E(w) ≤ E(u) . (4.4.1)

Often we need a better estimate on the energy of w; after all, if we have no information onu, E(u) could be arbitrarily large, so (4.4.1) would not really say much. We remedy this bycreating some map v equal to u and w on ∂B of which we know E(v). Often v is quite explicitlyconstructed. We then estimate E(w) by

E(w) ≤ E(v) . (4.4.2)

Though the method might seem too trivial to be given a name, it is often helpful keeping inmind that this trick is what is used down deep in the technical estimates. We will point outwhenever this is the case.

It is often hard to directly construct comparison maps with images in M . We will e.g. beinterested in connecting points in M with straight lines, but such lines may not be contained inM . We remedy this by using the embedding of M into Rm; we construct the maps in Rm andthen use a retraction to make sure that the image of the resulting map is contained in M . Forthis purpose, tubular neighborhoods are convenient. The notion of δ-tubular neighborhoods isdiscussed in Appendix B.2. In the following, we will let δ(M, g), Cδ(M, , g, δ) > 0 be constantssuch that there exists a δ-tubular neighborhood Uδ ⊂ Rm of M and a smooth nearest pointretraction r : Uδ →M with the property that, for any x ∈ Uδ,

‖dxr‖∞ ≤ 1 + Cδ‖x− r(x)‖Rm ≤√

2 . (4.4.3)

For technical reasons, we will shrink δ such that δ ≤ C−1δ . The existence of the claimed constants

is proved in Appendix B.2.

4.5 Continuity of HB

We will here establish continuity of the harmonic replacement map HB. We do this stepwisekeeping track of exactly which kind of convergence is involved in the different steps. In additionto making clear which assumptions we need at each step, it will be useful later on to be able torefer to the results separately. When this is done, we will discuss continuity when performingreplacement on a finite number of disjoint balls.

We start by proving uniform continuity of the energy of HB.

35


Lemma 4.5.1. There exists a constant CU (M, δ,Cδ) > 0 such that, for any u1, u2 ∈MB,

|E(HB(u1))− E(HB(u2))|

≤ CU (E(u1) + E(u2)) ‖u1 − u2‖C0(cl(B),Rm) + CU (E(u1) + E(u2))1/2 (E(u1 − u2))1/2 .

(4.5.1)

Furthermore, if HB(u2)(x) + (u1(x)− u2(x)) ∈ Uδ for all x ∈ B then letting v1 = r (HB(u2) +(u1 − u2)) we have

E(v1)≤ E(HB(u2))

+ CU (E(u1) + E(u2))‖u1 − u2‖C0(cl(B),Rm) + CU (E(u1) + E(u2))1/2 (E(u1 − u2))1/2 .

(4.5.2)

Proof. Set CU = max(1/δ, 3Cδ, 16) and let w1 = H(u1) and w2 = H(u2).Suppose first ‖u1 − u2‖C0(cl(B),Rm) ≥ δ. Then, since CU ≥ 1/δ,

|E(w1)− E(w2)| ≤ E(w1) + E(w2) ≤ E(u1) + E(u2)≤ CU‖u1 − u2‖C0(cl(B),M)(E(u1) + E(u2))

≤ CU‖u1 − u2‖C0(cl(B),M)(E(u1) + E(u2))

+ CU (E(u1 − u2)(E(u1) + E(u2))1/2

proving (4.5.1) in this case.If ‖u1 − u2‖C0(cl(B),Rm) < δ then w2(x) + (u1(x)− u2(x)) ∈ Uδ for all x ∈ B. We construct a

comparison map v1 by defining

v1 = r (w2 + (u1 − u2)) .

Note that

(1 + Cδ‖u1 − u2‖C0(cl(B),Rm))2 = 1 +

(Cδ‖u1 − u2‖C0(cl(B),Rm)

)2 + 2Cδ‖u1 − u2‖C0(cl(B),Rm)

≤ 1 + 3Cδ‖u1 − u2‖C0(cl(B),Rm) ≤ 4(4.5.3)

since we required δ ≤ C−1δ implying

Cδ‖u1 − u2‖C0(cl(B),Rm) < Cδδ ≤ 1 .

We will now estimate the energy of v1 with a series of inequalities. First,

E(v1) =∫B

12‖dr (w2 + (u1 − u2))‖2 ≤

∫B

12‖dw2+(u1−u2)r‖2∞‖d(w2 + (u1 − u2))‖2

36

Continuity of HB

by the chain rule and (2.2.2). Using Cauchy-Schwartz, (4.4.3) and that ‖w2 + (u1−u2)− r(w2 +(u1 − u2))‖Rm ≤ ‖w2 + (u1 − u2)− w2‖Rm since r is the nearest point projection, we have∫B

12‖dw2+(u1−u2)r‖2∞‖d(w2 + (u1 − u2))‖2

(4.4.3)

≤∫B

12

(1 + Cδ‖w2 + (u1 − u2)− r(w2 + (u1 − u2))‖Rm)2‖d(w2 + (u1 − u2))‖2

≤∫B

12

(1 + Cδ‖u1 − u2‖Rm)2‖d(w2 + (u1 − u2))‖2

C-S≤ (1 + Cδ‖u1 − u2‖C0(cl(B),Rm))

2

∫B

12(‖dw2‖2 + ‖d(u1 − u2)‖2 + 2‖dw2‖‖d(u1 − u2)‖

)= (1 + Cδ‖u1 − u2‖C0(cl(B),Rm))

2

(E(w2) + E(u1 − u2) +

∫B‖dw2‖‖d(u1 − u2)‖

).

(4.5.4)

By Cauchy-Schwartz and (4.5.3), we get

(1 + Cδ‖u1 − u2‖C0(cl(B),Rm))2

(E(w2) + E(u1 − u2) +

∫B‖dw2‖‖(u1 − u2)‖

)C-S≤ (1 + Cδ‖u1 − u2‖C0(cl(B),Rm))

2(E(w2) + E(u1 − u2) + 2 (E(w2)E(u1 − u2))1/2

)(4.5.3)

≤ (1 + 3Cδ‖u1 − u2‖C0(cl(B),Rm))(E(w2) + E(u1 − u2) + 2 (E(w2)E(u1 − u2))1/2

).

Using Corollary 2.4.3 and that E(w2) ≤ E(u2) ≤ E(u1) + E(u2), we have

(1 + 3Cδ‖u1 − u2‖C0(cl(B),Rm))(E(w2) + E(u1 − u2) + 2 (E(w2)E(u1 − u2))1/2

)= E(w2) + 3Cδ‖u1 − u2‖C0(cl(B),Rm)E(w2)

+ (1 + 3Cδ‖u1 − u2‖C0(cl(B),Rm))(E(u1 − u2)1/2 + 2E(w2)1/2

)E(u1 − u2)1/2

Cor. 2.4.3(4.5.3)

≤ E(w2) + 3Cδ‖u1 − u2‖C0(cl(B),Rm) (E(u1) + E(u2))

+ 4(

21/2(E(u1) + E(u2))1/2 + 2(E(u1) + E(u2))1/2)E(u1 − u2)1/2

≤ E(w2) + Cu‖u1 − u2‖C0(cl(B),Rm) (E(u1) + E(u2))

+ Cu(E(u1) + E(u2))1/2E(u1 − u2)1/2 .

Putting the inequalities together, we get (4.5.2). Then notice that v1 equals u1 and w1 on theboundary, and thus the comparison map method gives E(w1) ≤ E(v1). Hence, we have

E(w1)− E(w2)

≤ CU‖u1 − u2‖C0(cl(B),Rm)(E(u1) + E(u2)) + CU (E(u1) + E(u2))1/2E(u1 − u2)1/2 .

Now, redoing the argument with the comparison map

v2 = r (w1 + (u2 − u1))

37


we get

E(w2)− E(w1)

≤ CU‖u1 − u2‖C0(cl(B),Rm)(E(u1) + E(u2)) + CU (E(u1) + E(u2))1/2E(u1 − u2)1/2

proving (4.5.1).

Corollary 4.5.2. Let (uj), uj ∈ MB be a sequence converging to a map u ∈ MB. Definewj = H(uj) and w = H(u). Then E(wj − w)→ 0.

Proof. By assumption, both ‖uj − u‖C0(cl(B),Rm) → 0 and E(uj − u) → 0. Using the uniformconvergence we can, for sufficiently large j, define maps vj by

vj = r (w + (uj − u)) .

Note that wj equals vj on ∂B. By (4.5.2) of Lemma 4.5.1 we therefore have

E(vj)− E(wj) ≤ CU‖uj − u‖C0(cl(B),M)(E(uj) + E(u)) + CU(E(uj − u)(E(uj) + E(u)

)1/2so that E(vj)− E(wj)→ 0. By Lemma 2.5.2, we get

E(vj)− E(wj) ≥ 12E(vj − wj)

implying E(vj − wj)→ 0.By Proposition B.2.3, the map x 7→ dxr is Lipschitz. Thus, there exists L > 0 such that∫B‖dw − dvj‖2 =

∫B‖d(r w)− d(r (w − (uj − u)))‖2

=∫B‖dwrdw − dw−(uj−u)rd(w − (uj − u))‖2

=∫B‖(dwr − dw−(uj−u)r)dw − dw−(uj−u)rd(uj − u)‖2

Cor. 2.4.3(2.2.2)

≤∫B

2‖dwr − dw−(uj−u)r‖2F ‖dw‖2 + 2‖dw−(uj−u)r‖2∞‖d(uj − u)‖2

≤∫B

2L 2‖uj − u‖2Rm‖dw‖2 +∫B

2(1 + Cδ)2‖uj − u‖2Rm‖d(uj − u)‖2 .

In the last inequality we used that r is the nearest point projection as in (4.5.4). By theconvergence of uj , we get E(w−vj)→ 0. By the triangle inequality for energy (Corollary 2.4.3),

E(wj − w) ≤ 2E(wj − vj) + 2E(vj − w)→ 0 .

We will use the following lemma in proving uniform continuity of HB. The proof relies on theLocal Compactness Property of harmonic maps together with Theorem 2.6.2 and the uniquenessof low energy harmonic maps (Corollary 2.5.4). The reader may notice that the compactnessargument is quite similar to the argument we will later use in Theorem 5.1.3. There are twodifferences, though, which cause us to give the proof here instead of referring to the theorem;we will assume low energy of the entire map, which rules out the existence of so-called singularpoints, and the argument will be carried out on the non-compact manifold B.

38

Continuity of HB

Lemma 4.5.3. Let (wj), wj ∈ MB be a sequence of harmonic maps, and w ∈ MB an energyminimizing map, such that

limj→∞

‖wj |∂B − w|∂B‖C0(∂B,Rm) = 0

and such that, given ε > 0 and p ∈ ∂B, there exists r > 0 so that, for all j,

E(wj |B∩Br(p)) < ε .

Then wjC0(cl(B),Rm)→ w.

Proof. We assume there exists a subsequence (wjk) of (wj) such that, for some δ > 0,

‖wjk − w‖C0(cl(B),Rm) ≥ δ (4.5.5)

and derive a contradiction. From this we conclude that

‖wj − w‖C0(cl(B),Rm) → 0

which will prove the lemma.Assume (wj) has been replaced by a subsequence such that (4.5.5) holds. We construct a

sequence of compact sets K l, l ∈ N exhausting cl (B) and use the Local Compactness Propertyof harmonic maps to prove C1 convergence of a subsequence on each of K l. Then the assumptionon the energy close to ∂B will allow us to use Theorem 2.6.2 to prove equicontinuity of the setwj | j ∈ N

. The Arzela-Ascoli Theorem will then trigger the contradiction.

First, since ‖wj‖B is bounded by assumption, ‖wj‖W 1,2(B,Rm) is bounded. By Lemma 2.2.2,a subsequence converges weakly to a map w ∈W 1,2(B,M). Replace (wj) by this subsequence.

Since B ⊂ R2 is a ball, we let

K l = Λ−1(cl (B)rl (x)) ,

where x is the center of B and (rl), rl > 0 a sequence converging from below to the radiusr of B. Fix l ∈ N and use compactness of K l to choose a finite number of geodesic ballsBr1(x1), . . . , Brp(xp), Bri(xi) ⊂ S2 contained in B, such that the balls of half the radius coverK l. Then for each of these balls, use (H1) of Theorem 2.6.3 to replace (wj) by a subsequencesuch that we have C1(Bri/2(xi),Rm) convergence to a harmonic map wi : Bri/2(xi) → M for

each i. Since we ensured that wjW 1,2(B,Rm)

w, by Lemma 2.2.1 also

wj |Bri/2W 1,2(Bri/2,R

m) w|Bri/2 ,

and since limits in the weak topology are unique, wi = w|Bri/2(xi). Since this holds for alli ∈ 1, . . . , p, the balls cover K l, and all wi are harmonic, we get that w|Kl is harmonic byLemma 2.5.1.

We now replace (wj) by the diagonal subsequence; that is, the jth entry of (wj) is now thejth entry of the subsequence obtained by successively carrying out the above procedure for eachl = 1, . . . , j. Then (wj) converges in C1 to w on any compact subset of B. Furthermore, againusing Lemma 2.5.1 we get that w is harmonic.

We can now apply Theorem 2.6.2 to show that the setwj | j ∈ N

is equicontinuous. First,

since we assumed uniform convergence of wj on ∂B, the setwj |∂B | j ∈ N

is equicontinuous.

39


The equicontinuity onwj |B | j ∈ N

follows from the C1 convergence of Kj ; for any x ∈ B

eventually x is in the interior of Kj on which we have uniform convergence. The second propertyneeded to get the equicontinuity of

wj | j ∈ N

from Theorem 2.6.2 is exactly what we assumed.

The maps wj converge pointwise to a map w; on B they converge to w and on ∂B theyconverge to w. By the Arzela-Ascoli Theorem1, wj converges uniformly and w ∈ C0(cl (B) ,Rm).Since w|B = w, w is harmonic and it agrees with w on the boundary. By Corollary 2.5.4, w = w.But this contradicts (4.5.5);

0 = ‖w − w‖C0(cl(B),Rm) = limj→∞

‖wj − w‖C0(cl(B),Rm) ≥ δ > 0 .

Corollary 4.5.4. Let (uj), uj ∈ MB be a sequence converging to a map u ∈ MB. Definewj = H(uj) and w = H(u). Then ‖wj − w‖C0(cl(B),Rm) → 0.

Proof. We aim at using Lemma 4.5.3 on wj to get the result. Since w,wj equal u, uj respectivelyon the boundary, we have

limj→∞

‖wj |∂B − w|∂B‖C0(∂B,Rm) = 0 .

Furthermore, given ε > 0 and p ∈ ∂B choose r > 0 such that

E(w|B∩Br(p)) < ε/2 .

By Corollary 4.5.2, E(wj − w)→ 0. Thus we can choose J ∈ N such that, for all j ≥ J ,∣∣E(w|B∩Br(p))− E(wj |B∩Br(p))∣∣ < ε/2 .

Hence, for such j,E(wj |B∩Br(p)) < ε . (4.5.6)

After shrinking r such that (4.5.6) also holds for the finitely many j < J , we have proved the

second property needed in Lemma 4.5.3. By the lemma, wjC0(cl(B),Rm)→ w proving the result.

Theorem 4.5.5 (Continuity of HB). The harmonic replacement map HB is continuous.

Proof. Follows immediately from Corollary 4.5.2, Corollary 4.5.4, and the definition of the normon MB.

4.6 Continuity with scaling

Let γ ∈ ΩM and recall that the map s 7→ γ(·, s) is continuous. By Theorem 4.5.5, the maps 7→ HB(γ(·, s)) is also continuous. When constructing the sweepout tightening map, we needto scale the size of the ball that we perform harmonic replacement on. This rescaling needs tobe continuous as well. For r ∈ (0, 1] and u ∈MB we define

HrB(u)(x) =u(x) x 6∈ rBHrB(u|rB)(x) x ∈ rB .

1See [Roy88, Page 169].

40

Continuity with scaling

We then aim at proving continuity of the map (s, r) 7→ HrB(γ(·, s)) defined on [0, 1]× [0, 1] andmapping into MB.

In the arguments, we will use the following shorthand: for any r1, r2 ∈ [0, 1] we defineA(r1, r2) ⊂ S2 to be the set

max(r1, r2)B \min(r1, r2)B ,

which corresponds to an annulus in R2.

Lemma 4.6.1. Suppose u ∈MB. Then for all r1, r2 ∈ [0, 1],

E(Hr1B(u)−Hr2B(u)) ≤ 4E(u|A(r1,r2)) .

Proof. Assume r1 > r2, the other case being symmetric. We construct a comparison mapv : r2B →M to assess the energy of Hr2B(u).

Let C1 : R2 → R2 be the dilation (r, θ) 7→ (rr21/r

22, θ) and C2 the inversion (r, θ) 7→ (r2

2/r, θ).Both C1 and C2 are conformal. Thus, using the identification between B and B ⊂ R2, themaps Hr1B(u)|r1B and v1 = Hr1B(u) C1|(r22/r1)B have equal energy. This is also the case for themaps u|A(r1,r2) and v2 = u C2|A(r22/r1,r2). Note that v1 and v2 coincide on ∂r2

2/r1B. Hence, byLemma 2.2.3, we can glue them together to get the map

v(r, θ) =v1(r, θ) r ∈ [0, r2

2/r1]v2(r, θ) r ∈ [r2

2/r1, r2] .

Since, on ∂r2B, v equals the map Hr2B(u), which is minimizing on r2B, we have

E(Hr2B(u)|r2B) ≤ E(v) = E(v1) + E(v2)= E(Hr1B(u)|r1B) + E(u|A(r1,r2)) .

From this we get the upper bound

E(Hr2B(u)|r1B) = E(Hr2B(u)|r2B) + E(u|A(r1,r2))≤ E(Hr1B(u)|r1B) + 2E(u|A(r1,r2))

or equivalentlyE(Hr2B(u)|r1B)− E(Hr1B(u)|r1B) ≤ 2E(u|A(r1,r2)) .

Since Hr1B(u)|r1B is minimizing and E(u|B) < εL, by Lemma 2.5.2 and using that Hr2B(u) equalsHr1B(u) outside r1B, we get

E (Hr2B(u)−Hr1B(u)) = E (Hr2B(u)−Hr1B(u)|r1B) ≤ 4E(u|A(r1,r2)) . (4.6.1)

Lemma 4.6.2. Let γ ∈ ΩM with γ(·, s)|B ∈ MB for all s ∈ [0, 1]. Then the scaled harmonicreplacement map

(s, r) 7→ HrB(γ(·, s))

is continuous in energy in the sense that, given ε > 0 and (s0, r0), there exists δ > 0 such thatif |s− s0|, |r − r0| < δ then

E(HrB(γ(·, s))−Hr0B(γ(·, s0))) < ε .

41


Proof. Fix (s0, r0) ∈ [0, 1]× [0, 1] and let ε > 0. Choose δ1 > 0 be such that

E(γ(·, s0)|A(r0,r)) < ε (4.6.2)

whenever |r − r0| < δ1. Then use continuity of the map s 7→ γ(·, s) to choose δ2 > 0 such that,for all s ∈ [0, 1] with |s − s0| < δ2, E(γ(·, s) − γ(·, s0))1/2 < ε/(2ε1/2H ). In particular, for such sand all r ∈ [0, 1] using the triangle inequality and the upper bound on the energy of maps inMB,∣∣E(γ(·, s)|A(r0,r))− E(γ(·, s0)|A(r0,r))

∣∣=∣∣∣E(γ(·, s)|A(r0,r))

1/2 + E(γ(·, s0)|A(r0,r))1/2∣∣∣ ∣∣∣E(γ(·, s)|A(r0,r))

1/2 − E(γ(·, s0)|A(r0,r))1/2∣∣∣

≤ 2ε1/2H E(γ(·, s)− γ(·, s0)|A(r0,r))1/2 < ε .

Hence, when |s− s0| < δ2 and |r − r0| < δ1 using the bound (4.6.2),

E(γ(·, s)|A(r0,r)) < 2ε ,

so that Lemma 4.6.1 asserts

E(Hr0B(γ(·, s))−HrB(γ(·, s))) ≤ 4E(γ(·, s)|A(r0,r)) < 8ε .

By continuity of s 7→ γ(·, s) and Corollary 4.5.2, we can choose δ3 > 0 such that when |s− s0| <δ3,

E(Hr0B(γ(·, s0))−Hr0B(γ(·, s))) < ε .

By the triangle inequality for energy, we conclude

E(Hr0B(γ(·, s0))−HrBγ(·, s)))≤ 2E(Hr0B(γ(·, s0))−Hr0Bγ(·, s)) + 2E(Hr0B(γ(·, s))−HrBγ(·, s))< 18ε

when |r− r0| < δ1 and |s− s0| < min(δ2, δ3). Since ε > 0 and (s0, r0) was arbitrary, this finishesthe proof.

Theorem 4.6.3 (Continuity of (s, r) 7→ HrB(γ(·, s))). Let γ ∈ ΩM with γ(·, s)|B ∈ MB for alls ∈ [0, 1]. Then the scaled harmonic replacement map

(s, r) 7→ HrB(γ(·, s))

is continuous.

Proof. By Lemma 4.6.2 it suffices to control the C0(cl (B) ,Rm)-norm of the map.Fix (s0, r0) ∈ [0, 1]× [0, 1] and let (sj , rj), sj , rj ∈ [0, 1] be a sequence converging to (s0, r0).

Let ε > 0 be given and define γ0 = Hr0B(γ(·, s0)) and γj = HrjB(γ(·, sj)). We will findJ ∈ N such that, for all j ≥ J , ‖γj − γ0‖C0(cl(B),Rm) < 2ε. We do this by rescaling themaps HrjB(γ(·, sj))|rjB to be defined on r0B. We will then show uniform convergence of therescaled maps to Hr0B(γ(·, s0)). In turn this will then imply the uniform estimate of the mapsHrjB(γ(·, sj)).

42

Continuity with scaling

Since the set S2 × [0, 1] is compact, the continuity of γ implies uniform continuity. Hencegiven ε > 0 we can choose δ1 > 0 and J ∈ N such that, for j ≥ J and any p, p ∈ cl (B) withd(p, p) < δ1,

‖γ(p, s0)− γ(p, sj)‖Rm < ε and ‖γ0(p)− γ0(p)‖Rm < ε . (4.6.3)

Using the identification of B with B ⊂ R2 and possibly shrinking δ1, we can assume the sameestimate holds for p, p ∈ cl (B), |p − p| < δ1. For sufficiently large j, |rj/r0p − p| < δ1 for allp ∈ cl (B). Hence we can enlarge J such that, for all j ≥ J and p ∈ cl (B),

‖γ(p, s0)− γ(rj/r0p, sj)‖Rm < ε . (4.6.4)

For each j we define the rescaled maps γj(·) = γj(rj/r0·)|r0B. Since γj are minimizing onrjB, and by conformality of dilation, γj are minimizing. Since γj on ∂r0B are equal to γ(·, sj)on ∂rjB, (4.6.4) implies that (γj) converges uniformly to γ0 on ∂r0B.

We will apply Lemma 4.5.3 shortly and hence we need to control the energy of γj near ∂r0B.Given ε > 0 and p ∈ ∂r0B we can choose r > 0 such that

E(γ0|Br(p)∩B) < ε/2 .

By Lemma 4.6.2, for sufficiently large j,

E(γj |Br(p)∩B) < ε .

Also, again for sufficiently large j, the set B(r/2)(rj/r0)(rj/r0p) ∩ B is contained in Br(p) ∩ B.Hence

E(γj |B(r/2)(rj/r0)

(rj/r0p)∩B) < ε .

But the rescaling sends B(r/2)(rj/r0)(rj/r0p) ∩ rjB to Br/2(p) ∩ r0B, and by conformality we get

E(γj |Br/2(p)∩r0B) < ε

for all but finitely many j. This property, together with γj being harmonic, since they areminimizing, and uniform convergence of (γj) on ∂r0B, allows us to use Lemma 4.5.3 to getC0(cl (r0B) ,Rm) convergence of (γj) to γ0.

Now, again enlarge J such that, for all j ≥ J , ‖γj−γ0|cl(r0B)‖C0(cl(r0B),Rm) < ε. We claim thatfor such j, ‖γj − γ0|max(rj ,r0)B‖C0(max(rj ,r0)B,Rm) < 2ε. Since γj equals γ0 outside max(rj , r0)B,this will finish the proof. For any p ∈ rjB let p = r0/r

j p. Using (4.6.3), we have

‖γj(p)− γ0(p)‖Rm = ‖γj(p)− γ0(p)‖Rm ≤ ‖γj(p)− γ0(p)‖Rm + ‖γ0(p)− γ0(p)‖Rm< 2ε .

If rj ≥ r0, the argument is finished. If not, for p ∈ A(r0, rj) let p ∈ ∂r0B be such that p = tp

for some t > 0. We have

‖γj(p)− γ0(p)‖Rm = ‖γ(p, sj)− γ0(p)‖Rm ≤ ‖γ(p, sj)− γ(p, s0)‖Rm + ‖γ(p, s0)− γ0(p)‖Rm

= ‖γ(p, sj)− γ(p, s0)‖Rm + ‖γ0(p)− γ0(p)‖Rm(4.6.3)< 2ε .

43


4.7 Continuity of H(·, B)

Given a finite collection of disjoint projected balls B = B1, . . . ,Bk, we will extend the harmonicreplacement maps to maps defined on all of S2. We define

MB =u ∈ C0 ∩W 1,2(S2,M) |E(u|B) ≤ εH/2

and equip MB with a norm ‖ · ‖B similar to the norm defined on MB, when B is a ball. Thereason for requiring the energy of all the balls combined to be low, and the specific choiceof upper bound, will be apparent when we do repeated replacement (confer Lemma 4.8.1 andProposition 4.8.5/4.8.6).

We let H(u,B)(x) :MB →MB ⊂ C0 ∩W 1,2(S2,M) be the map defined by

H(u,B)(x) =HBi(u)(x) , x ∈ Bi ∈ B ,u(x) , otherwise .

Corollary 4.7.1 (Continuity of H(·,B)). Let B be a finite collection of disjoint projected balls.The harmonic replacement map H(·,B) :MB →MB is continuous.

Proof. This follows from Corollary 4.5.2, Corollary 4.5.4, and that B is finite; let (uj), uj ∈MB

be a sequence converging to u ∈MB. We have

‖H(uj ,B)−H(u,B)‖B = ‖H(uj ,B)−H(u,B)‖C0(S2,Rm) +(2E(H(uj ,B)−H(u,B))

)1/2= maxB∈B‖HB(uj)−HB(u)‖C0(cl(B),Rm) +

(2∑B∈B

E(HB(uj)−HB(u))

)1/2

→ 0 .

Corollary 4.7.2 (Continuity of (s, r) 7→ H(γ(·, s), rB)). Let B be a finite collection of disjointprojected balls and γ ∈ ΩM such that γ(·, s) ∈ MB for all s ∈ [0, 1]. Then the scaled harmonicreplacement map (s, r) 7→ H(γ(·, s), rB) :MB →MB is continuous.

Proof. Follows from Theorem 4.6.3 and B being finite exactly as in Corollary 4.7.1.

For ease of notation, we will in the following let B be the set of all finite collections of disjointprojected balls.

4.8 Energy decrease

We will be interested in the decrease in energy of harmonic replacement measured by the non-negative scalar E(u)−E(H(u,B)). In particular, we need to investigate what happens when weperform consecutive energy replacement on two collections of balls B1 and B2.

The main results of this section are Proposition 4.8.5 and Proposition 4.8.6. We will beginby considering repeated harmonic replacement and define the energy decrease. We will thenstate the two propositions and sketch the strategy that we will use in the proofs. After a seriesof technical results, the proofs will follow.

Since harmonic replacement is only possible on balls with low energy, to be able to performrepeated replacement, we will need to consider how much the energy can increase on balls, whenperforming harmonic replacement on another set of balls.

44

Energy decrease

Lemma 4.8.1. Let B1,B2 ∈ B and u ∈MB1 ∩MB2. Then, for each B ∈ B2,

E(H(u,B1)|B) ≤ εH . (4.8.1)

Proof. Let B ∈ B2 and define U = B ∩(∪B∈B1

B)

and V = B \ U . Then, since U ⊂ ∪B∈B1B,

E(H(u,B1)|U ) ≤ E(H(u,B1)|∪B∈B1B) ≤ E(u|∪B∈B1

B) ≤ εH/2

and, since V ∩(∪B∈B1

B)

= ∅,

E(H(u,B1)|V ) = E(u|V ) ≤ E(u|B) ≤ εH/2 .

Thus,E(H(u,B1)|B) = E(H(u,B1)|V ) + E(H(u,B1)|U ) ≤ εH .

As a consequence of the above lemma, for any u ∈ MB1 ∩MB2 , the map u 7→ H(u,B1,B2)is defined, and it is continuous by the results of the previous section.

Definition 4.8.2 (Energy decrease). Given a map u ∈ MB and a projected ball B, we definethe energy decrease Eδ(u,B) by

Eδ(u,B) = E(u)− E(H(u,B)) .

If B2 is another projected ball, we define

Eδ(u,B,B2) = E(u)− E(H(u,B,B2)) .

Similarly, if B and B2 are finite collections of disjoint projected balls, we define

Eδ(u,B) = E(u)− E(H(u,B))

andEδ(u,B,B2) = E(u)− E(H(u,B,B2)) .

Lemma 4.8.3. The energy decrease Eδ is non-negative and positive if u is not harmonic. Con-versely, if u is harmonic then Eδ(u, B) = 0 for all B ⊆ B.

Proof. Since harmonic replacement minimizes energy, clearly Eδ is non-negative. If Eδ(u,B) = 0then by Corollary 2.5.3, u|B = H(u,B) showing that u is harmonic. If u is harmonic thenEδ(u, B) = 0 as u is assumed to have low energy and thus is already minimal by Corollary 2.5.3.

The following lemma collects two additional minor facts about Eδ.

Lemma 4.8.4. Let B ∈ B and u ∈MB. Then

Eδ(u,B) = Eδ(u|∪B∈BB,B) =∑B∈B

Eδ(u,B) . (4.8.2)

If, in addition, also B2 ∈ B then

Eδ(u,B1,B2) = Eδ(u,B1) + Eδ(H(u,B1),B2) . (4.8.3)

45


Proof. Since H(u,B) equals u outside B, we have

Eδ(u,B) = E(u|B)− E(H(u,B)|B)= E(u|∪B∈BB)− E(H(u|∪B∈BB,B))= Eδ(u|∪B∈BB,B)

proving (4.8.2). Equation (4.8.3) follows directly by

Eδ(u,B1,B2) = E(u)− E(H(u,B1,B2))= E(u)− E(H(u,B1)) + E(H(u,B1))− E(H(H(u,B1),B2))= Eδ(u,B1) + Eδ(H(u,B1),B2) .

From (4.8.2) we immediately get that

Eδ(u,B) = Eδ(u|∪B∈BB,B) ≤ E(u|B) (4.8.4)

and from (4.8.3) that

Eδ(u,B1) = Eδ(u,B1,B2)− Eδ(H(u,B1),B2) ≤ Eδ(u,B1,B2) . (4.8.5)

Proposition 4.8.5. There exists a constant κ1(M, g) > 0 such that, when B1,B2 ∈ B havingno balls B1 ∈ B1, B2 ∈ B2 with B1 ∪ B2 = S2, and u ∈MB1 ∩MB2, we have the following lowerbound on the energy decrease with repeated replacement:

Eδ(u,B1,B2) ≥ κ1

(Eδ(u,

12

B2))2

. (4.8.6)

Proposition 4.8.6. There exists a constant κ2(M, g) > 0 such that, when B1,B2 ∈ B havingno balls B1 ∈ B1, B2 ∈ B2 with B1 ∪B2 = S2, and u ∈MB1 ∩MB2, for any µ ∈ (0, 1/2] we havethe following upper bound on the energy decrease with repeated harmonic replacement:

Eδ(H(u,B1), µB2) ≤ κ2Eδ(u,B1)1/2 + Eδ(u, 2µB2) . (4.8.7)

Remark 4.8.7. We note that if u ∈ MB1 ∩MB2 and E(u) ≥ εmin/2 then the condition thatno two balls B1 ∈ B1, B2 ∈ B2 with B1 ∪B2 = S2 exist is automatically true. The reason for thisis that such a pair would satisfy

εmin/2 ≤ E(u) ≤ E(u|B1) + E(u|B2) ≤ εH < εmin/2

by the choice of εH .

The proofs of both propositions will follow the same overall strategy; we start by noting thatwe may assume an upper bound on the energy E(u − H(u,B1)). This bound will be denotedεcc. We will then split B2 into two collections, B+

2 and B−2 , and discuss what happens whenperforming repeated harmonic replacement on each family. The balls in the former set will ina certain way be included in the balls of B1 making it easy to asses the energy decrease. Thelatter balls will be disjoint from the balls of B1 in such a way that u and H(u,B1) coincides atpoints on the boundaries of balls of lesser radii. To get this, we need some control of the size ofthe balls, thus requiring that no two balls make up all of S2. Using that u and H(u,B1) coincide,a carefull analysis will allow us to asses the energy decrease in this case too.

We begin by discussing the relation between the size of balls and when u and H(u,B1)coincide.

46

Energy decrease

Lemma 4.8.8. Let B1 and B2 be projected balls. Suppose µ ∈ (0, 1] and µB1 6⊂ B2. ThenB1 ∪ B2 = S2 or, for any r ∈ [µ, 1], ∂rB1 6⊂ B2.

Proof. Let B1 = (U,Λ) and let B1 = Λ(U) = Λ(B1) ⊂ R2 and B2 = Λ(B2). Let the radiusof B1 be R > 0 and the center c. It now suffices to prove that B1 ∪ B2 = R2 or, for allr ∈ [µR,R], ∂Br(c) 6⊂ B2. So suppose there exists r ∈ [µR,R] with ∂Br(c) ⊂ B2. Recall thatby Lemma 2.1.4, B2 is either a ball, the complement of a ball, or a half space. If B2 is a ball ora half space then by convexity, ∂BµR(c) ⊂ B2. Since this was supposed not to be the case, B2

must be the complement of a ball. Since, by assumption, ∂Br(c)∩(R2 \B2) = ∅, the ball R2 \B2

must be either contained in, or disjoint from, Br(c). Since ∂BµR(c) 6⊂ B2, ∂BµR(c) intersectsR2 \B2. Thus R2 \B2 ⊂ Br(c) so that B1 ∪B2 = R2.

Given B1 and B2 with the property of Lemma 4.8.8, we wish to find r such that ∂rB1 6⊂ B2.Thus we require that B1 ∪ B2 6= S2.

Corollary 4.8.9. Let B ∈ B and B be a projected ball such that for no B ∈ B, B ∪ B = S2. Ifµ ∈ (0, 1] and µB 6⊂ B for any B ∈ B then, for any r ∈ [µ, 1], ∂rB 6⊂ ∪B∈BB.

Proof. Let r ∈ [µ, 1]. As remarked above, the assumptions and Lemma 4.8.8 imply that ∂rB 6⊂ Bfor all B ∈ B. Define UB = ∂rB ∩ B. Then UB 6= ∂rB and UB is a closed subset of ∂rB. If wesuppose ∂rB ⊂ ∪B∈BB then ∪B∈BUB = ∂rB. Since the sets UB are disjoint, this contradicts that∂rB is connected.

Harmonic replacement fixes a map outside the balls on which the replacement takes place.Therefore, as a result of the corollary and with the assumptions on the balls, for any r ∈ [µ, 1],u and H(u,B) will coincide at a point of ∂rB.

We now turn to results concerning maps coinciding at a point in the boundary of a ball andthe energy on such a boundary. Since the energy is conformally invariant only in two dimensions,we must be careful when using the identification between B and B ⊂ R2. Therefore, some resultswill concern only balls B ⊂ R2.

We will work our way to constructing a comparison map which is used in Lemma 4.8.13 toget an energy estimate on a minimizing map. The derived Corollary 4.8.15 will be the centralkey to proving the two main propositions on the section. We start by proving interior estimateson maps defined on the boundary of a ball.

Lemma 4.8.10. Let B ⊂ R2 be a ball of radius R > 0, suppose u, v ∈ C0 ∩W 1,2(∂B,M), andu and v coincide at a point p ∈ ∂B. Then

maxy∈∂B

‖u(y)− v(y)‖2 ≤ 4πRE(u− v) (4.8.8)

and ∫∂B‖u− v‖2 ≤ π2R2E(u− v) . (4.8.9)

Proof. Write p = (R, θp) in polar coordinates centered at the center of B. Similarly, for anyy ∈ ∂B, let y = (R, θy) with θy ∈ [θp, θp + 2π). Use the Fundamental Theorem of Calculus

47


(Theorem 2.2.5) and that u(p) = v(p) to get

‖(u− v)(y)‖Rm = ‖(u− v)(y)− (u− v)(p)‖Rm

=

∥∥∥∥∥∫ θy

θp

d

dθ(u(R, θ)− v(R, θ)) dθ

∥∥∥∥∥Rm

≤∫ 2π

0

∥∥∥∥ ddθ (u(R, θ)− v(R, θ))∥∥∥∥

Rmdθ .

(4.8.10)

Since the metric in the chart θ 7→ (R, θ) has the single coefficient gθθ = R2, we have∫ 2π

0

∥∥∥∥ ddθ (u(R, θ)− v(R, θ))∥∥∥∥2

Rmdθ =

∫ 2π

0R2gθθ

∥∥∥∥ ddθ (u(R, θ)− v(R, θ))∥∥∥∥2

Rmdθ

=∫ 2π

0R2 ‖d(u(R, θ)− v(R, θ))‖2 dθ

=∫∂BR ‖d(u(R, θ)− v(R, θ)‖2

= 2RE(u− v) .

(4.8.11)

Then Cauchy-Schwartz implies

‖(u− v)(y)‖(4.8.10)

≤∫ 2π

0

∥∥∥∥ ddθ (u(R, θ)− v(R, θ))∥∥∥∥

Rmdθ

Cauchy-Schwartz≤

√2π

(∫ 2π

0

∥∥∥∥ ddθ (u(R, θ)− v(R, θ))∥∥∥∥2

Rmdθ

)1/2

(4.8.11)= 2

√π√RE(u− v)

implying (4.8.8).The inequality (4.8.8) is the result of Corollary 2.3.2.

The map constructed in the following lemma will be pasted between a map defined on anannulus and a map defined on a ball. We are interested in doing this while keeping control of theenergy of the inserted piece. Recall that δ > 0 is chosen so that Mδ is a tubular neighborhoodof M with retraction r : Mδ →M satisfying ‖dxr‖∞ ≤

√2 for each x ∈Mδ, confer (4.4.3).

Lemma 4.8.11. There exists C > 0 such that if B ⊂ R2 is a ball of radius R > 0 with centerc ∈ R2, u, v ∈ C0 ∩W 1,2(∂B,M) are non-equal with maxx∈∂B ‖u(x) − v(x)‖ < δ and coincideat a point p ∈ ∂B then there exist ρ ∈ (0, R/2] and w ∈ C0 ∩W 1,2(B \ cl (BR−ρ(c)) ,M) suchthat, for all θ ∈ [0, 2π) and using polar coordinates,

w(R, θ) = v(R, θ) , (4.8.12)w(R− ρ, θ) = u(R, θ) , (4.8.13)

andE(w) ≤ CR ((E(u) + E(v))E(u− v))1/2 . (4.8.14)

48

Energy decrease

Proof. Define

ρ =(

E(u− v)8(E(u) + E(v))

)1/2

(4.8.15)

and let ρ = Rρ. Note that, by the triangle inequality for energy (Corollary 2.4.3),

ρ ≤(

2E(u) + 2E(v)8(E(u) + E(v))

)1/2

=(

14

)1/2

=12

so that ρ ≤ R/2. In addition, ρ > 0; if not, E(u−v) = 0 so that (4.8.8) in Lemma 4.8.10 impliesthat u = v.

Let w : cl (B) \BR−ρ(c)→Mδ be the map defined by

w(r, θ) = u(R, θ) +r + ρ−R

ρ(v(R, θ)− u(R, θ)) .

The map is well-defined, because the image of w is in Mδ since u(R, θ) ∈M , |r + ρ−R|/ρ ≤ 1for r ∈ [R − ρ,R], and maxx∈∂B ‖u(x) − v(x)‖ < δ. Let now w = r w. Since r|M = IdM itis clear that (4.8.12) and (4.8.13) are satisfied. Since w is continuous and its domain bounded,w ∈ C0 ∩W 0,2(B \ cl (BR−ρ(c)) ,M). Thus we only need to obtain (4.8.14) to prove the lemma.Since

E(w) =12

∫B\cl(BR−ρ(c))

‖d(r w)‖2(2.2.2)

≤ 12

∫B\cl(BR−ρ(c))

‖dr‖2∞‖dw‖2

(4.4.3)

≤ 12

∫B\cl(BR−ρ(c))

2‖dw‖2 = 2E(w) ,(4.8.16)

we only need to consider the energy of w.By definition

E(w) =12

∫B\cl(BR−ρ(c))

‖dw‖2

=12

∫ R

R−ρ

∫ 2π

0

(grr⟨d

drw(r, θ),

d

drw(r, θ)

⟩+ gθθ

⟨d

dθw(r, θ),

d

dθw(r, θ)

⟩)rdθdr

=12

∫ R

R−ρ

∫ 2π

0

(⟨d

drw(r, θ),

d

drw(r, θ)

⟩+

1r2

⟨d

dθw(r, θ),

d

dθw(r, θ)

⟩)dθrdr .

(4.8.17)

Note thatd

drw(r, θ) =

1ρ

(v(R, θ)− u(R, θ)) ,

d

dθw(r, θ) =

d

dθu(R, θ) +

r + ρ−Rρ

(d

dθv(R, θ)− d

dθu(R, θ))

=−r +R

ρ

d

dθu(R, θ) +

r + ρ−Rρ

d

dθv(R, θ) .

(4.8.18)

From the convexity of energy and (4.8.18), we get

12

∫ 2π

0

1R2

⟨d

dθw(r, θ),

d

dθw(r, θ)

⟩Rdθ = E(

−r +R

ρu+

r + ρ−Rρ

v)

Lemma 2.4.2≤ −r +R

ρE(u) +

r + ρ−Rρ

E(v)

≤ E(u) + E(v) .

(4.8.19)

49


By (4.8.9) of Lemma 4.8.10,∫ 2π

0〈v(R, θ)− u(R, θ), v(R, θ)− u(R, θ)〉Rdθ ≤ π2R2E(u− v) . (4.8.20)

Thus

E(w)(4.8.17)

≤ 12

∫ R

R−ρ

∫ 2π

0

(⟨d

drw(r, θ),

d

drw(r, θ)

⟩+

1r2

⟨d

dθw(r, θ),

d

dθw(r, θ)

⟩)dθrdr

(4.8.19)

(4.8.20)

≤ 12

∫ R

R−ρ

(π2R

ρ2E(u− v) +

2Rr2

(E(u) + E(v)))rdr .

(4.8.21)

Observe that ∫ R

R−ρrdr =

12

(R2 − (R− ρ)2) =12

(2Rρ− ρ2) ≤ Rρ (4.8.22)

and ∫ R

R−ρ

1rdr = ln(R)− ln(R− ρ) = ln(R)− ln(R)− ln(1− ρ) = − ln(1− ρ) ≤ 2ρ (4.8.23)

since ρ ≤ 1/2.2 Thus, by integrating, we get

E(w)(4.8.21)

≤ 12

(π2R

ρ2E(u− v)

∫ R

R−ρrdr + 2R(E(u) + E(v))

∫ R

R−ρ

1rdr

)(4.8.22)

(4.8.23)

≤ 12

(π2R2

ρE(u− v) + 4Rρ(E(u) + E(v))

)=R

2

(π2

ρE(u− v) + 4ρ(E(u) + E(v))

).

(4.8.24)

By the choice of ρ,

E(w)(4.8.24)

≤ R

2

(π2

ρE(u− v) + 4ρ(E(u) + E(v))

)(4.8.15)

=R

2

(π281/2 ((E(u) + E(v))E(u− v))1/2 + 4

181/2

((E(u) + E(v))E(u− v))1/2

)=

21/2(2π2 + 1)R2

((E(u) + E(v))E(u− v))1/2 .

(4.8.25)

Then finally we obtain

E(w)(4.8.16)

≤ 2E(w)(4.8.25)

≤ 21/2(2π2 + 1)R ((E(u) + E(v))E(u− v))1/2 .

2 To see that − ln(1− x) ≤ 2x for x ∈ [0, 1/2] note that d/dx2x + ln(1− x) = 2 + 1/(1− x) ≥ 4 so that themap x 7→ 2x + ln(1− x) is monotonely increasing on [0, 1/2]. Since 2 · 0 + ln(1− 0) = 0 the map is non-negativeon [0, 1/2] which gives the result.

50

Energy decrease

We now relate the energy of maps defined on balls, to the energy of the maps restricted tothe boundary of possibly smaller balls.

Lemma 4.8.12. Let B ⊂ R2 be a ball of radius R and center c, and suppose u, v ∈ C0 ∩W 1,2(B,M). Then there exists r ∈ [3R/4, R) such that u|∂Br(c), v|∂Br(c) ∈ C0∩W 1,2(∂Br(c),M),

E(u|∂Br(c) − v|∂Br(c)) ≤9rE(u− v) (4.8.26)

andE(u|∂Br(c) + E(v|∂Br(c)) ≤

9r

(E(u) + E(v)) . (4.8.27)

Proof. It is the result of Lemma 2.2.4 that, for almost all r ∈ (0, R), u|∂Br(c), v|∂Br(c) ∈ C0 ∩W 1,2(∂Br(c),M). Let U ⊂ (0, R) be the set of those r’s.

Suppose for contradiction that, for all r ∈ [3R/4, R) ∩ U , one or both of (4.8.26) or (4.8.27)fail. Suppose that the set of r’s, such that (4.8.26) fails, is the one of largest measure; theargument is equivalent if not. Let V be that set so that, for all r ∈ V ,

12

∫∂Br(c)

‖du− dv‖2 > 9r

12

∫B‖du− dv‖2 . (4.8.28)

Note that, by assumption, the measure of V is no less that (R − 3/4R)/2 = R/8. Let χV bethe characteristic function of V . Then in polar coordinates centered at B we, for r ∈ V , inparticular have∫ 2π

0‖du(r, θ)− dv(r, θ)‖2rdθ

(4.8.28)>

9r

∫ R

0

∫ 2π

0χV (s)‖du(s, θ)− dv(s, θ)‖2sdθds . (4.8.29)

On the contrary,

9r

∫ R

0

∫ 2π

0χV (s)‖du(s, θ)− dv(s, θ)‖2sdθds

≥ 9r

(infs∈V

∫ 2π

0‖du(s, θ)− dv(s, θ)‖2sdθ

)∫ R

0χV (s)ds

≥ 9r

R

8infs∈V

∫ 2π


≥ 98

infs∈V

∫ 2π

0‖du(s, θ)− dv(s, θ)‖2sdθ .

(4.8.30)

Thus, for all r ∈ V ,

∫ 2π

0‖du(r, θ)− dv(r, θ)‖2rdθ

(4.8.29)

(4.8.30)>

98

infs∈V

∫ 2π


which is possible only if the infimum is zero. By (4.8.28), we get E(u − v) = 0, but then12

∫∂Br(c)

‖du− dv‖2 = 0 for almost all r ∈ (0, R) contradicting (4.8.28).

By cutting and pasting we now construct a comparison map estimating the energy of acertain minimizing map.

51


Lemma 4.8.13. There exist constants C, εcc(M, g) > 0 such that, if B is a projected ball,u, v ∈MB, for any r ∈ [3/4, 1), u|∂rB and v|∂rB coincide at a point, and

E(u− v) ≤ εcc (4.8.31)

then there exists r ∈ [3/4, 1) with

E(H(v,B)) ≤ E(v|B\rB) + E(H(u, rB)|rB)

+ C ((E(u) + E(v))E(u− v))1/2 .(4.8.32)

Proof. Let vm = H(v,B). Since vm is energy minimizing, it suffices to construct a comparisonmap vm ∈ C0 ∩W 1,2(B,M) with vm|∂B = vm|∂B and

E(vm) ≤ E(v|B\rB) + E(H(u, rB)|rB)

+ C ((E(u) + E(v))E(u− v))1/2 .(4.8.33)

We will piece vm together using two or three maps. Those will be v, a map constructed hereusing harmonic replacement, and, in the cases of using three maps, also an intermediate mapobtained from Lemma 4.8.11.

Let U and Λ be such that B = (U,Λ) and let B ⊂ R2 be the ball Λ(U). Let c ∈ R2 be thecenter of B and R > 0 the radius. We carry out the rest of the argument using B. Recall thatδ > 0 is the constant defining the tubular neighborhood Mδ with retraction satisfying (4.4.3).Let εcc ∈ (0, δ2/(36π)) so that we by (4.8.31) assume

E(u− v) <δ2

36π. (4.8.34)

Use Lemma 4.8.12 to pick r ∈ [3R/4, R) such that

E(u|∂Br(c)) + E(v|∂Br(c)) ≤9r

(E(u) + E(v)) (4.8.35)

and

E(u− v|∂Br(c)) ≤9rE(u− v)

(4.8.34)<

δ2

4πr. (4.8.36)

If u|∂Br(c) and v|∂Br(c) are equal, we can immediately glue the maps v|B\Br(c) andH(u,Br(c))|Br(c)together; using Lemma 2.2.3, we define

vm(t, θ) =v(t, θ) , t ∈ [r,R]H(u,Br(c))(t, θ) , t ∈ [0, r) .

We then directly get

E(vm) = E(v|B\Br(c)) + E(H(u,Br(c))|Br(c))

proving (4.8.33) and (4.8.32) in this case.If u|∂Br(c) and v|∂Br(c) are non-equal, we will need to introduce an intermediate map w

in addition to the map constructed using harmonic replacement. By (4.8.36) and using thatu|∂Br(c) and v|∂Br(c) coincide at a point, Lemma 4.8.10 implies

maxx∈∂Br(c)

‖u(x)− v(x)‖ ≤ (4πrE(u− v|∂Br(c)))1/2

(4.8.36)< δ . (4.8.37)

52

Energy decrease

Again using that u|∂Br(c) and v|∂Br(c) coincide at a point, Lemma 4.8.11 gives ρ ∈ (0, r/2] anda map w ∈ C0 ∩W 1,2(Br(c) \ cl (Br−ρ(c)) ,M) such that

w(r, θ) = v(r, θ) ,w(r − ρ, θ) = u(r, θ) ,

andE(w) ≤ C2r

((E(u|∂Br(c)) + E(v|∂Br(c)))E(u− v|∂Br(c))

)1/2 (4.8.38)

for a C2 > 0. Combining (4.8.38) with (4.8.35) and (4.8.36), we get C > 0 such that

E(w) ≤ C ((E(u) + E(v))E(u− v))1/2 . (4.8.39)

Let Dr : R2 → R2 be the conformal map

Dr(t, θ) =(

rt

r − ρ, θ

)in polar coordinates centered at c. The map is just a dilation centered at c. Then let h ∈C0 ∩W 1,2(Br−ρ(c),M) be the composition

H(u,Br(c)) Dr|Br−ρ(c) .

That is, h is obtained by performing harmonic replacement on Br(c) and then rescaling to letthe domain be Br−ρ(c). Note that, for all θ ∈ [0, 2π),

h(r − ρ, θ) = H(u,Br(c))(r, θ) = u(r, θ) = w(r − ρ, θ) ,

and since Dr is conformal,E(h) = E(H(u,Br(c))|Br(c)) . (4.8.40)

We are now able to construct vm by defining

vm(t, θ) =

v(t, θ) , t ∈ [r,R]w(t, θ) , t ∈ [r − ρ, r)h(t, θ) , t ∈ [0, r − ρ)

By Lemma 2.2.3, vm ∈ C0 ∩W 1,2(B,M). Calculating the energy of vm, we get

E(vm) = E(v|B\Br(c)) + E(w) + E(h)(4.8.40)

(4.8.39)

≤ E(v|B\Br(c)) + E(H(u,Br(c))|Br(c))

+C ((E(u) + E(v))E(u− v))1/2

proving (4.8.33) and hence (4.8.32).

We will need a bound on the energy difference between two maps.

Lemma 4.8.14. Let N be a manifold and u, v ∈W 1,2(N,Rm). Then∣∣E(u)− E(v)∣∣ ≤ ((2(E(u) + E(v)))E(u− v)

)1/2.

53


Proof. By the triangle inequality,∣∣‖du‖2 − ‖dv‖2∣∣ =∣∣‖du‖+ ‖dv‖‖du‖ − ‖dv‖

∣∣≤∣∣‖du‖+ ‖dv‖

∣∣‖du− dv‖ (4.8.41)

and thus ∣∣E(u)− E(v)∣∣ =

∣∣∣∣∫N

12

(‖du‖2 − ‖dv‖2)∣∣∣∣ ≤ ∫

N

12

∣∣‖du‖2 − ‖dv‖2∣∣(4.8.41)

≤∫N

12

(‖du‖+ ‖dv‖) ‖du− dv‖ .(4.8.42)

By Cauchy-Schwartz and the absorbing inequality (Lemma C.1.4), we then get

∣∣E(u)− E(v)∣∣ C-S

(4.8.42)

≤(∫

N

12

(‖du‖+ ‖dv‖)2

)1/2(∫N

12‖du− dv‖2

)1/2

Lem. C.1.4≤

((2(E(u) + E(v)))E(u− v)

)1/2.

Finally, we are ready to prove the key result used in the proof of Proposition 4.8.5 andProposition 4.8.6.

Corollary 4.8.15. Let εcc, C > 0 be the constants of Lemma 4.8.13. If B is a projected ball,u, v ∈MB, µ ∈ (0, 1/2], for any r ∈ [3/2µ, 2µ], u|∂rB and v|∂rB coincide at a point, and

E(u− v) ≤ εcc (4.8.43)

thenEδ(v, 2µB) ≥ Eδ(u, µB)− (C + 21/2) ((E(u) + E(v))E(u− v))1/2 . (4.8.44)

Proof. We apply Lemma 4.8.13 with u|2µB and v|2µB to get r ∈ [3/2µ, 2µ) with

E(H(v, 2µB)|2µB) ≤

E(v|2µB\rB) + E(H(u, rB)|rB) + C ((E(u) + E(v))E(u− v))1/2 .(4.8.45)

By Lemma 4.8.14, ∣∣E(u|rB)− E(v|rB)∣∣ ≤ (2(E(u) + E(v))E(u− v)

)1/2. (4.8.46)

Since r ≥ µ,

E(H(u, rB)|rB) ≤ E(H(u, µB)|rB) = E(u|rB\µB) + E(H(u, µB)|µB) . (4.8.47)

Combining (4.8.45), (4.8.46), and (4.8.47), we get

E(v)− E(H(v, 2µB))(4.8.45)

≥ E(v|rB)− E(H(u, rB)|rB)− C (E(u) + E(v)E(u− v))1/2

(4.8.46)

≥ E(u|rB)− E(H(u, rB)|rB)− (C + 21/2) (E(u) + E(v)E(u− v))1/2

(4.8.47)

≥ E(u|µB)− E(H(u, µB)|µB)− (C + 21/2) (E(u) + E(v)E(u− v))1/2 .

(4.8.48)

54

Energy decrease

If B ∈ B, u, v ∈MB with E(u−v) ≤ εcc, µ ∈ (0, 1/2], and u and v restricted to each ball of thecollection coincide at a point as in Corollary 4.8.15 then the corollary and the Cauchy-Schwartzinequality imply

Eδ(v, 2µB) =∑B∈B

Eδ(v, 2µB)

Cor. 4.8.15≥

∑B∈B

(Eδ(u, µB)− (C + 21/2) ((E(u|B) + E(v|B))E(u− v|B))1/2)

C-S≥ Eδ(u, µB)− (C + 21/2)

(∑B∈B

(E(u|B) + E(v|B))∑B∈B

E(u− v|B)

)1/2

≥ Eδ(u, µB)− (C + 21/2) ((E(u|B) + E(v|B))E(u− v|B))1/2 .

(4.8.49)

Proof of Proposition 4.8.5. Let εcc be the constant of Lemma 4.8.13. Suppose

E(u−H(u,B1)) > εcc . (4.8.50)

By (4.8.4) and since u ∈MB2 ,

Eδ(u,12

B2)) ≤ E(u| 12B2

) ≤ εH/2 ≤ 1/2 . (4.8.51)

Thus, using Lemma 4.8.4 and applying Lemma 2.5.2 to each of the balls in B1,

εccε2H

(Eδ(u,

12

B2))2 (4.8.51)

<εcc2

(4.8.50)<

12E(u−H(u,B1)) =

∑B∈B1

12E(u−H(u,B)|B)

Lemma 2.5.2≤

∑B∈B1

E(u|B)− E(H(u,B)|B) Lemma 4.8.4= Eδ(u,B1) .(4.8.52)

Applying harmonic replacement on B2 to H(u,B1) then gives

εccε2H

(Eδ(u,

12

B2)))2 (4.8.52)

≤ Eδ(u,B1)(4.8.5)

≤ Eδ(u,B1,B2) .

Thus, by choosing κ1 ≤ εcc/(ε2H) we get (4.8.6) in this case.Now, suppose

E(u−H(u,B1)) ≤ εcc . (4.8.53)

We will divide B2 into two disjoint collections B+2 and B−2 defined by letting

B+2 =

B ∈ B2 |

12B ⊂ B for some B ∈ B1

and B−2 = B2 \ B+

2 . If we find constants κ+1 and κ−1 such that

Eδ(u,B1,B+2 ) ≥ κ+

1

(Eδ(u,

12

B+2 ))2

, (4.8.54)

Eδ(u,B1,B−2 ) ≥ κ−1

(Eδ(u,

12

B−2 ))2

(4.8.55)

55


then from the equalities

Eδ(u,B1,B2)) = Eδ(u,B1,B+2 ) + Eδ(u,B1,B

−2 ) ,

Eδ(u,12

B2)) = Eδ(u,12

B+2 ) + Eδ(u,

12

B−2 )

we get

Eδ(u,B1,B2))

(4.8.55)

(4.8.54)

≥ κ+1

(Eδ(u,

12

B+2 ))2

+ κ−1

(Eδ(u,

12

B−2 ))2

≥ min(κ+1 , κ

−1 )

((Eδ(u,

12

B+2 ))2

+(Eδ(u,

12

B−2 ))2)

Lem. C.1.4≥ min(κ+

1 , κ−1 )

2

(Eδ(u,

12

B+2 ) + Eδ(u,

12

B−2 ))2

.

Thus (4.8.6) follows with κ1 = min(κ+1 /2, κ

−1 /2, εcc/ε

2H) if we establish (4.8.55) and (4.8.54).

The reason for treating the two collections separately is that the energy decrease obtained byperforming harmonic replacement on the balls in 1

2B+2 is easily controlled since they are con-

tained in B1, and, as we will see, we can apply Lemma 4.8.13 to the balls in B−2 and therebycontrol the energy decrease on B−2 .

Considering B+2 . Since each ball in 1

2B2 is contained in a ball in B1, harmonic replacementon the balls in 1

2B2 fixes u outside B1. Thus

E(H(u,12

B+2 ))− E(H(u,B1)) =

∑B∈B1

E(H(u,12

B+2 )|B)− E(H(u,B1)|B) .

Since harmonic replacement minimizes energy, each summand is positive. Thus

E(H(u,12

B+2 ))− E(H(u,B1)) ≥ 0 (4.8.56)

which, again using that harmonic replacement on B2 minimizes energy, implies

E(H(u,B1,B2)) ≤ E(H(u,B1))(4.8.56)

≤ E(H(u,12

B+2 )) . (4.8.57)

Thus

Eδ(u,B1,B2) = E(u)− E(H(u,B1,B2)(4.8.57)

≥ E(u)− E(H(u,12

B+2 )) = Eδ(u,

12

B+2 )) .

From the upper bound (4.8.51), we get

Eδ(u,B1,B2) ≥(Eδ(u,

12

B+2 ))2

so that (4.8.54) holds with κ+1 = 1.

Turning to B−2 , for each B ∈ B−2 , by Corollary 4.8.9 and the definition of B−2 , for all r ∈[1/2, 1], ∂rB 6⊂ ∪B1∈B1B1 and hence u and H(u,B1) agree at a point of ∂rB. Together with the

56

Energy decrease

bound (4.8.53), this enables us to apply Corollary 4.8.15 on each B ∈ B with v = H(u,B1)|Band µ = 1/2. From (4.8.49) with B = B−2 we get

Eδ(u,B1,B−2 ) ≥ Eδ(u,

12

B−2 )− (C + 21/2)(

(E(u|B−2 ) + E(H(u,B1)|B−2 ))E(u−H(u,B1)))1/2

.

(4.8.58)Note that

E(u−H(u,B1))Lem. 2.5.2≤ 2Eδ(u,B1)

(4.8.5)

≤ 2Eδ(u,B1,B−2 ) . (4.8.59)

Using that

Eδ(u,B1,B−2 ) Lem. 4.8.4= Eδ(u|B1∪B−2

,B1,B−2 )

(4.8.4)

≤ E(u|B1∪B−2)

≤ E(u|B1) + E(u|B−2 ) ≤ 1(4.8.60)

since u ∈MB1 ∩MB2 , we get

Eδ(u,12B−2 ))

Lem. 4.8.1(4.8.58)

≤ Eδ(u,B1,B−2 ) + (C +

√2)√

2εH (E(u−H(u,B1)))1/2

(4.8.59)

≤ Eδ(u,B1,B−2 ) + (C +

√2)√

2εH(2Eδ(u,B1,B

−2 ))1/2

(4.8.60)

≤ (1 + 2√εH(C +

√2))(Eδ(u,B1,B

−2 ))1/2

.

Thus (4.8.55) holds with

κ−1 =1

(1 + 2√εH(C +

√2))2

.

This concludes the proof.

Proof of Proposition 4.8.6. The proof will follow the same overall structure as the proof ofProposition 4.8.5. Let µ ∈ (0, 1/2] and let εcc be the constant of Lemma 4.8.13. Suppose

E(u−H(u,B1)) > εcc . (4.8.61)

Since u ∈ MB1 ∩MB2 , Eδ(H(u,B1), µB2)) ≤ εH . Using this and appying Lemma 2.5.2 to eachof the balls in B1, we get

εcc2ε2H

(Eδ(H(u,B1), µB2))2 ≤ εcc2

(4.8.61)<

12E(u−H(u,B1)) ≤ Eδ(u,B1) . (4.8.62)

Thus, choosing κ2 ≥ εH(2/εcc)1/2, we get (4.8.7) in this case.Now, suppose

E(u−H(u,B1)) ≤ εcc . (4.8.63)

We will again divide B2 into two disjoint collections B+2 and B−2 defined by letting

B+2 =

B ∈ B2 |µB ⊂ B for some B ∈ B1

and B−2 = B2 \ B+

2 . We will show that

Eδ(H(u,B1), µB+2 ) = 0 (4.8.64)

57


and find a constant κ−1 independent of µ such that

Eδ(H(u,B1), µB−2 ) ≤ κ−2 Eδ(u,B1)1/2 + Eδ(u, 2µB−2 ) . (4.8.65)

Then

Eδ(H(u,B1), µB2)(4.8.64)

= Eδ(H(u,B1), µB−2 )(4.8.65)

≤ κ−2 Eδ(u,B1)1/2 + Eδ(u, 2µB2) .

Thus (4.8.7) will follow with κ2 = max(κ−2 , εH(2/εcc)1/2).

By Lemma 4.8.3, Eδ(H(u,B1), µB+2 ) = 0 since each of the balls µB+

2 are contained in a ballin B1. Thus (4.8.64) holds.

Considering B−2 . Again, for all r ∈ [µ, 2µ], u and H(u,B1) agree at some point of ∂rB foreach B ∈ B−2 . Thus we can apply Corollary 4.8.15 and from (4.8.49) get

Eδ(u, 2µB−2 )

≥ Eδ(H(u,B1), µB−2 )− (C + 21/2)(

(E(H(u,B1)|B−2 ) + E(u|B−2 ))E(H(u,B1)− u))1/2

.

SinceE(u−H(u,B1))

Lem. 2.5.2≤ 2Eδ(u,B1) ,

we get

Eδ(H(u,B1), µB−2 ) ≤ (C + 21/2) (4εHEδ(u,B1))1/2 + Eδ(u, 2µB−2 ) .

Thus (4.8.65) holds with κ−2 = (C + 21/2)(4εH)1/2.

4.9 Construction of the map

We are now finally ready to construct a sweepout tightening map and thus prove Theorem 4.1.2.We first define a way to measure the maximal energy improvement by harmonic replacement.We will use this when deciding on which balls to do harmonic replacement. This will be done inLemma 4.9.5 below after having proved a result concerning the variation of the maximal energyimprovement. The proof of Theorem 4.1.2 will then follow.

Definition 4.9.1 (Maximal energy improvement). Given a sweepout γ ∈ ΩM , s ∈ [0, 1] andε ∈ (0, εH ] we define the maximal energy improvement E∆ of harmonic replacement on collectionsof projected balls of energy at most ε by

E∆(s, γ, ε) = supB∈B∈B |E(γ(·,s)|B)≤ε

Eδ(γ(·, s), 12

B) .

In the definition, we measure the energy decrease on balls of half the radius of those in B.We choose this to be able to use the maximal energy improvement in connection with (4.8.6) ofProposition 4.8.5.

Lemma 4.9.2. The maximal energy improvement E∆ is non-negative, and positive for s ∈ [0, 1]such that γ(·, s) is not harmonic.

58

Construction of the map

Proof. Since Eδ is non-negative, E∆ is non-negative.Suppose now s ∈ [0, 1], γ ∈ ΩM and ε ∈ (0, εH ] is such that E∆(s, γ, ε) = 0. For any p ∈ S2

we can let B be a sufficiently small projected ball containing p such that E(γ(·, s)|B) ≤ ε. By theassumption, Eδ(γ(·, s),B) = 0. By Lemma 4.8.3, γ(·, s)|B is harmonic. Thus, by Lemma 2.5.1,γ(·, s) is harmonic.

Let γ ∈ ΩM and recall that the map s 7→ γ(·, s) is in C0([0, 1], C0 ∩W 1,2(S2,M)). Define

cγ = maxs∈[0,1]

‖dγ(·, s)‖L2(S2,Rm) . (4.9.1)

Let V be an open or closed subset of S2. Then, using the triangle inequality of the L2-norm, weget

|E(γ(·, s)|V )− E(γ(·, s0)|V )|

=12

∣∣∣‖dγ(·, s)‖2L2(V,Rm) − ‖dγ(·, s0)‖2L2(V,Rm)

∣∣∣=

12

∣∣‖dγ(·, s)‖L2(V,Rm) + ‖dγ(·, s0)‖L2(V,Rm)

∣∣ ∣∣‖dγ(·, s)‖L2(V,Rm) − ‖dγ(·, s0)‖L2(V,Rm)

∣∣≤ cγ‖dγ(·, s)− dγ(·, s0)‖L2(S2,Rm)

(4.9.2)

showing that the map s 7→ E(γ(·, s)|V ) is continuous with the bound cγ is independent of V .The following lemma extends this to hold if harmonic replacement has been performed. Notethat the δ chosen in the lemma is independent of the collection B.

Lemma 4.9.3. Suppose γ ∈ ΩM . Then, for any ε > 0 and s0 ∈ [0, 1], there exists δ > 0 suchthat if |s− s0| < δ, B ∈ B, and

E(γ(·, s)|B), E(γ(·, s0)|B) ≤ εH (4.9.3)

then|E(H(γ(·, s),B))− E(H(γ(·, s0),B))| < ε . (4.9.4)

Proof. By (4.9.2) with V = S2 \ B we have∣∣E(γ(·, s)|S2\B)− E(γ(·, s0)|S2\B)∣∣ ≤ cγ‖dγ(·, s)− dγ(·, s0)‖L2(S2,Rm)

with cγ as defined in (4.9.1). Since we required (4.9.3), (4.5.1) in Lemma 4.5.1 applies to eachof the balls in B. Using Cauchy-Schwartz, we get∑

B∈B

|E(HB(γ(·, s)|B))− E(HB(γ(·, s0)|B))|

≤∑B∈B

(CU (E(γ(·, s)|B) + E(γ(·, s0)|B)) ‖γ(·, s)− γ(·, s0)‖C0(cl(B),Rm)

+ CU (E(γ(·, s)|B) + E(γ(·, s0)|B))1/2E(γ(·, s)− γ(·, s0)|B)1/2

)C-S≤ CU2εH‖γ(·, s)− γ(·, s0)‖C0(S2,Rm) + CU (2εH)1/2E(γ(·, s)− γ(·, s0))1/2 .

59


Hence

|E(H(γ(·, s),B))− E(H(γ(·, s0),B))|

≤∣∣E(γ(·, s)|S2\B)− E(γ(·, s0)|S2\B)

∣∣+∑B∈B

|E(HB(γ(·, s)|B))− E(HB(γ(·, s0)|B))|

≤ 2CεH‖γ(·, s)− γ(·, s0)‖C0(S2,Rm) + (Cε1/2H + cγ)‖γ(·, s)− γ(·, s0)‖L2(S2,Rm) . (4.9.5)

By the continuity of s 7→ γ(·, s), we can then choose δ > 0 such that (4.9.4) holds if |s− s0| < δ.

Corollary 4.9.4. Let s0 ∈ [0, 1], γ ∈ ΩM , and ε ∈ (0, εH ]. If E∆(s0, γ, ε) > 0 then there is anopen interval I ⊂ [0, 1] containing s0 such that

E∆(s, γ, ε/2) ≤ 2E∆(s0, γ, ε) (4.9.6)

for all s ∈ I.

Proof. By Lemma 4.9.3, we can let I ⊆ [0, 1] be an interval containing s0 such that∣∣∣∣E(H(γ(·, s), 12

B))− E(H(γ(·, s0),12

B))∣∣∣∣ ≤ E∆(s0, γ, ε)

2(4.9.7)

whenever s ∈ I and B ∈ B with

E(γ(·, s)| 12B), E(γ(·, s0)| 1

2B) ≤ εH . (4.9.8)

By (4.9.2) we can shrink I such that

|E(γ(·, s))− E(γ(·, s0))| ≤ E∆(s0, γ, ε)2

(4.9.9)

and|E(γ(·, s)|B)− E(γ(·, s0)|B)| ≤ ε

2(4.9.10)

for all s ∈ I and B ∈ B.Now fix s ∈ I and let B ∈ B with E(γ(·, s)|B) ≤ ε/2. Then, by (4.9.10), we have

E(γ(·, s0)|B)(4.9.10)

≤ ε

2+ E(γ(·, s)|B) ≤ ε (4.9.11)

and since ε ≤ εH , (4.9.8) holds. Thus∣∣∣∣Eδ(γ(·, s), 12

B)− Eδ(γ(·, s0),12

B)∣∣∣∣

≤ |E(γ(·, s))− E(γ(·, s0))|+∣∣∣∣E(H(γ(·, s), 1

2B))− E(H(γ(·, s0),

12

B))∣∣∣∣

(4.9.7)

(4.9.9)

≤ E∆(s0, γ, ε) .

(4.9.12)

Hence

Eδ(γ(·, s), 12

B)(4.9.12)

≤ E∆(s0, γ, ε) + Eδ(γ(·, s0),12

B)(4.9.11)

≤ 2E∆(s0, γ, ε) .

Since this holds for any B ∈ B with E(γ(·, s)|B) ≤ ε/2 and any s ∈ I, we conclude (4.9.6).

60


In the following lemma we determine on which balls to do harmonic replacement whenconstructing the sweepout tightening map. Note in particular that Property (3) of the lemmaensures a certain decrease in energy when performing the replacement. The balls on which wereplace are therefore of high energy, or “far” from being harmonic as indicated in the discussionin Section 4.1. Property (1) ensures that on each slice only two replacements will occur. Thiswill allow us to later use the estimates of Proposition 4.8.5 and Proposition 4.8.6. Harmonicreplacement will be well-defined because of Property (2).

Lemma 4.9.5. Let γ ∈ ΩM , suppose W ([γ]) > 0 and E∆(s, γ, εH/4) > 0 for all s ∈ [0, 1]with E(γ(·, s)) ≥ W ([γ])/2. Then there exists a finite collection of finite collections of ballsB1, . . . ,Bk, Bi ∈ B, and continuous maps r1, . . . , rk ∈ C0([0, 1], [0, 1]) such that

(1) for each s ∈ [0, 1] at most two of ri(s), i ∈ 1, . . . , k are positive,

(2) for each s ∈ [0, 1] and each i ∈ 1, . . . , k, E(γ(·, s)|ri(s)Bk) ≤ εH/2,

(3) for each s ∈ [0, 1] such that E(γ(·, s)) ≥W/2 there exists i ∈ 1, . . . , k such that

Eδ(γ(·, s), ri(s)/2Bi) ≥E∆(s, γ, εH/8)

8.

Proof. Let s0 ∈ [0, 1] be such that E(γ(·, s0)) ≥ W/2. By assumption E∆(s0, γ, εH/4) > 0.Hence there exists Bs0 ∈ B with E(γ(·, s0)|Bs0 ) ≤ εH/4 and

Eδ(γ(·, s0),12

Bs0) ≥ E∆(s0, γ, εH/4)2

. (4.9.13)

We now use Corollary 4.9.4, the continuity of s 7→ E(γ(·, s)), s 7→ E(H(γ(·, s), 12Bs0)), and

s 7→ E(γ(·, s)|Bs0 ) to choose an open interval Is0 ⊆ [0, 1] containing s0 so that, for all s ∈ Is0 ,

E∆(s, γ, εH/8) ≤ 2E∆(s0, γ, εH/4) , (4.9.14)∣∣∣∣Eδ(γ(·, s), 12

Bs0)− Eδ(γ(·, s0),12

Bs0)∣∣∣∣ ≤ E∆(s0, γ, εH/4)

4, (4.9.15)∣∣∣E(γ(·, s)|Bs0 )− E(γ(·, s0)|Bs0 )

∣∣∣ ≤ εH4. (4.9.16)

Combining the inequalities we get

Eδ(γ(·, s), 12

Bs0)(4.9.15)

≥ Eδ(γ(·, s0),12

Bs0)− E∆(s0, γ, εH/4)4

(4.9.13)

≥ E∆(s0, γ, εH/4)4

(4.9.14)

≥ E∆(s0, γ, εH/8)8

, (4.9.17)

E(γ(·, s)|Bs0 )(4.9.16)

≤ E(γ(·, s0)|Bs0 ) +εH4≤ εH

2. (4.9.18)

Then let Is0 be an open interval containing s0 such that cl(Is0

)⊂ Is0 .

Carrying out the above argument for each s0 in the set P = s ∈ [0, 1] |E(γ(·, s)) ≥W ([γ])/2,we get intervals Is covering the set. Since the map s 7→ E(γ(·, t)) is continuous and theset [W ([γ])/2,∞) closed, P is closed and hence compact. Thus, we can choose a finite set

61


s1, . . . , sk such that Isi cover P . By Lemma C.1.6, we can suppose each interval intersects atmost two other intervals which are disjoint.

For each i ∈ 1, . . . , k let ri : [0, 1] → [0, 1] be a continuous function such that ri(s) = 1for s ∈ Isi , and ri(s) = 0 for s ∈ [0, 1] \ Isi and when s is contained in some interval besides Isiand the intervals intersecting it. By the choice of the intervals and the construction of ri, (1)follows.

We claim that Bs1 , . . . ,Bsk will do as the collections of balls whose existence we wishto prove. Since E(γ(·, s)|ri(s)Bsi ) ≤ E(γ(·, s)|Bsi ) and ri(t) are zero outside Isi , we get (2) by(4.9.18). To see that (3) holds, for s ∈ P choose i ∈ 1, . . . , k such that s ∈ Isi . Then ri(s) = 1such that (4.9.17) gives the result.

We can now construct a sweepout tightening map (Ψ, ψ, εT ):

Proof of Theorem 4.1.2. For any γ ∈ ΩM , we define Ψ(γ) in the following way: Per assump-tion, W ([γ]) > 0, and, for any s ∈ [0, 1], γ(·, s) is either constant or non-harmonic. ThusLemma 4.9.2 implies that E∆(s, γ, εH/4) > 0 for all s ∈ [0, 1] with E(γ(·, s)) ≥ W ([γ])/2, andhence Lemma 4.9.5 gives finite collections B1, . . . ,Bk, Bi ∈ B and corresponding continuousmaps ri : [0, 1]→ [0, 1]. Let γ0 = γ and define γi : S2× [0, 1]→M , i ∈ 1, . . . , k inductively by

γi = (p, s) 7→ H(γi−1(·, s), ri(s)Bi)(p) .

Since, for each i, γ(·, s) ∈Mri(s)Bi and at most two of ri(s) are positive for each s, the harmonicreplacement is well-defined by Lemma 4.8.1.

We claim that each γi ∈ ΩM ; it is clear that for each s ∈ [0, 1], γi(·, s) ∈ C0 ∩W 1,2(S2,M).Furthermore, γi maps S2×0 and S2×1 to points since γ does. Finally, to see that the map

s 7→ γi(·, s) = H(γi−1(·, s), ri(s)Bi)

is continuous note that the continuity of ri implies that the map s 7→ (s, ri(s)) is continuous.Composing this with the scaled harmonic replacement map and applying Corollary 4.7.2 givesthe continuity.

Let Ψ(γ) = γk. Since γk ∈ ΩM , Ψ is well-defined. We need to establish the additionalproperties of sweepout tightening maps. First, to see that Ψ(γ) ∈ [γ], we construct a homotopyfrom γ to Ψ(γ). We do this by defining maps γi : S2 × [0, 1] × [0, 1] → M by letting γ0 =(p, s, t) 7→ γ(p, s) and inductively

γi = (p, s, t) 7→ H(γi−1(·, s, t), tri(s)Bi)(p) .

As above, we see that the maps γi are continuous. Since γk(p, s, 0) = γ(p, s) and γk(p, s, 1) =Ψ(γ)(p, s), γk is the desired homotopy.

Since, γi is constructed from γi−1 by performing harmonic replacement, we have E(γi(·, s)) ≤E(γi−1(·, s)) for each s. Thus E(Ψ(γ)(·, s)) ≤ E(γ(·, s)) for each s.

To establish the remaining properties, let εT = εH/12. We aim at finding a positive, contin-uous function ψ with ψ(0) = 0 and with the property that if B ∈ B and s ∈ [0, 1] is such thatE(γ(·, s)) ≥ W ([γ])/2 and E(Ψ(γ)(·, s)|B) ≤ εT then there exists an energy minimizing mapv : 1/8B→M equal to Ψ(γ) on ∂1/8B with

E(v −Ψ(γ(·, s))|1/8B) ≤ ψ(E(γ(·, s))− E(Ψ(γ)(·, s))) . (4.9.19)

Let B ∈ B and γ(·, s) have the above properties.

62


Again using that at any given s at most two ri(s)’s are positive, we have Ψ(γ)(·, s) =H(γ(·, s), ri(s)Bi, rj(s)Bj) for some i, j. We directly see that

E(γ(·, s))− E(Ψ(γ)(·, s)) ≥ Eδ(γ(·, s), ri(s)Bi) ≥ Eδ(γ(·, s), ri(s)/2Bi) . (4.9.20)

By Proposition 4.8.5 and Remark 4.8.7 using that E(γ(·, s)) ≥W ([γ])/2 ≥ εmin/2, we also havethat

E(γ(·, s))− E(Ψ(γ)(·, s)) ≥ κ1

(Eδ(γ(·, s), rj(s)

2Bj))2

. (4.9.21)

By Property (3) of Lemma 4.9.5, one of

Eδ(γ(·, s), ri(s)2

Bi) ≥E∆(s, γ, εH/8)

8,

Eδ(γ(·, s), rj(s)2

Bj) ≥E∆(s, γ, εH/8)

8

(4.9.22)

must hold. Combining (4.9.20), (4.9.21), and (4.9.22), we get

max

(8 (E(γ(·, s))− E(Ψ(γ)(·, s))) , 8

(E(γ(·, s))− E(Ψ(γ)(·, s))

κ1

)1/2)

≥ E∆(s, γ, εH/8) .

(4.9.23)

Suppose for a moment that we have a constant c > 0, independent of γ and s, such thatE(γ(·, t))− E(Ψ(γ)(·, s)) ≥ c. Then

E(v −Ψ(γ(·, s))|1/8B) ≤ 2E(v) + 2E(Ψ(γ(·, s))|1/8B) ≤ 4εT ≤4εTc

(E(γ(·, s))− E(Ψ(γ)(·, s))

).

Thus, in this case, we have established (4.9.19) with ψ(s) = s(4εT )/c. We will show that if theenergy of γ(·, s) or γi(·, s) on B is large, such a constant exists.

Suppose first E(γi(·, s)|B) > εH/10. Then using Lemma 2.5.2 and the triangle inequality forthe L2-norm, we have

E(γ(·, s))− E(Ψ(γ)(·, s)) ≥ E(γi(·, s)|rj(s)B2)− E(Ψ(γ)(·, s)|rj(s)B2

)Lem. 2.5.2≥ 1

2E(γi(·, s)−Ψ(γ)(·, s)|rj(s)B2

)

=12E(γi(·, s)−Ψ(γ)(·, s))

≥ 12

(E(γi(·, s)|B)1/2 − E(Ψ(γ)(·, s)|B)1/2

)2

>12

((εH/10)1/2 − (εH/12)1/2

)2.

Thus, in this case, we have a positive lower bound; denote it c1. Therefore, as long as we letψ(s) ≥ (s4εT )/c1, we can assume E(γi(·, s)|B) ≤ εH/10. By a similar calculation, we get alower bound c2 > 0 when supposing E(γ(·, s)|B) > εH/8. Thus, we can suppose E(γ(·, s)|B) ≤εH/8. We are now ready to apply Proposition 4.8.6 to finish the proof. As above, Remark4.8.7 shows that the conditions for using the proposition are satisfied. Let µ = 1/8 and v =H(Ψ(γ)(·, s), 1/8B) so that

E(v −Ψ(γ(·, s))|1/8B)Lem. 2.5.2≤ 2Eδ(Ψ(γ(·, s)), 1/8B)

Prop. 4.8.6≤ 2κ2Eδ(H(γ(·, s), ri(s)Bi), rj(s)Bj)1/2 + 2Eδ(H(γ(·, t), ri(s)Bi), 1/4B) .

(4.9.24)

63


Then letting µ = 1/4, we get

Eδ(H(γ(·, s), ri(s)Bi), 1/4B)Prop. 4.8.6≤ κ2Eδ(γ(·, s), ri(s)Bi)1/2 + Eδ(γ(·, s), 1/2B) . (4.9.25)

Since Eδ(γ(·, s), 1/2B) ≤ E∆(γ, t, εH/8) and

Eδ(H(γ(·, s), ri(s)Bi), rj(s)Bj) ≤ E(γ(·, s))− E(Ψ(γ)(·, s)) ,Eδ(γ(·, t), ri(s)Bi) ≤ E(γ(·, s))− E(Ψ(γ)(·, s)) ,

we have

E(v −Ψ(γ(·, s))|1/8B)

(4.9.24)

(4.9.25)

≤ 4κ2 (E(γ(·, s))− E(Ψ(γ)(·, s)))1/2 + 2E∆(γ, s, εH/8) .

After inserting (4.9.23), we have bounded E(v−Ψ(γ(·, s))|1/8B) by ψ1(E(γ(·, s))−E(Ψ(γ)(·, s)),where ψ1 is a continuous, positive map being ψ1(0) = 0. Then letting

ψ(s) = max(ψ1(s), s(4εT )/c1, s(4εT )/c2)

we get (4.9.19).

Notes

As noted, the notion of sweepout tightening maps is in [CM07a] called energy decreasing maps.The idea of sweeping out hypersurfaces with loops and performing harmonic replacement onthose is based on [CM07b]. The discussion of continuity of harmonic replacement is a muchdetailed exposition of the proof of [CM07a, Corollary 3.4]. In particular, the results concerningcontinuity with scaling are not covered in [CM07a] and thus written by the author of this thesis.The proofs of Propositions 4.8.5 and 4.8.6 are again detailed versions of the proof of [CM07a,Lemma 3.11], except that [CM07a] does not rule out the case of balls covering all of S2 whichimplies that Corollary 4.8.9 cannot be used. In broad terms, the rest of the chapter follows[CM07a].

64

5

Bubble convergence

A central point in the proof of Theorem 1 will be that, given a sequence of slices of sweepoutswhich have already been tightened by a sweepout tightening map, a subsequence converges ina certain sense to a collection of minimal spheres in M . This convergence will be the theme ofthis chapter.

We will divide the discussion into four parts. First, we prove a theorem extending local con-vergence of subsequences to global convergence away from a finite set of singular points. We usethis to prove compactness of harmonic maps. We then turn to developing an appropriate notionof almost harmonic maps from the 2-sphere. The singular points mentioned correspond to lostenergy. In the third part we will “blow up” a region around each singular point, iterate the argu-ment and thereby keep track of the lost energy. This will prove the main result, Corollary 5.3.15.In the last part of the chapter, we will relate bubble convergence to the convergence of certainintegral limits. The result, Theorem 5.4.1, will be used together with Corollary 5.3.15 in theproof of Theorem 3.3.1 in the next chapter.

Throughout the chapter, we will let (M, g) be a compact manifold isometrically embedded inRm. We will work with sequences of maps - often denoted (uj) - and take subsequences of those.When taking such a subsequence, we will replace the original sequence by the subsequence,i.e. such that (uj) now denotes the subsequence instead of the original sequence. When doingthis, sequences related to uj and indexed by j will implicitly be replaced by the correspondingsubsequences as well.

5.1 Compactness

A common characterization of compact (metric) sets is the property that, given a sequence ofelements of the set, a subsequence converges to an element of the set. Let Σ be a compactsurface and (uj), uj ∈W 1,2(Σ,Rm) a sequence. If we, for some E0 ≥ 0, suppose E(uj) ≤ E0 forall j then, by weak compactness of unit balls in Banach spaces, a subsequence of (uj) convergesweakly to some u ∈W 1,2(Σ,Rm) with E(u) ≤ E0. Thus, the set

u ∈W 1,2(Σ,Rm) |E(u) ≤ E0

is compact in the weak topology. If we furthermore require uj to be harmonic, it will be naturalto ask if the limit map u is also harmonic. As we will see, this is the case. We will not, though,be directly interested in compactness of harmonic maps, but rather in the weaker notion ofalmost harmonic maps that we will develop in the next section. Thus, we make sure to provethe compactness result in a sufficiently general setting allowing us to use it for almost harmonicmaps. Compactness of harmonic maps will be an easy corollary of the result. We include it tomotivate the techniques used.

65

5. Bubble convergence

The convergence we will show will take place in a topology stronger than the weak topology;the convergence will be strong away from a finite set of singular points.

5.1.1 Special cover

To prove the compactness result, we will need to cover a compact surface Σ with open sets whilekeeping control of the energy on all the sets combined. We do this by constructing covers withan upper bound on the number of sets a point can possibly be contained in.

Lemma 5.1.1. Let Σ be a compact surface with metric h. Then there exists K(Σ) ∈ N suchthat, for any r > 0, Σ can be covered by a finite number of open sets V1, . . . , Vk of diameterat most r in such a way that no point of Σ is contained in more than K of the sets.

Proof. Note first that, for any s > 0, we can cover R2 by balls of radius s such that eachpoint is included in at most four balls; the cover Bs(sx, sy) |x, y ∈ Z will suffice. Then, usingcontinuity of the metric, for each p ∈ Σ, choose rp > 0 such that B2rp(p) is a geodesic ball,and such that, in normal coordinates (x1, x2) around p, maxi,j∈1,2 |hij(q)− δij | < 1/2 for eachq ∈ B2rp(p). Then use compactness of Σ to extract a finite subcover Br1(p1), . . . , Brl(pl).

Now, given r > 0, let s = min(r, r1, . . . , rl)/√

8 and define Vi,(x,y) = exppi Bs(x, y). We claimthat the collection

V =Vi,(sx,sy) | i ∈ 1, . . . , l, x, y ∈ Z with ‖(sx, sy)‖ < ri + s

possesses the required properties. First, if, for each i, the sets Vi,(sx,sy), x, y ∈ Z, ‖(sx, sy)‖ <ri+s cover Bri(pi), then clearly V covers Σ. It is sufficient to show that the sets Bs(sx, sy), x, y ∈Z, ‖(sx, sy)‖ < ri + s cover Bri(0) ⊂ R2. For any q ∈ Bri(0), pick x, y ∈ Z such that ‖(1/s)q −(x, y)‖ < 1. Then ‖q − (sx, sy)‖ < s so that q ∈ Bs(sx, sy). In addition, the triangle inequalityimplies that ‖(sx, sy)‖ ≤ ‖q‖+ s < ri + s. Thus, q is contained in one of the above sets.

Since, for each i, at most four of the sets Vi,(sx,sy), x, y ∈ Z contain each point of R2, at mostK = 4l of the sets in V contain each point of Σ. Finally, let Vi,(sx,sy) ∈ V and q1, q2 ∈ Vi,(sx,sy).Let qi = exp−1

pi (qi) ∈ R2, γ be the path t 7→ q2t + (1 − t)q1, and γ = expp γ. Since d(q1, q2) ≤L (γ), it suffices to bound L (γ) to bound the diameter of Vi,(sx,sy). Since q1, q2 ∈ B2ri(0) by thechoice of s, disabling the Einstein summation convention,

|E(γ)− E(γ)| ≤ 12

∫ 1

0

∣∣‖γ′(t)‖2 − ‖γ′(t)‖2∣∣ dt ≤ 12

∫ 1

0

∑i,j∈1,2

∣∣(hij − δij)dtγidtγj∣∣ dt<

14

∫ 1

0

(∣∣dtγ1∣∣+∣∣dtγ2

∣∣)2 dt ≤ 12

∫ 1

0(dtγ1)2 + (dtγ2)2dt = E(γ) .

Since E(γ) = L (γ)2 < (2s)2, we have L (γ) ≤ E(γ)1/2 <√

8s ≤ r. Since q1 and q2 werearbitrary in Vi,(sx,sy), this shows that the diameter is at most r.

5.1.2 Compactness

As mentioned, compactness of harmonic maps is the motivating example for the results of thissection. Recall from Theorem 2.6.3, Property (H1) that a subsequence of a sequence of harmonicmaps converges uniformly to a harmonic map on small, low-energy subsets. This was observed

66

Compactness

in [SU81] and used there to prove compactness of harmonic maps. The following definitioncaptures this property, allowing us to prove Theorem 5.1.3 along the same lines as the proofof [SU81]. After the proof, we show that harmonic maps indeed fulfill the requirements of thedefinition allowing us to prove compactness of harmonic maps as a consequence of the theorem.

Definition 5.1.2. A sequence of W 1,2(Σ,M) maps (uj) is said to be ε-locally converging toharmonic maps if, for any point x ∈ Σ contained in an open set Vx ⊆ Σ with E(uj |Vx) ≤ ε forall j, there exist an open set Ux ⊆ Σ containing x, a subsequence (ujk), and a harmonic mapux : Ux →M such that

ujk |UxW 1,2(Ux,M)−→ ux .

Confer Definition 2.2.6 for the definition of the convergence used in the theorem.

Theorem 5.1.3 (Compactness of maps ε-locally converging to harmonic maps).Suppose Σ is a compact surface, (uj), uj ∈W 1,2(Σ,M) a sequence of maps ε-locally convergingto harmonic maps for some ε > 0, and suppose E(uj) ≤ E0 for an E0 ≥ 0. Then there exist aharmonic map u : Σ → M and a finite collection of points S ⊂ Σ such that a subsequence of(uj) converges strongly to u except on S.

Proof. We will carry out the proof in three steps.

Weak convergenceSince E(uj) is bounded, Σ and M compact, we have

‖uj‖W 1,2 =∫

Σ‖uj‖2 + (2E(uj))1/2 ≤ Area(Σ) sup

x∈M‖x‖2 + (2E0)1/2

and thus ‖uj‖W 1,2 is bounded. By weak compactness of the unit ball in W 1,2(Σ,Rm)(Lemma 2.2.2), a subsequence converges weakly to a map u ∈W 1,2(Σ,Rm). Replace (uj)by this subsequence. By Lemma 2.4.4, E(u) ≤ E0.

Excluding high-energy subsetsUse Lemma 5.1.1 to get K ∈ N such that, for each k ≥ 1, the lemma gives a cover Vk ofΣ consisting of sets of diameter no more than 1/k such that no point is contained in morethan K of the sets. Fix k ∈ N and let Vk = V1, . . . , Vl. Then, for each j ∈ N,

l∑i=1

E(uj |Vi) ≤ KE0 .

It follows that, for each j, E(uj |Vi) > ε for at most dKE0/εe 1 of the sets. Let Ikj be theset of integers such that

Vkj =Vi ∈ Vk | i ∈ Ikj

⊆ Vk

denotes the collection of these “high-energy” sets. As noted, |Ikj | ≤ dKE0/εe.

Still for fixed k, replace (uj) by a subsequence still denoted (uj) such that |Ikj | is constantin j. This is possible due to the uniform bound on |Ikj |. Then replace (uj) by yet anothersubsequence, such that Ikj is constant in j; this is possible since the discrete set ∪j∈NI

kj is

finite and thus compact. Let Ik denote the resulting set and Vk the corresponding set of1dxe of x ∈ R being the smallest n ∈ Z such that n ≥ x.

67


high-energy sets now also constant in j. For each V ∈ Vk we pick a point xV ∈ V which

we call the center point of V . Let Sk =xV | V ∈ Vk

be the collection of the center

points.

Note that the sets in Vk \ Vk cover Σ \ ∪V ∈Vk , and each V ∈ Vk \ Vk satisfies E(uj |V ) ≤ ε

for each j.

We now replace (uj) by the diagonal subsequence; we let the jth entry of (uj) be the jthentry of the subsequence obtained by successively carrying out the above procedure foreach k = 1, . . . , j.

Then again replace (uj) by a subsequence such that the number of high-energy balls isconstant in j; that is |Sj | = |Vj | = l for all j and some l ∈ 1, . . . , dKE0/εe. Then usecompactness of Σ to take a further subsequence such that the center points Sj converges;that is, the ith entry of Sj converges to some si ∈ Σ for each i ∈ 1, . . . , l. Let S =s1, . . . , sl be the limit center points. Note that the diameter of the sets in Vj is still atmost 1/j.

Strong convergence on compact setsOur strategy will be to prove strong W 1,2 convergence of a subsequence of (uj) on eachelement of a sequence of compact sets exhausting Σ \ S in the limit. We easily get such asequence (Ck) by letting

Ck = Σ \ ∪s∈SB1/k(s) .

Taking the diagonal subsequence will then prove strong convergence of (uj) on all Ck.Since any compact set C ⊂ Σ \ S is contained in Ck for sufficiently large k, we get strongconvergence on any such C.

Fix k ∈ N. The key is that each x ∈ Ck is contained in a “low-energy” set; that is, x ∈ Vxwith E(uj |Vx) ≤ ε for all j. To see this, choose a j ≥ 2k such that d(Sj , S) < 1/(2k).Suppose for contradiction that x ∈ ∪V ∈Vj V . Let s ∈ Sj and s ∈ S satisfy d(x, Sj) = d(x, s)and d(x, S) = d(x, s). Then, using the upper bound 1/j on the diameter of the sets in Vj ,we have

1/k ≤ d(x, s) ≤ d(x, s) + d(s, s) <1j

+12k≤ 1k.

We conclude that x ∈ Σ \ ∪v∈Vj V and then, since the sets in Vj \ Vj cover Σ \ ∪V ∈Vj V ,x ∈ ∪V ∈Vj\VjV . Thus, there exists Vx ∈ Vj \ Vj containing x.

For each x ∈ Ck use the ε-locally converging to harmonic maps property of (uj) to getsets Ux. Since Ck is compact, we can extract at finite set x1, . . . , xp of points in Ck suchthat the collection Ux1 , . . . , Uxp covers Ck.

Then, for each i ∈ 1, . . . , p, replace (uj) by a subsequence, such that

uj |UxiW 1,2(Uxi ,M)−→ ui

for harmonic maps ui : Uxi → M . Since we began by ensuring that ujW 1,2(Σ,Rm)

u, by

Lemma 2.2.1 also uj |UxiW 1,2(Uxi ,R

m) u|Uxi . Thus, since limits in the weak topology are

unique, ui = u|Uxi . Since this holds for all i, the collection Ux1 , . . . , Uxp covers Ck, andall ui are harmonic, we get that u|Ck is harmonic by Lemma 2.5.1.

68

Almost harmonic maps

As described above, we now replace (uj) by the diagonal subsequence; that is, the jthentry of (uj) is now the jth entry of the subsequence obtained by successively carryingthe above procedure for each k = 1, . . . , j. Then (uj) converges strongly on any compactsubset of Σ \ S. Hence u(x) ∈ M for almost every x ∈ Σ implying u ∈ W 1,2(Σ,M).Furthermore, again using Lemma 2.5.1, we see that u|Σ\S is harmonic. Applying theRemovable Singularities Property (H2) of Theorem 2.6.3 to each s ∈ S, we get that u|Σ\Sextends to a harmonic map on all of Σ.

The proof can be carried out by taking fewer subsequences than done here. The presentedstep-by-step version was chosen to make the strategy clearer. In the following sections we willlet Vj , Vj and S refer to the collections defined in the above proof. The collection S may containduplicate points. If this is the case, we remove the duplicates. We will refer to the sets in Vjas high-energy sets and the sets in Vj \ Vj as low-energy sets. The points in S will be calledsingular points.

Corollary 5.1.4 (Compactess of harmonic maps). Suppose Σ is a compact surface and (uj),uj : Σ→ M a sequence of harmonic maps with E(uj) ≤ E0 for some E0 ≥ 0. Then there exista harmonic map u : Σ→M and a finite set S ⊂ Σ such that (uj) converges strongly to u excepton S.

Proof. By the Local Compactness Property (H1) of Theorem 2.6.3, for any geodesic ball Br(x)with E(uj |Br(x)) ≤ εSU for all j, there exists a subsequence with C1(Br/2(x),M) convergence toa harmonic map u : Br/2(x)→M . For any x ∈ Σ and Vx containing x with E(uj |Vx) ≤ εSU forall j, we can choose rx > 0 such that Brx(x) ⊂ Vx and hence E(uj |Brx (x)) ≤ εSU for all j. ThusBrx/2(x) will do as the set Ux showing that (uj) is εSU -locally converging to harmonic maps.The result then follows from Theorem 5.1.3.

5.2 Almost harmonic maps

We will be interested in a notion of maps being almost harmonic. When defining this we wishto obtain three properties. Firstly, we wish the set of almost harmonic maps with boundedenergy to be compact. Thus, it will be convenient if sequences of almost harmonic maps areε-locally converging to a harmonic map. Secondly, we wish sweepout tightening maps to producesweepouts with almost harmonic slices. This will allow us to actually use the definition. Thirdly,the set of almost harmonic maps S2 →M should be invariant under dilation; that is, if u : S2 →M is almost harmonic, then, for any dilation Dp,λ, u Dp,λ should be so too. We will see whythis is essential in the following section.

In light of Corollary 5.1.4, the first requirement can be fulfilled if we locally require a sequenceof almost harmonic maps to become progressively closer to harmonic maps. Such a definitionwould satisfy the second as well, confer Property (4) of Definition 4.1.1. In fact, we will only beinterested in sequences of harmonic maps. We will therefore make a definition just concerningsequences. The third property will be hard to satisfy if the definition applies to general domainsas for such domains we have no notion of dilations. Therefore, we restrict our attention to mapson S2. The set of projected balls is dilation invariant - this is the reason for their slightly crypticdefinition - and a definition based on them will make it easy to satisfy the second requirement.We arrive at the following definition.

69


Definition 5.2.1 (ε-almost harmonic sequences). Suppose ε > 0. A sequence of W 1,2(S2,M)maps (uj) is said to be ε-almost harmonic if there exists r ∈ (0, 1] such that, for any j ∈ N andany projected ball B with E(uj |B) ≤ ε, there exists a harmonic map vj : rB → M equal to uj

on the boundary ∂rB with

E(vj − uj |rB) ≤ 1j.

Note that a subsequence (ujk) of an ε-almost harmonic sequence (uj) is ε-almost harmonic.This follows easily by noting that

E(vjk − ujk |rB) ≤ 1jk≤ 1k.

The following results show that the definition satisfies the three sought for properties.

Lemma 5.2.2. Let (uj), uj : S2 → M be a sequence of ε-almost harmonic maps. Then (uj) isalso min(ε, εSU )-locally converging to harmonic maps.

Proof. Let x ∈ Σ and suppose Vx is an open set containing x such that E(uj |Vx) ≤ min(ε, εSU )for all j. Choose R > 0 such that BR(x) is a geodesic ball contained in Vx. Then E(uj |BR(x)) ≤E(uj |Vx) ≤ min(ε, εSU ) for all j.

Since B = (BR(x),Πx) is a projected ball, we can use the ε-almost harmonic property of (uj)to choose harmonic maps vj : rB →M coinciding with uj on ∂rB such that

E(vj − uj |rB) ≤ 1j. (5.2.1)

Let r > 0 be the geodesic radius of rB such that rB = Br(x). Choose a subsequence of (ujk)such that (vjk) converges in C1(Br/2(x),M) to a harmonic map v : Br/2(x) → M . Then, byCorollary 2.4.3 and (5.2.1), we get

E(ujk − v|Br/2(x))Cor. 2.4.3≤ 2E(ujk − vjk |Br/2(x)) + 2E(vjk − v|Br/2(x))→ 0 .

Using the same approach and Theorem 2.3.1, noting that vj − uj ∈W 1,20 (Br(x),M), we get∫

Br/2(x)‖ujk − v‖2 ≤ 2

∫Br/2(x)

‖ujk − vjk‖2 + 2∫Br/2(x)

‖vjk − v‖2

≤ 2∫Br(x)

‖ujk − vjk‖2 + 2∫Br/2(x)

‖vjk − v‖2

Thm. 2.3.1≤ r2

∫Br(x)

‖dujk − dvjk‖2 + 2∫Br/2(x)

‖vjk − v‖2

→ 0 .

Thus ujkW 1,2(Br/2(x),M)

−→ v so that Br/2(x) will do as the set Ux whose existence we sought for.

Corollary 5.2.3. let (uj), uj : S2 →M be a sequence of ε-almost harmonic maps with E(uj) ≤E0. Then there exist a harmonic map u : S2 → M and a finite set S ⊂ Σ such that (uj)converges strongly to u except on S.

70

Energy preservation

Proof. Follows directly from Lemma 5.2.2 and Theorem 5.1.3.

Lemma 5.2.4. Let (Ψ, ψ, εT ) be a sweepout tightening map, (σj), σj ∈ ΩM a sequence ofsweepouts, and (sj), sj ∈ [0, 1] a sequence such that E(σj(·, sj)) ≥ W[σj ]/2 and E(σj(·, sj)) −E(Ψ(σj)(·, sj))→ 0. Then a subsequence of (Ψ(σj)(·, sj)) is εT -almost harmonic.

Proof. Since E(σj(·, sj)) − E(Ψ(σj)(·, sj)) → 0, ψ is continuous and ψ(0) = 0, we can let(Ψ(σjk)(·, sjk)) be a subsequence such that

ψ(E(σjk(·, sjk))− E(Ψ(σjk)(·, sjk))) ≤ 1/k .

Given a projected ball B and k ∈ N such that E(Ψ(σjk)(·, sjk)|B) ≤ εT , let vk : rB → M be anenergy minimizing map equal to Ψ(σjk)(·, sjk) on ∂rB. By Property (4) of sweepout tighteningmaps,

E(vk −Ψ(σjk)(·, sjk)|rB) ≤ ψ(E(σjk(·, sjk))− E(Ψ(σjk)(·, sjk))) ≤ 1/k .

Lemma 5.2.5. Let (uj), uj : S2 →M be a sequence of ε-almost harmonic maps and (Dpj ,λj ) asequence of dilations. Then (uj Dpj ,λj ) is ε-almost harmonic.

Proof. Let B = (U,Λ) be a projected ball with E(uj Dpj ,λj |rB) ≤ ε. Since B = (Dpj ,λj (U),Λ D−1pj ,λj

) is a projected ball and E(uj |Dpj,λj

(U)) = E(ujDpj ,λj |U ), the ε-almost harmonic property

of (uj) gives a harmonic map vj : rB →M equal to uj on ∂rB with

E(vj − uj |rB) ≤ 1j.

Thus vj = vj Dpj ,λj equals uj Dpj ,λj on rB and

E(vj − uj Dpj ,λj |rB) ≤ 1j.

5.3 Energy preservation

In Corollary 5.2.3 we saw that a sequence (uj) of almost harmonic maps converges to a harmonicmap u. By weak lower semicontinuity of the energy (Lemma 2.4.4),

lim supj→∞

E(uj) ≥ lim infj→∞

E(uj) ≥ E(u) . (5.3.1)

The inequality might indeed be strict, the reason being that the energies E(uj |V ) on the high-energy sets V ∈ Vj are not accounted for in the map u. The point of the proof of Theorem 5.1.3was exactly to exclude those sets in the limit. It will be essential for the result of the nextsection to keep track of this lost energy. This section will concern the problem.

We will start by introducing the notion of bubble convergence designed for the purpose ofkeeping track of the energy. Following this, we will show that a sequence of almost harmonicmaps bubble converges to a finite collection of harmonic maps, but we will leave out the mainenergy preservation property of bubble convergence. To establish that property, we will needproperties on the sequence in addition to being almost harmonic. This will be described in thelast subsection.

As indicated in the discussion in the previous section, the use of dilations will be importantin the constructions. We therefore again restrict the results to sequences of maps defined on S2.

71


5.3.1 Bubble convergence

Let us try, intuitively, to develop a way to keep track of the energy possibly lost in equation(5.3.1). Let (uj) be a sequence of almost harmonic maps. As mentioned, in Theorem 5.1.3 weidentify the set of singular points S and the high energy sets Vj which each get progressivelysmaller and close in on the singular points. It is the energy of those we loose. Imagine these setsas being balls Bj , and, for each j, use a dilation Dj : S2 → S2 to blow up the size of the ball tobe, say, the southern hemisphere. While the dilated maps uj (Dj)−1 have the same energy asuj , the energy of S2 \Bj will be located in the northern hemisphere after the dilation. Since Bj

gets progressively smaller, we might expect this energy to be concentrated at a singular pointat the north pole, and, in the limit, lost. Using Theorem 5.1.3 on uj (Dj)−1, we could thusexpect to get a limit map having the energy of Bj except for some amount lost in a new setof singular points. The argument could then be iterated. If the iteration at some point stopsbecause no more singular points arise, we hope to have all the energy accounted for; what is lostat the north pole of each iteration is accounted for in the previous iteration.

The blowing up of a high-energy set can be pictured as attaching a bubble to S2. In theabove sketch, this would be attaching a southern hemisphere to S2 by gluing the boundary of Bj

to the boundary of the hemisphere. The construction we will use is slightly different in that thebubbles will converge to being full spheres instead of just hemispheres. In the limit, the sphereand a bubble can be thought of as a pair of spheres touching each other in just a singular point.

The below definition of bubble convergence is inspired by the ideas above. The definitionby itself says nothing about almost harmonic sequences, but we will prove that such sequenceswith an additional property has a subsequence which bubble converge. We will then use thebubble convergence to prove the result needed in Theorem 1. In the definition, the sets Ωj

i canbe regarded as comprising each bubble except for the high-energy sets of the bubble. The mapsDji have the role of blowing up the “small” sets Ωj

i to bubbles converging to full spheres. Thesets Ωj

0 are the original domain except for the high-energy sets.

Definition 5.3.1 (Bubble convergence). A sequence of maps (vj), vj ∈ W 1,2(S2,M) is said tobubble converge to a finite collection of maps u, u1, . . . , ul, ui ∈ W 1,2(S2,M) if the followingconditions are satisfied:

(BC1) Weak limit. (vj) converges strongly to u except on a finite set S = x1, . . . , xk ⊂ S2.

To ease notation, we sometimes refer to u and S as u0 and S0 respectively. Define alsoDj

0 = IdS2 .

(BC2) Bubbles. For each i ∈ 1, . . . , l there exists conformal diffeomorphisms Dji : S2 → S2

such that (vj Dji ) converges strongly to ui except on a finite set Si ⊂ S2. Furthermore,

for each j ∈ N, there exists l+1 disjoint, open or closed sets Ωj0, . . . ,Ω

jl such that, for each

i ∈ 0, . . . , l, (Dji )−1(Ωj

i ) exhausts S2 \ Si in the limit and the limits limj→∞E(vj |Ωji

)exist.

(BC3) Energy preservation. The limit energy of (vj) is accounted for in the maps u, u1, . . . , ul:

limj→∞

E(vj) = E(u) +l∑

i=1

E(ui) .

72

Bubble compactness with energy loss

Our notion of bubble convergence is derived from the bubble tree construction developed in[PW93] and [Par96] and the bubble convergence of [CM07a]. The main difference from [PW93]and [Par96] is that we leave out information regarding the tree structure obtained from havingbubbles on bubbles etc. Compared to the bubble convergence defined in [CM07a], our definitionis slightly weaker in that no information regarding the radius and center points of the bubblesis included.

5.3.2 Bubble compactness with energy loss

We now prove that, for each sequence of almost harmonic maps, there exist a subsequence anda finite collection of harmonic maps such that the first two properties of bubble convergenceare satisfied. In the proof of Theorem 5.3.2, we describe the iteration process leaving out somedetails to lemmata following the proof. The construction is illustrated in Figure 5.1.

Theorem 5.3.2. For ε > 0 and E0 ≥ 0 let (uj), uj : S2 →M be a sequence of ε-almost harmonicmaps with E(uj) ≤ E0. Then there exist a finite collection of harmonic maps u, u1, . . . , ul,u, ui : S2 → M and a subsequence of (uj) such that the properties (BC1) and (BC2) of bubbleconvergence are satisfied.

Proof. Let ε0 = min(ε, εSU ). We start by using Corollary 5.2.3 to replace (uj) with a subsequenceconverging strongly to a harmonic map u except on a finite set of singular points S.

For each point in S we spread out the energy on a bubble in the following way: Fix ani ∈ 1, . . . , |S| and let si be the ith point of S. Choose ρi > 0 such that B2ρi(si) containsno points of S besides si and E(u|Bρi (si)) ≤ ε0/3. The factor of two on the radius ensuresthat a similar choice of ball for another s ∈ S will be disjoint from Bρi(si). Replace (uj) bya subsequence such that limj→∞E(uj |Bρi (si)) exists and use the Lemmata 5.3.4 and 5.3.5 to

replace (uj) by a further subsequence such that we have sequences (εji ), εji ∈ (0, ρi], (rji ), r

ji ∈

(0, ρi] and (yji ), yji ∈ S2 satisfying

(1) εji → 0, rji → 0 and yji → si,

(2) εji/rji →∞ and d(yji , si)/ε

ji → 0,

(3) for each j, no r ∈ (0, rji ) and y ∈ Bρi−r(si) exist such that E(uj |Bρi (si)\Br(y)) ≤ ε0/2,

(4) limj→∞E(uj |Bρi (si)\Brj

i

(yji )) = ε0/2,

(5) limj→∞E(uj |Bεji

(si)) ≥ ε0,

(6) limj→∞E(uj − u|Bρ(si)\Bεji

(si)) = 0 ,

where all of the limits exist. We do this for all i.Let the map ec : S → R be defined by

ec(si) = limj→∞

E(uj |Bρi (si))− E(u|Bρi (si)) . (5.3.2)

We denote it the energy concentration map. The map measures the energy build-up around thesingular points. Note that ρi has no special role in the definition; for any r ∈ (0, ρi), the strong

73


rij

ij

si y

ij

Dij~ D

i,i1j~

S2

...

Figure 5.1: The sphere and bubbles.

convergence on compact sets away from S implies

ec(si) = limj→∞

(E(uj |Bρi (si)\Br(si)) + E(uj |Br(si))

)− E(u|Bρi (si)\Br(si))− E(u|Br(si))

= limj→∞

E(uj |Br(si))− E(u|Br(si)) .(5.3.3)

By (6) and the triangle inequality for the L2-norm, we have∣∣∣∣E(uj |Bρ(si)\Bεji

(si))1/2 − E(u|Bρ(si)\B

εji

(si))1/2

∣∣∣∣ ≤ E(uj − u|Bρ(si)\Bεji

(si))1/2 → 0 .

Since limj→∞E(u|Bεji

(si)) = 0 this implies limj→∞E(uj |Bρ(si)\Bεji

(si)) = E(u|Bρi (si)). We then

get the following additional way of measuring the energy concentration:

ec(si) = limj→∞

E(uj |Bρi (si))− E(u|Bρi (si)) = limj→∞

E(uj |Bεji

(si)) .

In particular, by (5), we have ec(s) ≥ ε0 for all s ∈ S. It is the result of Lemma 5.3.6 thatlimj→∞E(uj) = E(u) +

∑s∈S ec(s).

Use Lemma 5.3.7 to get dilations Dji : S2 → S2 satisfying

(7) Dji (S2−) = B

rji(yji ),

(8) uj Dji are still ε-almost harmonic,

(9)((Dj

i )−1(B

εji(si))

)exhausts S2 \ p+ in the limit.

For each i ∈ 1, . . . , |S|, repeat the above construction on the maps uj Dji replacing (uj)

by a subsequence such that uj Dji converges strongly to a harmonic map ui : S2 → M except

on a finite set Si. We then again get maps eci : Si → R. Before iterating the constructionfurther, we need additional observations to ensure only a finite number of iterations are needed.Namely, Lemma 5.3.8 shows that E(ui) +

∑s∈Si\p+ eci(s) ≤ ec(si). Note that we exclude the

north pole from the sum. Then Lemma 5.3.9 asserts that

eci(s) ≤ ec(si)− ε0/6 (5.3.4)

74


for all s ∈ Si \ p+.We now for each i iterate the construction on all points of Si \ p+ getting sets Si,i1 ,

i1 ∈ 1, . . . , |Si|, then on all points of Si,i1 \ p+ getting sets Si,i1,i2 etc. For ease of notation,at this point we exclude the north pole from each of the sets of singular points. Then, sinceec(s) ∈ [ε0, E0] for any singular point at any iteration, the energy decrease (5.3.4) implies thatonly a finite number of iterations can occur. The end result is a finite collection of harmonic mapsu, ui,i1,i2,... | ik ∈ 1, . . . |Si,i1,...,ik−1

|

together with sequences (Dji,i1,i2,...

), (εji,i1,i2,...), (rji,i1,i2,...),and (yji,i1,i2,...).

We now only need a few modifications of these sequences to get the properties (BC1) and(BC2) of bubble convergence. Since each Dj

i,i1,i2,...is conformal, the compositions

Dji,i1,i2,...

= Dji D

ji,i1 Dj

i,i1,i2 · · ·

will suffice as the conformal diffeomorphisms of (BC2). Define Dj0 = IdS2 . We define sets

Ωj0,Ω

ji,i1,i2,...

by

Ωj0 = S2 \ ∪i∈1,...,|S|Bεji (si) = S2 \ ∪i∈1,...,|S|D

j0(B

εji(si)) ,

for each i ∈ 1, . . . , |S|,

Ωji = Dj

0(Bεji

(si)) \ ∪i1∈1,...,|Si|Dji (Bεji,i1

(si,i1))

and inductively, for each ik ∈ 1, . . . , |Si,i1,...,ik−1|,

Ωji,i1,...,ik

= Dji,i1,...,ik−1

(Bεji,i1,...,ik

(si,i1,...,ik)) \ ∪ik+1∈1,...,|Si,1,...,ik |Dji,i1,...,ik

(Bεji,i1,...,ik+1

(si,i1,...,ik+1)) .

Since Ωji,i1,...,ik

may not be open or closed, we replace the sets by their interior. For fixedj ∈ N and each k, the sets Dj

i,i1,...,ik−1(B

εji,i1,...,ik(si,i1,...,ik)), ik ∈ 1, . . . , |Si,i1,...,ik−1

| are disjoint

by the choice of ρi,i1,...,ik . Also, since((Dj

i,i1,...,ik)−1(B

εji,i1,...,ik(si,i1,...,ik))

)exhausts S2 \ p+

by (9), Bεji,i1,...,ik+1

(si,i1,...,ik+1) is contained in (Dj

i,i1,...,ik)−1(B

εji,i1,...,ik(si,i1,...,ik)) except for at

finite number of j’s which we disregard. Hence Dji,i1,...,ik

(Bεji,i1,...,ik+1

(si,i1,...,ik+1)) is contained in

Dji,i1,...,ik−1

(Bεji,i1,...,ik

(si,i1,...,ik)) and then, by the definition of Ωji,i1,...,ik

, the sets are disjoint for

fixed j.Since εji → 0, (Ωj

0),Ωj0 = (Dj

0)−1(Ωj0) exhausts S2 \ S. In the same way, for each i ∈

1, . . . , |S|, the sets

(Dji )−1(Ωj

i ) = (Dji )−1(B

εji(si)) \ ∪i1∈1,...,|Si|Bεji,i1 (si)

exhaust S2 \ Si, since((Dj

i )−1(B

εji(si))

)does and εji,i1 → 0. Performing this argument for

each i, i1, . . . , ik, we get that((Dj

i,i1,...,ik)−1(Ωj

i,i1,...,ik))

exhaust S2 \ Si,i1,...,ik−1. Since we took

subsequences as to ensure that all the limits in question exist, also the limits

limj→∞

E(uj |Ωji,i1,...

) = limj→∞

E(uj Dji,i1,...

|(Dji,i1,...

)−1(Ωji,i1,...))

75


exist.Summing up: We chose u such that uj converges to u strongly except on S. By construction of

Dji,i1,...

and ui,i1,..., (uj Dji,i1,...

) converges strongly to ui,i1,... except on Si,i1,.... By the discussionabove, after relabeling ui,i1,..., D

ji,i1,...

, Ωji,i1,...

, and Si,i1,... we get that (BC1) and (BC2) aresatisfied.

The following lemmata suppose (uj) is an ε-almost harmonic sequence converging stronglyto a harmonic map u except on a finite set S = s1, . . . , sk. We let ε0 = min(ε, εSU ) andec : S → R be the energy concentration map

ec(si) = limj→∞

E(uj |Bρi (si))− E(u|Bρi (si)) ,

where ρi > 0 is as chosen in the proof of Theorem 5.3.2 and we suppose the limits exist.We need the following result in the proof of Lemma 5.3.4.

Lemma 5.3.3. Let f ∈ L1(S2,R) and Brx(x0) ⊆ S2 for some x0 ∈ S2 be a geodesic ball. Then,for any ε ∈ [0,

∫Brx (x0) |f |), there exist y0 ∈ Brx(x0) and a minimal ry ∈ (0, rx] attaining

infr∈(0,rx],y∈cl(Brx−r(x))

∫Brx (x0)\Br(y)

|f | = ε .

Proof. We claim that the maps F : [0, rx]× S2 → R and F : [0, rx]→ R given by

F (r, y) =∫Brx (x0)\Br(y)

|f |

F (r) = infy∈cl(Brx−r(x))

F (r, y)

are continuous. If so, the preimage F−1(ε) is compact, and non-empty since F (0) =∫Brx (x0) |f |] >

ε, F (rx) = 0, and [0, rx] is connected. Thus, there exists a minimal ry in F−1(ε). Since F (0) > ε,ry > 0. The continuity of F and compactness of cl (B)rx−ry (x) imply that a y0 ∈ cl (B)rx−ry (x)exists attaining the infimum in the expression for F (ry). This will prove the lemma.

Suppose (rj , yj)→ (r, y). The maps χBrx (x0)\Brj

(yj)|f | converge pointwise to χBrx (x0)\Br(y)|f |and are dominated by the integrable map |f |. Thus, by the Lebesque Dominated ConvergenceTheorem (see [Roy88, Page 92]),

limj→∞

∫Brx (x0)

χBrx (x0)\Brj

(yj)|f | =∫Brx (x0)

χBrx (x0)\Br(y)|f |

proving the continuity of F .Since [0, rx]×S2 is compact, F is then uniformly continuous. Thus, given ε > 0 and r ∈ [0, rx]

choose δ > 0 such that for all y ∈ S2, |F (r′, y)− F (r, y)| < ε, whenever |r′ − r| < δ. For such r′,again using the continuity of F , let yr′ and yr be such that F (r′, yr′) = F (r′) and F (r, yr) = F (r).Suppose F (r) > F (r′). Using that F (r, yr) ≤ F (r, yr′) by the choice of yr, we have

F (r)− F (r′) = F (r, yr)− F (r, yr′) + F (r, yr′)− F (r′, yr′)< F (r, yr)− F (r, yr′) + ε

≤ ε .

76


We get a similar inequality in the case F (r) < F (r′). Thus we conclude that |F (r)− F (r′)| < εproving the continuity of F .

Lemma 5.3.4. Let si ∈ S and ρi > 0 be such that E(u|Bρi (si)) ≤ ε0/3. Then there exist

sequences (rji ), rji ∈ (0, ρi] and (yji ), y

ji ∈ S2 satisfying

(1) rji → 0 and yji → si,

(2) for each j, no r ∈ (0, rji ) and y ∈ Bρi−r(si) exist such that E(uj |Bρi (si)\Br(y)) ≤ ε0/2,

(3) limj→∞E(uj |Bρi (si)\Brj

i

(yji )) = ε0/2,

except for a finite number of j’s.

Proof. Let V ji ∈ Vj denote the jth high-energy set with points converging to si and vanish-

ing radii as j increases. Disregarding a finite number of j’s, we have V ji ⊆ Bρi(si). Thus,

E(uj |Bρi (si)) ≥ E(uj |V ji

) ≥ ε0 and hence we can use Lemma 5.3.3 to choose rji > 0 and yji ∈ S2

such that rji is the least radius with

infy∈B

ρi−rji

(x)E(uj |Bρi (si)\Brj

i

(y)) = ε0/2

andE(uj |

Bρi (si)\Brji

(yji )) = ε0/2 .

From the choice of rji and yji , we get property (2) and (3).We claim that rji → 0 and yji → si. To see this, recall that uj converges strongly to u on

compact sets K ⊂ S2 \ S. Thus, for any δ > 0, we can choose Jδ such that for all j ≥ Jδ,

E(uj |Bρi (si)\Bδ(si)) < E(u|Bρi (si)\Bδ(si)) + ε0/6≤ E(u|Bρi (si)) + ε0/6≤ ε0/2 .

Since E(uj |Bρi (si)) ≥ ε0, we get E(uj |Bδ(si)) > ε0/2 and hence we can apply Lemma 5.3.3 to getr ∈ (0, δ] and y ∈ S2 with

E(uj |Bδ(si)\Br(y)) = ε0/2− E(uj |Bρi (si)\Bδ(si)) .

Then

E(uj |Bρi (si)\Br(y)) = E(uj |Bρi (si)\Bδ(si)) + ε0/2− E(uj |Bρi (si)\Bδ(si)) = ε0/2 .

By the choice of rji , rji ≤ r ≤ δ for j ≥ Jδ. Since this holds for any δ > 0, we get rji → 0.

If we suppose yji 6→ si then, by compactness of cl (Bρi(si)), there exists a subsequence of (yji )converging to some x ∈ cl (Bρi(si)) \ si. Since rji → 0, eventually B

rji(yji ) ⊂ Bρi(si) \ si.

Thus, for large j, the high-energy sets V ji are disjoint from B

rji(yji ) implying that

E(uj |V ji

) ≤ E(uj |Bρi (si)\Brj

i

(yji )) = ε0/2

contradicting that E(uj |V ji

) ≥ ε0.

77


Lemma 5.3.5. Let si ∈ S and (rji ), (yji ) be the sequences constructed in Lemma 5.3.4. Byreplacing (uj) with a subsequence,2 we obtain a sequence (εji ), ε

ji ∈ (0, ρi] satisfying

(1) εji → 0,

(2) εji/rji →∞ and d(yji , si)/ε

ji → 0,

(3) limj→∞E(uj |Bεji

(si)) ≥ ε0,

(4) limj→∞E(uj − u|Bρ(si)\Bεji

(si)) = 0 ,

where all of the limits exist.

Proof. Let V ji ∈ Vj denote the jth high-energy set with points converging to si and vanishing

radii as j increases. Let (εki ) be a sequence converging to zero such that 0 < εki ≤ min(ρi, 1).Recall that uj converges strongly to u on compact sets K ⊂ S2 \ S. We use this to find asubsequence of (uj) with the properties listed below. We will index this subsequence directly toavoid confusion on the indexes. For each k choose jk ≥ k such that, for all j ≥ jk,

(a) the high-energy sets V ji ⊆ Bεki (si),

(b) the balls Brji

(yji ) ⊆ B(εki )2(si), and hence rji , d(yji , si) ≤ (εki )2.

(c)∣∣∣∣E(uj − u|Bρ(si)\Bεk

i(si))

∣∣∣∣ < εki .

In addition, when choosing the subsequence, we can ensure limk→∞E(ujk |Bρ(si)\Bεki

(si)) and

limk→∞E(ujk |Bεki

(si)) exist.

By construction (εki ) satisfies property (1). After shifting indexes on (rji ) and (yji ), Property(2) will hold since

εki

rjki≥ εki

(εki )2=

1εki→∞

and equivalently for d(yjki , si)/εki . Property (3) holds since V jk

i ⊆ Bεki (si) implies E(ujk |Bεki

(si)) ≥

E(ujk |Vjki

) ≥ ε0. The equality in Property (4) follows immediately from∣∣∣∣E(ujk − u|Bρ(si)\Bεki

(si))∣∣∣∣ < εki → 0

by (c).Replacing (uj) with (ujk), (rji ) with (rjki ) and (yji ) with (yjki ) finishes the lemma.

Lemma 5.3.6. Let I1, I2 be a partition of 1, . . . , |S|. Then, for any compact K ⊆ S2 with∪i∈I1si ⊂ int (K) and ∪i∈I2si ∩K = ∅, we have

limj→∞

E(uj |K) = E(u|K) +∑i∈I1

ec(si) .

2As usual we will also replace (rji ) and (yji ) with the corresponding subsequences.

78


In particularlimj→∞

E(uj) = E(u) +∑s∈S

ec(s) .

Proof. Let ri ∈ (0, ρi) be such that Bri(si) ⊂ K. Since uj |K\∪i∈I1Bri (si) converges strongly tou|K\∪i∈I1Bri (si), Lemma 2.4.6 implies that

limj→∞

E(uj |K\∪i∈I1Br(si)) = E(u|K\∪i∈I1Bri (si)) .

Thus

limj→∞

E(uj |K) = limj→∞

E(uj |K\∪i∈I1Bri (si)) + limj→∞

E(uj |∪i∈I1Bri (si))

(5.3.3)= E(u|K\∪i∈I1Bri (si)) + E(u|∪i∈I1Bri (si)) +

∑i∈I1

ec(si)

= E(u|K) +∑i∈I1

ec(si) .

Lemma 5.3.7. Let (eji ), (rji ) and (yji ) be the sequences constructed in Lemma 5.3.4 and 5.3.5.Then there exist dilations Dj

i : S2 → S2 such that

(1) Dji (S2−) = B

rji(yji ),

(2) (uj Dji ) is still ε-almost harmonic,

(3)((Dj

i )−1(B

εji(si))

)exhausts S2 \ p+ in the limit.

Proof. We let Dji be the inverse of a dilation (Πp−)−1 Dλj Π

yjiwhere λj > 0 is such that the

image of Brji

(yji ) is S2−. Thus (1) follows by construction. That Property (2) holds is the result

of Lemma 5.2.5.We show Property (3) by showing that the distances from the circles ∂Π

yji

((Dj

i )−1(B

εji(si))

)⊂

R2 to the origin diverge. We cannot just consider the radii of the circles since the circlesmight not be centered at the origin. Let pj ∈ ∂B

εji(si)) be such that Π

yji(pj) attain the de-

scribed distances. We then wish to show that ‖Πp− (Dji )−1(pj)‖R2 diverges. By definition,

Πp− (Dji )−1 = Dλj Πp− R, where R ∈ O(3) is a rotation taking yji to p−. Since R is an

isometry and Dji sends S2

− to Brji

(yji ), (2.1.2) implies λj = (1 + cos rji )/ sin rji . Using this and(2.1.2) again,

‖Πp− (Dji )−1(pj)‖R2 =

1 + cos rjisin rji

sin d(pj , yji )

1 + cos d(pj , yji ).

Since rji → 0, we only need to show that sin d(pj , yji )/ sin rji diverge. By Property (2) ofLemma 5.3.5,

d(pj , yji )

rji≥d(pj , si, )− d(si, y

ji )

rji=εjirji

(1−

d(si, yji )

εji

)→∞

79


so that Taylor expanding sin and using that rji , d(pj , yji )→ 0 implies

sin d(pj , yji )

sin rji=d(pj , yji )

rji

1− d(pj ,yji )2

3! + d(pj ,yji )4

5! − · · ·

1− (rji )2

3! + (rji )4

5! − · · ·→ ∞ .

Lemma 5.3.8. Let Dji be the maps constructed in Lemma 5.3.7 and si ∈ S. Suppose (uj Dj

i )converges strongly to ui except on Si and that the energy concentration map eci is defined. Then

E(ui) +∑

s∈Si\p+

eci(s) ≤ ec(si) .

Proof. Given ε > 0 let K be a compact subset of S2\p+ with E(ui|S2\K) < ε and ∪s∈Si\p+scontained in int (K). By the result of Lemma 5.3.7, for sufficiently large j, K ⊆ (Dj

i )−1(B

εji(si))

and thus E(uj Dji |K) ≤ E(uj |B

εji

(si)). By Lemma 5.3.6,

E(ui) +∑

s∈Si\p+

eci(s) < E(ui|K) + ε+∑

s∈Si\p+

eci(s)

Lem. 5.3.6= limj→∞

E(uj Dji |K) + ε

≤ limj→∞

E(uj |Bεji

(si)) + ε

= ec(si) + ε .

Since this holds for any ε > 0, the result follows.

Lemma 5.3.9. Let si ∈ S and Dji be the maps constructed in Lemma 5.3.7. Suppose (uj Dj

i )converges strongly to ui except on Si, that the energy concentration map eci is defined, and thatLemma 5.3.5 has been applied to (uj Dj

i ). Let ρi > 0 be such that E(u|Bρi (si)) ≤ ε0/3. Then

eci(s) ≤ ec(si)− ε0/6

for all s ∈ Si \ p+.

Proof. Let si,i1 ∈ Si\p+. Though we can make the argument without considering the positionof si,i1 on S2, we divide it into cases to make the strategy clearer. We let (rji ), (yji ) be thesequences of Lemma 5.3.4, and (eji,i1) the sequence constructed when applying Lemma 5.3.5 to(uj Dj

i ) so that eci(si,i1) = limj→∞E(uj Dji |B

εji,i1

(si,i1 )). Recall that eci(si,i1) ≥ ε0.

First, we claim that si,i1 ∈ cl(S2−); if not, eventually B

εji,i1(si,i1) ⊂ S2

+ and Dji (Bεji,i1

(si,i1)) ⊂

Bρi(si) \Brji (yji ). Since B

εji,i1(si,i1) has high energy we obtain a contradiction from Property (3)

of Lemma 5.3.4;

ε0/2 = limj→∞

E(uj |Bρi (si)\Brj

i

(yji )) ≥ lim

j→∞E(uj |

Dji (Bεji,i1

(si,i1 ))) = eci(si,i1) ≥ ε0 .

80

Ruling out energy loss

Now, if si,i1 ∈ int(S2−)

eventually Bεi,i1 (si,i1) ⊂ S2−. Thus Dj

i (Bεi,i1 (si,i1)) ⊂ Brji

(yji ) and, againusing Property (3) of Lemma 5.3.4, we get

eci(si,i1) = limj→∞

E(uj |Dji (Bεi,i1

(si,i1 )))

= limj→∞

E(uj |Bρi (si))− limj→∞

E(uj |Bρi (si)\D

ji (Bεi,i1

(si,i1 )))

Lem. 5.3.4(5.3.2)

≤ ec(si) + E(u|Bρi (si))− ε0/2≤ ec(si)− ε0/6

proving the result.If si,i1 is on the equator of S2 the above argument does not work; the balls Bεi,i1 (si,i1) could

intersect both S2+ and S2

− possibly having ε0/2 energy on the former part of the intersectionand ec(si) − ε0/2 on the latter. We rule out this case by using the minimality of rji ; supposeeci(si,i1) > ec(si)− ε0/6. Then

limj→∞

E(uj |Bρi (si)\D

ji (Bεi,i1

(si,i1 ))) = lim

j→∞E(uj |Bρi (si))− lim

j→∞E(uj |

Dji (Bεi,i1(si,i1 ))

)

= ec(si) + E(u|Bρi (si))− eci(si,i1)< ε0/2 .

Hence, eventually E(uj |Bρi (si)\D

ji (Bεi,i1

(si,i1 ))) ≤ ε0/2. Since rji were chosen to be the least radii

of balls with this property, it suffices to show that Dji (Bεi,i1 (si,i1)) has radius strictly less than

rji to obtain a contradiction. Since si,i1 is on the equator of S2, the center of Dji (Bεi,i1 (si,i1))

is in ∂Brji

(yji ). If the radii of Dji (Bεi,i1 (si,i1)) are always at least rji , the distances between

Dji (Bεi,i1 (si,i1)) and yji are zero. Hence, the distances between Bεi,i1 (si,i1) and the south pole of

S2 are zero. But this contradicts that si,i1 is on the equator and εi,i1 → 0.

5.3.3 Ruling out energy loss

By using the construction in the previous subsection and the notation introduced there, andafter having applied Theorem 5.3.2 to a sequence (uj), we have

limj→∞

E(uj |Bεji

(si)) = ec(si)

andlimj→∞

E(uj |S2\∪si∈SBεji

(si)) = E(u) .

In other words, the energy is accounted for in u and ec(si). To get Property (BC3) of bubbleconvergence, we thus need to ensure that, for each i, the energy ec(si) is accounted for in themap ui and the singular points in Si. What we need to rule out is that energy “disappears atp+”, that is, that uj Dj

i concentrates more energy than ui can account for in progressivelysmaller neighborhoods of p+. Assuming this does not happen, we have

ec(si) = E(ui) +∑

s∈Si\p+

ec(s)

81


and inductively we getlimj→∞

E(uj) = E(u) +∑i

E(ui) .

The rough idea to rule out energy loss at p+ is as follows. Excess energy concentrationaround p+ implies - using Dj

i - that energy concentrates in annuli around yji . It turns out thatsuch energy concentration, under the circumstances present here, is not possible for conformalmaps. In fact, even maps getting more and more conformal in the way defined below cannotintroduce such energy concentration. We thus require our maps to be conformal in this way,rule out energy concentration in those annuli, and thus energy loss at p+.

The impossibility of energy concentration in annuli will be proved using conformal equiva-lence between cylinders and annuli, almost conformality of the maps in question, and an analyticproperty of almost harmonic maps from cylinders. We discuss those three properties, beforeturning to the main proof.

For x ∈ R2 and r2 > r1 > 0 we let an annulus Ar1,r2(x) ⊂ R2 be the set Br2(x) \ cl (Br1(x)).We let the manifold S1 × R ⊂ R3 be Riemannian with the induced flat metric from R3. Acylinder Ch1,h2 ⊂ S1 × R, h1 < h2 is a subset ∪h∈(h1,h2)S1 × h.

Let the map C : S1 × R→ R2 \ 0 be given by

(θ, h) 7→ (θ, eh) ,

using the chart (θ, h) 7→ (sin θ, cos θ, h) on S1×R and polar coordinates on R2. Then clearly theimage of C applied to a cylinder is an annulus. Furthermore, C is conformal; the chart abovedefines conformal coordinates, and since ∂θC = eh(cos θ∂x − sin θ∂y) and ∂hC = eh(sin θ∂x +cos θ∂y) we have 〈∂θC, ∂θC〉 = 〈∂hC, ∂hC〉 = e2h and 〈∂θC, ∂hC〉 = 0. Using C we can pull backmaps Ar1,r2(x) → M to maps Cln r1,ln r2 → M . Since C is conformal, the energy is unaffectedby this. We can, in addition, define annuli Ar1,r2(p) = Br2(p) \ cl (Br1(p)) ∈ S2 for p ∈ S2 andr2 > r1 > 0. Using a stereographic projection, such annuli are conformally equivalent to annuliin R2. Thus, for p ∈ S2 we will let Cp : S1 × R → S2 \ p be the map Π−1

p C. Using Cp, wecan pull back maps u : Ar1,r2(p)→M to cylinders Ch1,h2 . Again, the energy is invariant underthe pullback.

The following definition of almost conformal sequences is inspired from the fact that Area(u) ≤E(u) with equality if and only if u is conformal almost everywhere, confer Lemma 2.4.1. Recallthat, if u : Ch1,h2 → M is conformal almost everywhere, since (θ, h) are conformal coordinates,〈∂θu, ∂θu〉M is equal to 〈∂hu, ∂hu〉M almost everywhere. As we will see, the definition allows usto show that 〈∂θu, ∂θu〉M and 〈∂hu, ∂hu〉M cannot differ much on cylinders, if u is the pullbackof a map in an almost conformal sequence.

Definition 5.3.10 (Almost conformal sequences). A sequence of W 1,2(S2,M) maps (uj) is saidto be almost conformal if, for all j,

0 ≤ E(u)−Area(uj) ≤ 1j.

Recall that both area and energy are conformally equivalent. Thus, composing with dila-tions, as done in the iteration steps when proving bubble convergence, preserves the almostconformality of a sequence.

Lemma 5.3.11. Assume (uj) is almost conformal and let C, δ > 0 with C ≥ δ. Then, forsufficiently large j, there exist no cylinder Ch1,h2 and pullback uj : Ch1,h2 → M , uj = uj Cpj

82


with δ ≤ E(uj) ≤ C and12

∫Ch1,h2

‖∂θuj‖2 <13E(uj) .

Proof. Fix j ∈ N with δ ≤ E(uj) ≤ C and suppose δ ≤ E(uj) ≤ C and 12

∫Ch1,h2

‖∂θuj‖2 <13E(uj). We will find an upper bound for j.

Note that ∂θ and ∂h constitute an orthonormal frame and hence ‖∂θuj‖2 +‖∂huj‖2 = ‖duj‖2.Using this we have

E(uj)−Area(uj) =∫Ch1,h2

12‖duj‖2 − Juj

=∫Ch1,h2

12‖duj‖2 −

(‖∂θuj‖2‖∂huj‖2 −

⟨∂θu

j , ∂huj⟩2)1/2

≥∫Ch1,h2

12‖duj‖2 − ‖∂θuj‖‖∂huj‖

=∫Ch1,h2

12‖duj‖2 − 1

2(‖∂θuj‖2 + ‖∂huj‖2) +

12

(‖∂θuj‖ − ‖∂huj‖)2

=∫Ch1,h2

12

(‖∂θuj‖ − ‖∂huj‖)2 .

(5.3.5)

By the upper bound on the energy,∫Ch1,h2

12

(‖∂θuj‖+ ‖∂huj‖)2Lem. C.1.4≤

∫Ch1,h2

‖∂θuj‖2 + ‖∂huj‖2 ≤ 2C . (5.3.6)

Using that ‖∂θuj‖2 − ‖∂huj‖2 = (‖∂θuj‖ + ‖∂huj‖)(‖∂θuj‖ − ‖∂huj‖) and ‖∂θuj‖ + ‖∂huj‖ isnon-zero whenever ‖∂θuj‖2 − ‖∂huj‖2 is, we get, using Cauchy-Schwartz,

E(uj)−Area(uj)(5.3.5)

≥∫Ch1,h2

12

(‖∂θuj‖ − ‖∂huj‖)2

(5.3.6)

≥ 14C

∫Ch1,h2

(‖∂θuj‖+ ‖∂huj‖)2 12

∫Ch1,h2

(‖∂θuj‖2 − ‖∂huj‖2

‖∂θuj‖+ ‖∂huj‖

)2

C-S≥ 1

8C

(∫Ch1,h2

‖∂θuj‖2 − ‖∂huj‖2)2

.

(5.3.7)

Conversely, by the assumption,

12

∫Ch1,h2

‖∂θuj‖2 <13E(uj) =

16

∫Ch1,h2

‖duj‖2 =16

∫Ch1,h2

‖∂θuj‖2 + ‖∂huj‖2 .

Hence∫Ch1,h2

‖∂θuj‖2 < 12

∫Ch1,h2

‖∂huj‖2 implying 4δ3 <

∫Ch1,h2

‖∂huj‖2 using the lower δ energybound. Thus ∫

Ch1,h2

‖∂huj‖2 − ‖∂θuj‖2 >12

∫Ch1,h2

‖∂huj‖2 >2δ3. (5.3.8)

Combining (5.3.7) and (5.3.8), we get

E(uj)−Area(uj)(5.3.7)

≥ 18C

(∫Ch1,h2

‖∂θuj‖2 − ‖∂huj‖2)2

(5.3.8)>

δ2

18C.

83


But the sequence (uj) is almost conformal. Thus, using the conformality of Cpj ,

0 ≤ E(uj)−Area(uj) ≤ E(uj)−Area(uj) < 1/j .

Hence j < 18Cδ2

.

The following lemma contrasts the above result, by showing that, for sequences of almostharmonic maps from cylinders,

⟨∂θu

j , ∂θuj⟩M

and⟨∂hu

j , ∂huj⟩M

do in fact differ. The ac-tual proof is given in [CM07a, Proposition B.29]. Here we only argue to make that result fitour framework. In particular, we make the choice of not defining a separate notion of almostharmonic maps from cylinders as [CM07a] does.

Lemma 5.3.12. There exist εC(M, g) > 0 and l(M, g) ≥ 1 so that, for any µ > 0, thereexists J(M, g, µ) ∈ N with the following property: if m ∈ N, (uj), uj : S2 → M is a ε-almostharmonic sequence with (uj), uj = uj Cpj corresponding pullbacks to a cylinder C−(m+3)l,3l,and µ ≤ E(uj) ≤ min(ε, εC) then, for j ≥ J ,

12

∫C−ml,0

‖∂θuj‖2 ≤16E(uj) . (5.3.9)

Proof. Let δ = 1/42. By [CM07a, Proposition B.29] there exist ν(M, g, δ), εC(M, g, δ) > 0 andl(M, g, δ) ≥ 1 so that if m ∈ N and u ∈ W 1,2(C−(m+3)l,3l,M) with E(u) ≤ εC is a map withthe property that for any finite collection of disjoint closed balls B in C(C−(m+3)l,3l) ⊂ R2 thereexists an energy minimizing map v : rB→M 3 equaling u C−1 on ∂rB with

E(u C−1 − v|rB) ≤ ν

2E(u) , (5.3.10)

then ∫C−ml,0

‖∂θu‖2 ≤ 7δ∫C−(m+3)l,3l

‖du‖2 . (5.3.11)

Here we used the shorthand rB for the collection of balls in B having their radii shrunk by r. Ifm,µ, (uj), (uj) have the required properties then since any such B corresponds to a collection ofprojected balls and E(uj Π−1

pj|C(C−(m+3)l,3l)) ≤ ε by assumption, for large enough j, there exists

a minimizing map vj : rB→M equaling uj Π−1pj

on ∂rB with

E(uj Π−1pj− vj |rB) ≤ νµ

2.

Thus, by the lower bound on E(uj), (5.3.10) is satisfied. Plugging in δ = 1/42 in (5.3.11) givesthe result.

Notice that the two cylinders over which we integrate in (5.3.9) are of different lengths. Weremedy this in the following corollary by showing that if the energy in one end of the cylinderis low, and the θ-energy on the entire cylinder is high, then the overall energy is great.

3In [CM07a] the proof is carried out with r = 1/8, but this particular choice of r is not important.

84


Corollary 5.3.13. Let µ > 0, m1,m2 ∈ N, and suppose uj satisfies the assumptions ofLemma 5.3.12 with m = m1+3m2. Suppose in addition that E(uj |C−(m1+3)l,3l

) ≥ µ, E(uj |C0,3l) ≤

12E(uj |C−m1l,0

), and, for any i ∈ 1, . . . ,m2,

12

∫C−(m1+3(i−1))l,0

‖∂θuj‖2 ≥13E(uj |C−(m1+3(i−1))l,0

) . (5.3.12)

Then

E(uj |C−(m1+3m2)l,0) ≥

(32

)m2

E(uj |C−m1l,0) .

Proof. We will prove that

E(uj |C−(m1+3i)l,0) ≥ 3

2E(uj |C−(m1+3(i−1))l,0

)

for any i ∈ 1, . . . ,m2. This will prove the result.From Lemma 5.3.12 we get

12

∫C−(m1+3(i−1))l,0

‖∂θuj‖2(5.3.9)

≤ 16E(uj |C−(m1+3i)l,3l

) . (5.3.13)

By this, (5.3.12) and since, by assumption, E(uj |C0,3l) ≤ 1

2E(uj |C−m1l,0) ≤ 1

2E(uj |C−(m1+3(i−1))l,0),

∫C−(m1+3(i−1)),0

‖∂θuj‖2(5.3.13)

(5.3.12)

≤ 13E(uj |C−(m1+3i)l,−(m1+3(i−1))l

) +12

∫C−(m1+3(i−1))l,0

‖∂θuj‖2

+16E(uj |C−(m1+3(i−1)l),0

) .

(5.3.14)

Hence

12

∫C−(m1+3(i−1)),0

‖∂θuj‖2(5.3.14)

≤ 13E(uj |C−(m1+3i)l,−(m1+3(i−1))l

)+16E(uj |C−(m1+3(i−1))l,0

) . (5.3.15)

Again using the assumptions,

13E(uj |C−(m1+3(i−1)),0

)

(5.3.15)

(5.3.12)

≤ 13E(uj |C−(m1+3i)l,−(m1+3(i−1))l

) +16E(uj |C−(m1+3(i−1))l,0

) (5.3.16)

so that12E(uj |C−(m1+3(i−1))l,0

)(5.3.16)

≤ E(uj |C−(m1+3i)l,−(m1+3(i−1))l) . (5.3.17)

Thus finally,

E(uj |C−(m1+3i)l,0) = E(uj |C−(m1+3i)l,−(m1+3(i−1))l

) + E(uj |C−(m1+3(i−1))l,0)

(5.3.17)

≥ 32E(uj |C−(m1+3(i−1))l,0

) .

85


The main ingredients for ruling out energy loss are now ready, though we postpone identifyingannuli, on which energy concentrates, to a lemma following the proof. We will need scaledversions of Cp given by

Cλp = Dp,λ Cp ,

where λ > 0 is chosen so that Cλp(θ, 0) ∈ ∂Bλ(p). These maps are also conformal and work just

as well as Cp in the above lemmata. We let εC be the constant of Lemma 5.3.12.

Theorem 5.3.14. For ε > 0 and E0 ≥ 0 let (uj), uj : S2 → M be a sequence of ε-almostharmonic maps with E(uj) ≤ E0. Let ε = min(ε, εC) so that (uj) is also ε-almost harmonic.Suppose (uj) has been replaced by the subsequence constructed in Theorem 5.3.2 so that thereexists a finite collection of harmonic maps u, u1, . . . , ul, u, ui : S2 →M satisfying the properties(BC1) and (BC2) of bubble convergence. Assume in addition that (uj) is almost conformal. Then

ec(si) = E(ui) +∑

s∈Si\p+

eci(s) (5.3.18)

for all singular points si ∈ S.

Proof. Let (eji ), (rji ) and (yji ) be the sequences constructed in Lemma 5.3.4 and 5.3.5. Weargue by contradiction. Thus, suppose si ∈ S and (5.3.18) fails. Use Lemma 5.3.16 to get asubsequence (ujk) of (uj) such that we have annuli Abk,ak(yjki ) ⊂ S2 and δ > 0 satisfying

(1) ak, bk → 0 ,

(2) bk/rjki →∞ ,

(3) E(ujk − u|Bρi (y

jki )\B

ak(yjki )

)→ 0 ,

(4) E(ujk |Abk,ak

(yjki )

) ≥ δ .

Since ak → 0, for k large, Cak

yjki

(C0,3l)) ⊂ Bρi(yjki ) \ Bak(yjki ). Again, since ak → 0 also

E(u|Cak

yjki

(C0,3l))→ 0 and hence Property (3) implies that, for large k,

E(ujk |Cak

yjki

(C0,3l)) ≤ δ/2 . (5.3.19)

For all k let mk1 ∈ N be the least such that Abk,ak(yjki ) ⊂ Cak

yjki

(C−mk1 l,0). Then

E(ujk |Cak

yjki

(C−(mk1+3)l,3l)) ≥ E(ujk |

Cak

yjki

(C−mk1 l,0)) ≥ E(ujk |

Aak,bk

(yjki )

) ≥ δ . (5.3.20)

If we, for some large k and mk2 ∈ N, suppose

E(ujk |Cak

yjki

(C−(mk1+3mk2+3)l,3l)) ≤ ε (5.3.21)

and12

∫C−(mk1+3(i−1))l,0

‖∂θ(ujk Cak

yjki

)‖2 ≥ 13E(ujk |

Cak

yjki

(C−(mk1+3(i−1))l,0)) (5.3.22)

86


for all i ∈ 1, . . . ,mk2, then (5.3.19), (5.3.20), (5.3.21), and (5.3.22) will allow us to use

Corollary 5.3.13 to get

E(ujk |Cak

yjki

(C−(mk1+3mk2)l,0))

Cor. 5.3.13≥

(32

)mk2E(ujk |

Cak

yjki

(C−mk1 l,0))

(5.3.20)

≥(

32

)mk2δ . (5.3.23)

Recall that Lemma 5.3.4 implies that E(ujk |Cak

yjki

(C−(mk1+3mk2+3)l,3l)) ≤ ε/2 when

Cak

yjki

(C−(mk1+3mk2+3)l,3l) ⊆ Bρi(si) \Brjki(yjki ) .

Thus, using that bk/rjki → ∞ and since mk1 was chosen to be the least with the required

properties, we can, for any n ∈ N, choose Kn ∈ N such that, for all k ≥ Kn, (5.3.21) holds withmk

2 = n. But for mk2 large, (5.3.23) then contradicts the upper bound. Thus, (5.3.22) must fail

for such k and therefore there exists ik ∈ 1, . . . ,mk2 such that

12

∫C−(mk1+3(ik−1))l,0

‖∂θ(ujk Cak

yjki

)‖2 < 13E(ujk |

Cak

yjki

(C−(mk1+3(ik−1))l,0)) .

Note that stillδ ≤ E(uj |

Abk,ak

(yjki )

) ≤ E(uj |Cak

yjki

(C−(mk1+3ik)l,0)) ≤ ε/2 .

Since (uj) is almost conformal, this contradicts Lemma 5.3.11. Thus, the assumption that(5.3.18) fails, cannot hold.

Using Theorem 5.3.14 at the singular points of each bubble, we get the bubble convergenceof a subsequence:

Corollary 5.3.15 (Bubble compactness). Let (M, g) be a compact manifold. For E0 ≥ 0 andε > 0 let (uj), uj : S2 → M be an almost conformal sequence of ε-almost harmonic mapswith E(uj) ≤ E0. Then a subsequence bubble converges to a finite collection of harmonic mapsu, u1, . . . , ul, u, ui : S2 →M .

Proof. Use Theorem 5.3.2 to replace (uj) with a subsequence such that (uj) almost bubbleconverges to a collection of harmonic maps u, u1, . . . , ul, u, ui : S2 →M in the sence that theproperties (BC1) and (BC2) of bubble convergence are satisfied.

By Lemma 5.3.6, limj→∞E(uj) = E(u) +∑

si∈S ec(si), and from Theorem 5.3.14 we get foreach i ∈ 1, . . . , |S| that ec(si) = E(ui) +

∑s∈Si\p+ eci(s). Hence

limj→∞

E(uj) = E(u) +∑si∈S

E(ui) +∑

s∈Si\p+

eci(s)

.

Applying Theorem 5.3.14 inductively to the energy concentrations of Si, Si,i1 etc., we get

limj→∞

E(uj) = E(u) +∑

E(ui,i1,...) .

This equation is exactly Property (BC3).

87


Lemma 5.3.16. Suppose (uj) is an ε-almost harmonic sequence converging strongly to a har-monic map u except on a finite set S = s1, . . . , sk. Let si ∈ S, Dj

i be the maps constructed inLemma 5.3.7. Suppose (uj Dj

i ) converges strongly to ui except on Si, that the energy concen-tration map eci is defined, and that Lemma 5.3.5 has been applied to (uj Dj

i ). Let (rji ), (yji ) bethe sequences of Lemma 5.3.4. If we assume

ec(si) 6= E(ui) +∑

s∈Si\p+

eci(s) (5.3.24)

then there exist a subsequence (ujk), annuli Abk,ak(yjki ) ⊂ S2 and δ > 0 satisfying

(1) ak, bk → 0 ,

(2) bk/rjki →∞ ,

(3) E(ujk − u|Bρi (y

jki )\B

ak(yjki )

)→ 0 ,

(4) E(ujk |Abk,ak

(yjki )

) ≥ δ .

Proof. We first explicitly construct a subsequence (ujk) on which we prove the first three prop-erties. The last will follow from (5.3.24) when passing to a further subsequence. We let (eji,i1)be the sequence constructed when applying Lemma 5.3.5 to (uj Dj

i ).We begin by choosing a sequence of balls (Bk) in S2 centered at p− and exhausting S2\p+.

One can let Bk = (Dki )−1(B√

rki(yki )), but the particular choice is not important. Note that by

Lemma 5.3.7,((Dj

i )−1(B

εji(si))

)also exhausts S2 \ p+. Use this and Lemma 5.3.6 to, for each

k ∈ N, choose jk ≥ k such that for all j ≥ jk,

Bk ⊆ (Dji )−1(B

εji(si)) (5.3.25)

and ∣∣∣∣E(uj Dji |Bk)−

(E(ui|Bk) +

∑s∈Si\p+

eci(s))∣∣∣∣ < εki .

For any µ > 0 choose Kµ ∈ N such that, for all k ≥ Kµ, E(ui|S2\Bk) < µ/2 and εki < µ/2. Then,for k ≥ Kµ,∣∣∣∣E(ujk Djk

i |(Djki )−1(Bεjki

(si)))−

(E(ui) +

∑s∈Si\p+

eci(s))∣∣∣∣

≤ E(ujk Djki |(Djki )−1(B

εjki

(si))\Bk)

+∣∣∣∣E(ujk Djk

i |Bk)−(E(ui|Bk) +

∑s∈Si\p+

eci(s))∣∣∣∣+ E(ui|S2\Bk)

≤ E(ujk Djki |(Djki )−1(B

εjki

(si))\Bk) + µ .

88

Varifold convergence

Since this holds for any µ > 0, we conclude that∣∣∣∣ limk→∞

E(ujk Djki |(Djki )−1(B

εjki

(si)))−

(E(ui) +

∑s∈Si\p+

eci(s))∣∣∣∣

≤ lim supk→∞


εjki

(si))\Bk) .

By (5.3.24), we thus have lim supk→∞E(ujk Djki |(Djki )−1(B

εjki

(si))\Bk) ≥ 2δ for some δ > 0.

Now, let ak = εjki + d(si, yjki ) such that B

εjki

(si) ⊆ Bak(yjki ). Then also

lim supk→∞


ak(yjki ))\Bk) ≥ 2δ . (5.3.26)

Since both εjki → 0 and d(si, yjki ) → 0, ak → 0. The reason for using the ak’s is that the balls

(Djki )−1(Bak(yjki )) and Bk now both are centered in p−. By (5.3.25), the latter is a subset of

the former. We apply Djki to the balls, getting annuli

Abk,ak(yjki ) = Bak(yjki ) \ Djki (Bk) .

Since ak → 0 also bk → 0. Since Bk exhausts S2 \ p+ and bk is the radius of the ball Djki (Bk),

we get bk/rjki → ∞. In Lemma 5.3.5, we saw that limk→∞E(ujk − u|Bρ(si)\Bεjki

(si)) = 0. Since

Bεjki

(si)) ⊆ Bak(yjki ) we have E(uj − u|Bρ(si)\Bak (y

jki )

) ≤ E(uj − u|Bρ(si)\Bεki

(si)) and hence

limj→∞E(uj − u|Bρ(si)\Bak (y

jki )

) = 0. By the lower bound (5.3.26), there exists a subsequence

of (ujk) satisfying Property (4). This concludes the proof.

5.4 Varifold convergence

We now aim at using bubble convergence to show a result regarding convergence of integrallimits which will be used to prove (3.3.1) and (3.3.2) in Theorem 3.3.1. We will need to considerthe Grassmannian bundle G2M , confer Appendix B.1.

Let Σ be a surface and v ∈ W 1,2(Σ,M). We define a map V : p ∈ Σ | Jv(p) 6= 0 → G2Mby p 7→ dv(TpS2). If h : G2M → R is a map such that

∫p∈Σ | Jv(p) 6=0(h V )Jv is defined, it

makes sense to consider the integral∫vh V =

∫Σ

(h V )Jv

since the integral is independent of a chosen extension of V to Σ. Let Σ2 be another surface.A sequence (vj), vj ∈ W 1,2(Σ,M) is said to varifold converge to a map v : Σ2 → M if, for allh ∈ C0(G2M,R),

limj→∞

∫Σ

(h V j)Jvj =∫

Σ2

(h V )Jv (5.4.1)

with V j defined similarly to V . Note that we do not need the surfaces to be connected for thedefinition to make sense. Varifolds, a varifold metric, and the related topology can be definedin a more general setting than needed here. See [CM07a, Page 5] for more details.

89


In Theorem 3.3.1 we will use varifold convergence in the following setting. We will letΣ = S2 and Σ2 =

∐li=0 S2 be a disjoint union of a finite number of spheres. We consider a

sequence (vj), vj ∈ C0 ∩W 1,2(S2,M) bubble converging to a finite collection of harmonic mapsu, u1, . . . , ul, ui : S2 → M . These correspond in a natural manner to the map v : Σ2 → M .We will show that vj varifold converges to v and use this convergence with two particular choicesof h.

We prove the varifold convergence leaving out some details to lemmata following the proof.Note that the energy preservation Property (BC3) of bubble convergence plays a crucial role inthe proof; we use Lemma 5.4.3 and Lemma 5.4.5 which depend on it.

Theorem 5.4.1 (Bubble convergence implies varifold convergence). Let (M, g) be a compactmanifold isometrically embedded in Rm. Suppose the sequence (vj), vj ∈ W 1,2(S2,M) bubbleconverges to a finite collection of maps u, u1 . . . , ul, ui ∈ C1(S2,M). Let V j , Ui be the mapsp 7→ dvj(TpS2), p 7→ dui(TpS2) respectively. Suppose h : G2M → R is continuous. Then

limj→∞

∫vjh V j =

∑i

∫ui

h Ui . (5.4.2)

Proof. Let (Ωj0, . . . ,Ω

jl ), Ωj

i ⊆ S2 be the sequence of collections of disjoint sets that havetheir existence asserted by Property (BC2) of bubble convergence. Since M is compact, G2Mis compact, so that ‖h‖∞ = supW∈G2M h(W ) < ∞. Using Lemma 5.4.3 to bound the energyoutside ∪iΩj

i , we have∣∣∣∣∣∫

S2\∪iΩji(h V j)Jvj

∣∣∣∣∣ ≤∫

S2\∪iΩji

∣∣(h V j)Jvj∣∣ ≤ ‖h‖∞Area(vj |S2\∪iΩji

)

≤ ‖h‖∞E(vj |S2\∪iΩji)→ 0 .

Thus, limj→∞∫

S2(h V j)Jvj = limj→∞∫∪iΩji

(h V j)Jvj and hence (5.4.2) reduces to

limj→∞

∫∪iΩji

(h V j)Jvj =∑i

∫S2

(h Ui)Jui , (5.4.3)

which follows if, for each i,

limj→∞

∫Ωji

(h V j)Jvj =∫

S2

(h Ui)Jui . (5.4.4)

For ease of notation, let vji be the maps vj Dji , V

ji the maps p 7→ dvji (TpS2) = dvj(T

Dji (p)S2),

and Jvji

the maps Jvjiχ

(Dji )−1(Ωji )

. Since((Dj

i )−1(Ωj

i ))

exhausts S2 \ Si in the limit and by

Lemma 5.4.3, limj→∞E(vj |Ωji

) = E(ui), Corollary 5.4.6 gives Jvji

L1(S2,R)→ Jui . Let Ωεi be the

setsx ∈ S2 | Jui(x) ≥ ε

. By Lemma 5.4.7 it suffices to show that

limj→∞

∫Ωεi

(h V ji )J

vji=∫

Ωεi

(h Ui)Jui (5.4.5)

for all ε > 0 to obtain (5.4.4) and complete the proof. The reason for doing this trick is thatlower bounds on the Jacobian determinant enable us to control the distance between V j

i and Ui

90


in G2M ; as shown in Lemma B.1.2, if Ju(x), Jvji

(x) ≥ ε/2 then there exists C(ε) ∈ R such that

dG2M (V ji (x), Ui(x)) ≤ C

(‖vji (x)− ui(y)‖+ ‖dvji (x)− dui(x)‖2

)1/2. (5.4.6)

Thus, let ε > 0. We will need the notion of convergence in measure, confer Definition C.2.1. Weaim at showing that dG2M (V j

i , Ui)measure→ 0 on Ωi,ε and then applying Theorem C.2.5. Thus,

given δ, µ > 0 we want J ∈ N such that the measure of the setsx ∈ Ωi,ε | dG2M (V j

i (x), Ui(x)) > δ

(5.4.7)

is less than µ for j ≥ J . We do this by considering different subsets separately.

First, since Jvji

L1(S2,R)→ Jui , by Lemma C.2.2, Jvji

measure→ Jui on S2. Thus, we can choose

J ∈ N such that the measure of the setsx ∈ Ωi,ε | Jvji (x) < ε/2

are less than µ/3 for all j ≥ J .

Then let K be a compact subset of S2 \ Si such that the measure of S2 \ K is less than µ/3.

Now use that vji |KW 1,2(K,M)−→ ui|K to enlarge J such that, for all j ≥ J , the measure ofx ∈ K | ‖vji (x)− ui(x)‖2 + ‖dvji (x)− dui(x)‖2 > δ2/C2

is less than µ/3. Then, for all j ≥ J , the measure of (5.4.7) is less than µ plus the measure ofthe setx ∈ Ωi,ε | dG2M (V j

i (x), Ui(x)) > δ, Jvji

(x) ≥ ε

2, ‖vji (x)− ui(x)‖2 + ‖dvji (x)− dui(x)‖2 ≤ δ2/C2

.

But from (5.4.6) we see that this set is empty.Since δ > 0 was arbitrary, we conclude that dG2M (V j

i , Ui)measure→ 0 on Ωi,ε. Then, by

Lemma C.2.4, hV ji

measure→ hUi. By Lemma C.2.3 using that ui ∈ C1(S2,M), (hV ji )J

vji

measure→(h Ui)Jui . Then Theorem C.2.5 gives (5.4.5) finishing the proof.

The following lemma proves lower semicontinuity of the energy when restricting to an ex-hausting sequence of sets.

Lemma 5.4.2. Let Σ be a surface. Suppose (vj), vj ∈ W 1,2(Σ,M) converges strongly to u ∈W 1,2(Σ,M) except on a set S ⊂ Σ of measure zero. Let (V j), V j ⊂ Σ \S be a sequence of openor closed sets exhausting Σ \ S in the limit. Then

lim infj→∞

E(vj |V j ) ≥ E(u) .

Proof. Since (V j) exhausts Σ \ S in the limit and S has measure zero, Lemma 2.2.8 implies∫Σ ‖du‖

2 = limj→∞∫V j ‖du‖

2, In addition, for all k ∈ N,∫V k‖du‖2 = lim

j→∞

∫V k‖dvj‖2 ≤ lim inf

j→∞

∫V j‖dvj‖2

since V k ⊆ V j for sufficiently large j. Given ε > 0 use these observations to pick K such that∫Σ‖du‖2 −

∫V K‖du‖2 < ε

2, (5.4.8)

91


and JK ≥ K such that, for all j ≥ JK ,∫V K‖du‖2 −

∫V j‖dvj‖2 < ε

2. (5.4.9)

Then, for all j ≥ JK ,∫Σ‖du‖2 −

∫V j‖dvj‖2

(5.4.8)<

∫V K‖du‖2 +

ε

2−∫V j‖dvj‖2

(5.4.9)< ε .

Since this holds for all ε > 0, the result follows.

By the result of the following lemma, we can restrict our attention to the energy on the setsΩji . Note that energy preservation is used.

Lemma 5.4.3. Suppose (vj), vj ∈ W 1,2(S2,M) bubble converges to a finite collection of mapsu, . . . , ul, ui ∈W 1,2(S2,M). Then no energy is lost outside ∪li=0Ωj

i in the limit:

limj→∞

E(vj |S2\∪iΩji) = 0 .

In addition, for each i ∈ 0, . . . , l, limj→∞E(vj |Ωji

) = E(ui).

Proof. From the bubble convergence we get∑i

∫S2

‖dui‖2(BC3)

= limj→∞

∫S2

‖dvj‖2 = limj→∞

∫S2\∪iΩji

‖dvj‖2 + limj→∞

∫∪iΩji‖dvj‖2 . (5.4.10)

Note that by Property (BC2) of bubble convergence, the right hand side limits exist. It is theresult of Lemma 5.4.2 that, for each i,

limj→∞

∫Ωji

‖dvj‖2 ≥∫

S2

‖dui‖2 . (5.4.11)

Then combining (5.4.10) and (5.4.11),

limj→∞

∫∪iΩji‖dvj‖2 ≥ lim

j→∞

∫S2\∪iΩji

‖dvj‖2 + limj→∞

∫∪iΩji‖dvj‖2

and hence limj→∞∫

S2\Ωji‖dvj‖2 ≤ 0 proving the first assertion. Now putting together the equal-

ity∑

i limj→∞∫

Ωji‖dvj‖2 =

∑i

∫S2 ‖dui‖2 and (5.4.11), we get limj→∞

∫Ωji‖dvj‖2 =

∫S2 ‖dui‖2

for each i proving the second assertion.

We need an estimate on the connection between energy and area.

Lemma 5.4.4. Let Σ be a surface, u, v ∈W 1,2(Σ,M) and V an open or closed subset. Then∫V|Ju − Jv| ≤ 2E(u− v|V )1/4(E(u|V ) + E(v|V ))3/4 .

92


Proof. Choosing orthonormal coordinates at p ∈ S2 and letting dv, du be the Jacobian matrices,Lemma C.1.5 gives

|Ju − Jv| =∣∣∣det(duTdu)1/2 − det(dvTdv)1/2

∣∣∣ ≤ ∣∣det(duTdu)− det(dvTdv)∣∣1/2

≤ 21/2‖du− dv‖1/2 max(‖du‖3/2, |dv‖3/2) .

By the Holder inequality ([LL01, Theorem 2.3]) with p = 4 and q = 4/3, we have∫V‖du− dv‖1/2 max(‖du‖3/2, |dv‖3/2)

≤(∫

V

(‖du− dv‖1/2

)4)1/4(∫

V

(max(‖du‖3/2, |dv‖3/2)

)4/3)3/4

≤ (2E(u− v|V ))1/4 (2E(u|V ) + 2E(v|V ))3/4

giving the result.

In the following lemma, energy preservation is again crucial.

Lemma 5.4.5. Let Σ be surface. Suppose (vj), vj ∈ W 1,2(Σ,M) converges strongly to u ∈W 1,2(Σ,M) except on a set S ⊂ Σ of measure zero. Let (V j), V j ⊂ Σ \S be a sequence of openor closed sets exhausting Σ \ S in the limit. If limj→∞E(vj |V j ) = E(u) then

limj→∞

E(vj − u|V j )→ 0 .

Proof. Let ε > 0. Use Lemma 2.2.8 to choose K such that, for all j ≥ K,

E(u|Σ\V j ) < ε . (5.4.12)

Because vjW 1,2(V K ,M)−→ u, we can choose JK ≥ K such that, for all j ≥ JK ,

E(vj − u|V K ) < ε (5.4.13)

and ∣∣E(vj |V K )− E(u|V K )∣∣ < ε . (5.4.14)

Finally, use the assumption to enlarge JK such that, for all j ≥ JK ,

|E(vj |V j )− E(u)| < ε . (5.4.15)

Combining (5.4.12), (5.4.14), and (5.4.15), we get∣∣∣E(vj |V j\V K )− E(u|V j\V K )∣∣∣ ≤ ∣∣E(vj |V j )− E(u)

∣∣+ E(u|Σ\V j ) +∣∣E(vj |V K )− E(u|V K )

∣∣< 3ε .

(5.4.16)

Using this, Corollary 2.4.3, and (5.4.12),

E(vj − u|V j\V K ) ≤ 2E(vj |V j\V K ) + 2E(u|V j\V K )(5.4.16)< 4E(u|V j\V K ) + 6ε

(5.4.12)< 10ε .

(5.4.17)

93


Summing (5.4.13) and (5.4.17), we get E(vj − u|V j ) < 11ε. Since ε > 0 was arbitrary thisconcludes the proof.

Corollary 5.4.6. Let Σ be a surface. Suppose (vj), vj ∈W 1,2(Σ,M) converges strongly to u ∈W 1,2(Σ,M) except on a set S ⊂ Σ of measure zero. Let (V j), V j ⊂ Σ \S be a sequence of openor closed sets exhausting Σ \ S in the limit with limj→∞E(vj |V j ) = E(u). Let Jvj ∈ L1(Σ,M)be the maps JvjχV j . Then ∫

Σ|Jvj − Ju| → 0 .

Proof. It is the result of Lemma 5.4.5 that E(vj − u|V j ) → 0. By Lemma 2.2.8,∫

Σ\V j Ju → 0.Using Lemma 5.4.4, we get∫

Σ|Jvj − Ju| ≤

∫Σ\V j

|Ju|+∫V j|Jvj − Ju|

Lem. 5.4.4≤

∫Σ\V j

|Ju|+ 2E(vj − u|V j )1/4(E(vj |V j ) + E(u|V j ))3/4 → 0 .

Lemma 5.4.7. Let (hj), (J j), hj , J j ∈ L1(S2,R) be sequences with ‖hj‖∞ ≤ C for some C ≥ 0,

and J j ≥ 0. Let h, J ∈ L1(S2,R) with ‖h‖∞ ≤ C, J ≥ 0, and suppose J jL1(S2,R)→ J . For ε > 0

let Ωε =p ∈ S2 | J(p) ≥ ε

, and suppose for all ε > 0, limj→∞

∫ΩεhjJ j =

∫ΩεhJ . Then

limj→∞

∫S2

hjJ j =∫

S2

hJ .

Proof. Using that for any a, b, c, d ∈ R, |ab − cd| ≤ |a − c||d| + |a||b − d|, we get, for any j ∈ Nand ε > 0,∣∣∣∣∣

∫S2\Ωεi

(hjJ j − hJ

)∣∣∣∣∣ ≤∫

S2\Ωεi

∣∣hjJ j − hJ∣∣ ≤ ∫S2\Ωεi

∣∣(hj − h)J∣∣+∫

S2\Ωεi

∣∣hj(J j − J)∣∣

≤ 2Cε4π + C

∫S2\Ωεi

∣∣J j − J∣∣ .Now, given any µ > 0, let ε > 0 be such that 2Cε4π < µ/3 and J such that, for all j ≥ J ,∣∣∣∫Ωεi

hjJ j −∫

ΩεihJ∣∣∣ < µ

3 and∫

S2

∣∣J j − J∣∣ < µ3C . Then, for all j ≥ J ,

∣∣∣∣∫S2

hjJ j − hJ∣∣∣∣ ≤ 2Cε4π + C

µ

3C+µ

3< µ .

94


Notes

The special cover Lemma 5.1.1 is written by the author of this text. The proof of Theorem 5.1.3is based on [SU81] and [Par96] where the result is proved for harmonic maps. The definition ofbubble convergence and the proof Theorem 5.3.2 is a composition of the compactness results of[Par96] and [CM07a]; to prove the details not covered in [CM07a] we use the ideas of [Par96] andmodify them to suit the current setting. The lemmata following Theorem 5.3.2 are also provedin this way. Ruling out energy loss is a detailed exposition of the proof of [CM07a, Proposition2.2]. We choose to describe only a minimal amount of the theory of varifolds. The proof ofTheorem 5.4.1 and the following lemmata are based on [CM07a, Proposition A.3].

95

6

Combining the results

The existence of a sweepout tightening map combined with the bubble compactness result andvarifold convergence of bubble converging sequences, will now allow us to prove Theorem 3.3.1.Since this result is the only missing part of the proofs of Theorem 1 and Corollary 2, this willcomplete the proofs of the main results of the thesis.

The proof of the Theorem 3.3.1 proceeds as follows: by definition there exists a sequence ofsweepouts with maximal slice energy converging to the width. We apply a sweepout tighteningmap to each sweepout of the sequence. The tightening has the effect of making the slices of eachsweepout of the resulting sequence close to being harmonic. Then the bubble compactness resultasserts the existence of a finite number of minimal spheres to which a subsequence converges.This implies varifold convergence which will conclude the proof.

We will use Theorem 5.4.1 with h being constant and h(W ) = S−Ric(nW , nW ) where nW isa unit normal to W ∈ G2M . Since Ric(−W,−W ) = Ric(W,W ), the latter map is independentof choice of normal. As discussed in Section 5.4, the integrals in (3.3.2) are well-defined eventhough such normals may not exist everywhere.

Proof of Theorem 3.3.1. By definition of the width, there exists a sequence γj ∈ [σ] withmaxs∈[0,1]Eg(γj(·, s)) → Wg([σ]). If Wg([σ]) = 0 then Areag(γj(·, s)) → 0 and it is easilyverified that the sequence (γj), γj = γj satisfies the required properties. Thus, we can supposeWg([σ]) > 0.

By Corollary 4.2.2, we can replace each sweepout of the sequence with a sweepout γj ∈ [σ]close in energy and without non-constant harmonic slices. Using the sweepout tightening map(Ψ, ψ, ε) of Theorem 4.1.2, we define the sequence (γj), γj = Ψ(γ). We claim that (γj) satisfiesthe properties.

Since γj are close to γj in energy, and Ψ is energy decreasing, Property (1) holds. Suppose oneor both of (3.3.1) or (3.3.2) fail. The integrals in both inequalities are of the form

∫f,g hdf(TpS2)

where f is either ui or γj(·, s) and h a smooth, real-valued map from the Grassmannian bundleG2M . Thus the assumption implies that there exist such a map h, ε > 0, a subsequence (γjk)of (γj), and a sequence (sk), sk ∈ [0, 1] so that

Areag(γjk(·, sk)) > Wg([σ])− 1/k (6.1.1)

and, for all finite collection of harmonic maps u1, . . . , uk, ui : S2 →M ,∣∣∣∣∣∫γjk (·,sk),g

h dγjk(·, sk)(TpS2)−∑i

∫ui,g

h dui(TpS2)

∣∣∣∣∣ ≥ ε . (6.1.2)

97

6. Combining the results

The inequality (6.1.1) implies that Areag(γjk(·, sk)) converges to Wg([σ]). Since Eg(γjk(·, sk)) ≥Areag(γjk(·, sk)) and Eg(γjk(·, sk)) ≤ Eg(γjk(·, sk)) ≤ maxs∈[0,1]Eg(γj(·, s))→Wg([σ]), we haveEg(γjk(·, sk))−Eg(γjk(·, sk))→ 0. Combining (6.1.1) with Property (1), we get Eg(γjk(·, sk))−Areag(γjk(·, sk)) → 0. By this and Lemma 5.2.4, a subsequence of γjk(·, sk) is both ε-almostharmonic and almost conformal. Since the energies are clearly bounded, by Corollary 5.3.15we get bubble convergence of a subsequence of (γjk) to a finite collection of harmonic mapsu, u1, . . . , ul, ui : S2 → M . But then Theorem 5.4.1 excludes the existence of h such that(6.1.2) holds. Thus, our assumption, that one or both of (3.3.1) or (3.3.2) does not hold, fails.

Notes

We avoid referring to the theory of varifolds as much as possible. Except for this, the proof ofthis Chapter is based on the proof of [CM07a, Theorem 1.14].

98

Appendix A

Proofs of foundational material

Proof of Lemma 2.1.1. Since (Πp,S2 \p) are charts, both statements will follow after showingthat Π−1

p is conformal. We can suppose p = p+ since the possible extra rotation map is isometric.The result then follows by computing ∂x and ∂y. Indeed

〈∂x, ∂y〉 =4

(x2 + y2 + 1)2= 〈∂y, ∂x〉

and〈∂x, ∂y〉 = 0

as required.

By a circle on S2 we mean a non-empty intersection between S2 ⊂ R3 and a plane.

Lemma A.1.8. Let p ∈ S2 and p be its antipole. Then Πp : S2 \ p → R2 sends circles on S2

to circles on R2 or lines, and Π−1p sends lines and circles to circles on S2. Lines correspond to

circles intersecting p and lines through the origin correspond to circles intersecting p and p.

Proof. We can suppose p = p− and p = p+ as the initial rotation of S2 sends circles to circlesand vice versa.

Let C ⊂ S2 be a circle. By definition there exist kx, ky, α, β ∈ R where not all of kx, ky, αare zero such that

C =

(x, y, z) ∈ R3 | (x, y, z) ∈ S2, kxx+ kyy + αz + β = 0.

Recall that a circle in the plane is a set(x, y) ∈ R2 | (x− cx)2 + (y − cy)2 = r2

for cx, cy ∈ R and r > 0.

Suppose first Π−1p− (S) is a circle C for a set S ⊆ R2. Using the expression (2.1.1) for Π−1

p− , weget for constants kx, ky, α, β ∈ R with

kx2x

x2 + y2 + 1+ ky

2yx2 + y2 + 1

+ αx2 + y2 − 1x2 + y2 + 1

+ β = 0 (A.1.1)

for all (x, y) ∈ S. This reduces to

x2(α+ β) + y2(α+ β) + 2kxx+ 2kyy = α− β . (A.1.2)

99

Appendix A. Proofs of foundational material

Suppose α+ β = 0. Then2kxx+ 2kyy = α− β = 2α . (A.1.3)

If α = 0 then, by the definition of a plane, not both of kx and ky are zero. If α 6= 0, (A.1.3)again forces one of ky and ky to be non-zero. Supposing kx 6= 0 we have

x =α

kx− kykxy

defining a line. The case ky 6= 0 follows equivalently.Now, if α+ β 6= 0 we get

(x+kx

α+ β)2 + (y +

kxα+ β

)2 =α2 − β2 + k2

x + k2y

(α+ β)2(A.1.4)

implying that S is a circle in R2 with center(− kxα+β ,−

kyα+β

)and radius

√α2−β2+k2

x+k2y

α+β . Summing

up, if C is a circle then S = Πp−(Π−1p− (S)) = Πp−(C) is a circle if α+ β 6= 0 and a line if not.

Now, let S be a subset of R2. We suppose S is either a line or a circle and prove thatC = Π−1

p− (S) is a circle. Suppose first that

S =

(x, y) | (x− cx)2 + (y − cy)2 = r2

for constants cx, cy ∈ R and r > 0. Choose α, β ∈ R with either α 6= 0 or β 6= 0 such that

α(r2 − c2x − c2

y − 1) = −β(r2 − c2x − c2

y + 1) . (A.1.5)

Note that α+ β 6= 0 since otherwise

r2 − c2x − c2

y − 1 = r2 − c2x − c2

y + 1 .

Now choose kx, ky ∈ R such thatcx = − kx

β+α

cy = − kyβ+α .

We claim that (A.1.1) is satisfied for (x, y) ∈ S. To see this, note that (A.1.5) implies

(α+ β)(r2 − c2x − c2

y) = α− β

so that

r2 − c2x − c2

y =α2 − β2

(α+ β)2.

Plugging in the equations involving cx and cy we get

r2 =α2 − β2 + c2

x(α+ β)2 + c2y(α+ β)2

(α+ β)2=α2 − β2 + k2

x + k2y

(α+ β)2.

Thus, (x, y) ∈ S satisfies (A.1.4) and by reversing the calculations leading from (A.1.1) to (A.1.4)we get (A.1.1). Since one of α and β is non-zero, Πp−((x, y)) is in the circle


100

Suppose S is the line (x, y) ∈ R2 | kxx+ kyy = α

for kx, ky, α ∈ R and one of kx or ky non-zero. Then letting β = −α we get immediately that(A.1.2) and hence (A.1.1) is satisfied for (x, y) ∈ S. Thus again Πp−((x, y)) is in the circle


Finally, if p+ is in the circle C then, inserting the coordinates (0, 0, 1) in the equation of thecircle, we see that α + β = 0 showing that Πp−(C) is a line. Similarly, if S is a line, we sawΠ−1p− (S) is a circle with α+ β = 0, which therefore intersects p+. Similarly, the line S passes the

origin if and only if α = β = 0 which happens if and only if the corresponding circle intersectp−.

Corollary A.1.9. Let B ⊂ S2 be a geodesic ball. Then there is a stereographic projection Πp,p ∈ S2 such that Πp(B) is a ball in R2. Conversely, if B is ball in R2, then Π−1

p (B) is a geodesicball for any stereographic projection Πp.

Proof. Let B be a geodesic ball of radius r centered at p ∈ S2. By rotating we can assumep = p−. Then ∂B is the circle

C =

(x, y,− cos r) ∈ R3 |x2 + y2 = sin2 r.

By Lemma A.1.8, Πp−(C) is a circle C. Let B ⊂ R2 be a ball such that ∂B = C. We claim thatΠp−(B) = B. Since Πp− |S2\p− is a diffeomorphism, Πp− |S2\(p+∪C) is a diffeomorphism ontoits image R2 \ C. Thus each of the two components B and S2 \ (p+ ∪C ∪B) maps to exactlyone of the components R2 \ (C ∪ B) and B. But B is simply connected and R2 \ (B ∪ C) is not,thus we conclude that Πp−(B) = B.

Letting B ⊂ R2 be a ball, Lemma A.1.8 ensures that for any p ∈ S2, Π−1p (∂B) is a circle C.

Arguing as above, we see that C = ∂B for some geodesic ball B. Again, using the topology ofR2 \ ∂B we get that Πp−(B) = B.

Proof of Proposition 2.1.3. Follows from Corollary A.1.9.

Proof of Lemma 2.1.4. By Corollary A.1.9, U1 is a geodesic ball. Using Lemma A.1.8, we seethat Λ2(∂U1) is a circle or a line in R2. Suppose it is a circle C. Let B ⊂ R2 be a ball suchthat ∂B = C. The component U1 of S2 \ ∂U1 maps to either R2 \ (C ∪ B) or B. Thus, Λ2(U1)is either a ball or the complement of the closure of a ball. Suppose Λ2(∂U1) is a line. Then U1

maps to the half space on either side of the line. The case for Λ1(U2) is symmetric.

Proof of Lemma 2.1.6. See [Top06, Theorem 3.2.1].

Proof of Lemma 2.2.1. The first statement follows since

‖uj − u‖2W 1,2(V,M) =∫V‖uj − u‖2 +

∫V‖duj − du‖2 ≤

∫N‖uj − u‖2 +

∫N‖duj − du‖2

= ‖uj − u‖2W 1,2(N,M) → 0 .

101


For the second, we first show that uj |VL2(V,Rm) u|V and duj |V

L2(V,Rm) du|V , that is,∫

V

⟨uj , ϕ

⟩Rm →

∫V〈u, ϕ〉Rm ,

∫V

⟨duj , ϕ

⟩Rm →

∫V〈du, ϕ〉Rm (A.1.6)

for all ϕ ∈ L2(V,Rm). To see this, note that the maps Lϕ, Lϕ : W 1,2(N,Rm)→ R given by

Lϕ(u) =∫V〈u, ϕ〉Rm , Lϕ(u) =

∫V〈du, ϕ〉Rm

are linear, and continuous since, by Cauchy-Schwartz,

|Lϕ(u)| ≤∫V| 〈u, ϕ〉Rm | ≤ ‖u‖L1,2(V,Rm)‖ϕ‖L2(V,Rm) ≤ ‖u‖W 1,2(N,Rm)‖ϕ‖L2(V,Rm) ,

|Lϕ(u)| ≤∫V| 〈du, ϕ〉Rm | ≤ ‖du‖L1,2(V,Rm)‖ϕ‖L2(V,Rm) ≤ ‖u‖W 1,2(N,Rm)‖ϕ‖L2(V,Rm) .

Thus, the weak convergence of (uj) in particular applies to the functionals Lϕ and Lϕ. Hence∫V

⟨uj , ϕ

⟩Rm = Lϕ(uj)→ Lϕ(u) =

∫V〈u, ϕ〉Rm ,∫

V

⟨duj , ϕ

⟩Rm = Lϕ(uj)→ Lϕ(u) =

∫V〈du, ϕ〉Rm

showing that (A.1.6) holds.The lemma is proved if for all ϕ ∈W 1,2(V,M),⟨

uj |V , ϕ⟩W 1,2(V,Rm)

=∫V

⟨uj , ϕ

⟩+∫V

⟨duj , dϕ

⟩→∫V〈u, ϕ〉+

∫V〈du, dϕ〉

= 〈u|V , ϕ〉W 1,2(V,Rm) .

(A.1.7)

Since both ϕ, dϕ ∈ L2(V,Rm), (A.1.7) follows by (A.1.6).

Proof of Lemma 2.2.2. Weak compactness of the unit ball in W 1,2(N,Rm) is a direct conse-quence of being a Hilbert space, confer [Jos05, Theorem A.1.9]. Suppose M is closed as a subset

of Rm, N ⊂ Rn, uj ∈ W 1,2(N,M) and ujW 1,2(N,Rm)

u. Let Aj be the set of points x ∈ Nof which uj(x) 6∈ M . Then, by subadditivity, the measure of ∪jAj is zero. Hence uj(x) ∈ Mfor almost every x ∈ N and all j. By [LL01, Corollary 8.7], a subsequence (ujk) of (uj) con-verges pointwise almost everywhere to u. Thus, since M is closed, for almost every x ∈ N ,u(x) = limj u

j(x) ∈M . Hence u ∈W 1,2(N,M).

Proof of Lemma 2.2.3. The standard Gluing Lemma for continuous maps (see e.g. [Lee00,Lemma 3.8]) asserts the existence of w ∈ C0(R2,M) with the required properties. Since∫

R2

‖w‖2 =∫B‖u‖2 +

∫R2\B

‖v‖2 <∞ ,

w ∈ L2(R2,M). Now define

dwji (x) =∂xju

i(x) , x ∈ B ,∂xjv

i(x) , x ∈ R2 \B .

102

We claim that dwji are weak partial derivatives of w. Let ν = (νx1 , νx2) be the unit outer normalof ∂B, and ϕ ∈ C∞c (R2,R). Then by [EG92, Theorem 4.3.1],∫

Bui∂xjϕ = −

∫B∂xju

iϕ+∫∂Bνxiu

iϕ (A.1.8)

and ∫R2\B

vi∂xjϕ = −∫

R2\B∂xjv

iϕ+∫∂B−νxiviϕ . (A.1.9)

Thus, using that u equals v on ∂B,

∫R2

wi∂xjϕ =∫Bui∂xjϕ+

∫R2\B

vi∂xjϕ

(A.1.8)

(A.1.9)= −

∫B∂xju

iϕ−∫

R2\B∂xjv

iϕ

= −∫ 2

Rdwjiϕ

proving the claim. It is clear that∫

R2 ‖dw‖2 <∞ concluding the proof.

Proof of Lemma 2.2.4. We will use [EG92, Theorem 4.9.1] which in particular concernsW 1,2loc (R2)

maps. We let u : (0, R) × (0, 2π) → Rm be the map (r, θ) 7→ u(r cos θ, r sin θ). We extend uto R2 using [EG92, Theorem 4.4.1]. Then [EG92, Theorem 4.9.2(i)] asserts that, for almost allr ∈ (0, R), the maps

ur = θ 7→ u(r, θ)

are absolutely continuous and have L2 weak derivative. By [EG92, Theorem 4.9.1(ii)], ur ∈W 1,2(R,M), proving the first assertion. The second follows immediately.

Proof of Theorem 2.2.5. By [EG92, Theorem 4.9.2(i)], u has a representative, which is absolutelycontinuous on compact subsets of (a, b). Then [Roy88, Corollary 5.15] gives the result.

Proof of Lemma 2.2.8. Since a point set x is compact, eventually every point in N \ S isincluded in V j , and hence (uχV j ) converges pointwise to u. Since u ∈ L2(N \ S,M), theLebesque Dominated Convergence Theorem (see [Roy88, Page 92]) gives the result.

Proof of Theorem 2.3.1. For each i ∈ 1, . . . ,m it is the result of [Mor66, Therem 3.2.1] that∫U|ui|2 ≤ r2

2

∫U‖dui‖2 .

The result then follows by noticing that∫U‖u‖2 =

m∑i=1

∫U|ui|2 ≤

m∑i=1

r2

2

∫U‖dui‖2 =

r2

2

∫U‖du‖2 .

103


Proof of Corollary 2.3.2. Choose polar coordinates on R2 such that ∂B = (r, θ) | θ ∈ [0, 2π)and p = (r, 0). Since u vanishes at p, we have u ∈W 1,2

0 (∂B \ p,Rm). Since (0, 2π) = Bπ(π) ⊂R, we get by Theorem 2.3.1 that∫

∂B‖u‖2 =

∫ 2π

0‖u(r, θ)‖2rdθ

Thm. 2.3.1≤ π2

2

∫ 2π

0‖ ddθu(r, θ)‖2rdθ

=π2r2

2

∫ 2π

0gθθ‖ d

dθu(r, θ)‖2rdθ =

π2r2

2

∫∂B‖du‖2 .

Proof of Lemma 2.4.1. Let p ∈ Σ. In normal coordinates at p we have

Ju(p) =

√det((∑

i ∂xαui∂xβu

i)βα

)=

√det((〈du(∂xα), du(∂xβ )〉

)βα

)=√‖du(∂xα)‖2‖du(∂xβ )‖2 − 〈du(∂xα), du(∂xβ )〉2

≤ ‖du(∂xα)‖‖du(∂xβ )‖ =12

(‖du(∂xα)‖2 + ‖du(∂xβ )‖2 − (‖du(∂xα)‖ − ‖du(∂xβ )‖)2

)≤ 1

2(‖du(∂xα)‖2 + ‖du(∂xβ )‖2

)=

12‖du‖2 .

It follows that Area(u) ≤ E(u) with equality if and only if 〈du(∂xα), du(∂xβ )〉 and ‖du(∂xα)‖ −‖du(∂xβ )‖ vanishes almost everywhere.

Proof of Lemma 2.4.2. Note that

E(tu+ (1− t)v) =12

∫N‖tdu+ (1− t)dv‖2 =

12

∫Nt2‖du‖2 + (1− t)2‖dv‖2 + 2t(1− t) 〈du, dv〉 .

By Cauchy-Schwartz,

12

∫Nt2‖du‖2+(1−t)2‖dv‖2+2t(1−t) 〈du, dv〉 ≤ 1

2

∫Nt2‖du‖2+(1−t)2‖dv‖2+2t(1−t)‖du‖‖dv‖ ,

and by Lemma C.1.4,

12

∫Nt2‖du‖2+(1−t)2‖dv‖2+2t(1−t)‖du‖‖dv‖ ≤ 1

2

∫Nt‖du‖2+(1−t)‖dv‖2 = tE(u)+(1−t)E(v) .

Proof of Corollary 2.4.3. By Lemma 2.4.2 we have

E(v + u) = 4E(12v +

12u) ≤ 4

(12E(v) +

12E(u)

)= 2E(v) + 2E(v) .

Proof of Lemma 2.4.4. Follows from lower semicontinuity of norms in Banach spaces, confer e.g.[Jos05, Theorem A.1.9].

104

Proof of Lemma 2.4.5. This follows directly:

‖uj − u‖2W 1,2(Σ,Rm) =∫

Σ‖uj − u‖2 + ‖duj − du‖2 ≤ Area(Σ)

(‖uj − u‖2∞ + ‖duj − du‖2∞

)= Area(Σ)‖uj − u‖2C1(Σ,Rm) → 0 .

Proof of Lemma 2.4.6. Using the strong convergence except on S we have∫K‖uj − u‖2 +

∫K‖duj − du‖2 → 0 .

In particular E(uj − u|K) → 0. By the Minkowski inequality (confer [Eva98, Page 623]) alsoE(uj |K)→ E(u|K).

Proof of Lemma 2.4.7. Writing the integrals locally using a partition of unity, the equations(2.4.1) and (2.4.2) give the result. Alternatively, one can refer to [Jos05, Corollary 8.2.4].

Proof of Theorem 2.4.9. Using Lemma 2.2.2, [Mor66, Theorem 3.3.1] gives the result.

Proof of Lemma 2.5.2. See [CM07a, Appendix C].

Proof of Corollary 2.5.3. From Lemma 2.5.2 we get

0 = E(u)− E(v) ≥ 12E(v − u) ≥ 0 .

Thus E(v − u) = 0 and since u, v ∈W 1,20 (B,M), by Theorem 2.3.1 we have u = v.

Proof of Corollary 2.5.4. From Lemma 2.5.2 we get

E(u)− E(v) ≥ 0

andE(v)− E(u) ≥ 0 .

Thus E(v) = E(u) and the assertion follows by Corollary 2.5.3.

Proof of Proposition 2.5.5. See [Sam78, Theorem 1].

Proof of Theorem 2.6.1. See [Hel02, Theorem 4.1.1].

105


Proof of Theorem 2.6.2. The first statement is [Qin93, Theorem 1].The second statement follows by inspecting the dependence of the constants used in the

proof of [Qin93, Theorem 1]. It is shown that for u(p) ∈ ∂B and ε > 0 there exists Br(p) suchthat

supx∈Br(p)∩B

‖u(x)− u(p)‖Rm < ε.

We claim that r is independent of u. Notice that r ≥ R/2 where R is chosen so that for the ε1of the proof,

(1) E(u|BR(p)) <ε12 ,

(2) supx∈∂B∩BR(p) ‖u(x)− u(p)‖Rm < ε1 .

Here u is assumed extended to R2 as shown in [Qin93, Proposition 1]. Since ε1 is independentof u and because of the assumptions, we can choose R such that the two conditions are satisfiedfor all u ∈ U , we see that r can be chosen independently of u.

Proof of Theorem 2.6.3. The properties are the results of [SU81, Lemma 4.2] and [SU81, Theo-rem 3.6] respectively.

Proof of Proposition 2.6.4. See [Jos05, Corollary 8.2.5].

Proof of Lemma 2.7.1. We first construct the map ψ. Let G2M be the Grassmannian bundleof 2-planes, confer Appenidx B.1. We define a map Ψ : I × I ×G2M → R given by

(s, t, Vx) 7→det((gs(vi, vj)(x))ji

)det((gt(vi, vj)(x))ji

) ,where v1, v2 is basis for the 2-plane Vx ⊂ TxM . First, Ψ is well-defined because a differentchoice of basis vi yields

det((gs(vi, vj)(x))ji


) =(detS)2 det

((gs(vi, vj)(x))ji

)(detS)2 det

((gt(vi, vj)(x))ji

) =det((gs(vi, vj)(x))ji


)with S being the 2× 2 shift of basis matrix. We claim that Ψ is continuous. To see this suppose(sk, tk, V k)→ (s, t, v) with V k ⊂ TxkM and V ⊂ TxM . Let vi be an orthonormal basis for Vwith respect to gt, and likewise, for each k let vki be an orthonormal basis for V k with respectto gtk . Then∣∣∣Ψ(sk, tk, V k)−Ψ(s, t, V )

∣∣∣ =∣∣∣det

((gsk(vki , v

kj )(xk))ji

)− det

((gs(vi, vj)(x))ji

)∣∣∣ . (A.1.10)

Now extend vi and vki to bases for all of TxM , TxkM , such that the appended basis vectorsare orthonormal with respect to gs and gsk respectively. In addition, we can suppose gtk(vkj , v

ki ) =

0 and gt(vj , vi) = 0, for j = 1, 2 and i > 2. Then, extending the matrices on the right hand sideof (A.1.10) to include all basis vectors does not change the determinants. Let Sk be the shift ofbasis matrices between vki and vi. Then∣∣∣Ψ(sk, tk, V k)−Ψ(s, t, V )

∣∣∣ =∣∣∣(detSk)2 det

((gsk(vi, vj)(xk))

ji

)− det

((gs(vi, vj)(x))ji

)∣∣∣ .(A.1.11)

106

By continuity of the determinant and the family of metrics, and since (detSk)2 converges to 1 1,the right hand side of (A.1.11) converges to 0 proving the continuity.

Now use compactness of G2M to define

ψ(s, t) = maxV ∈G2M

∣∣∣1−Ψ(s, t, V )1/2∣∣∣ .

By compactness of I×I×G2M , ψ is continuous. We use ψ to prove the result of the lemma in thefollowing way. Let p ∈ Σ and pick normal coordinates (xα) centered at p. Let f ∈ W 1,2

loc (Σ,M)and q = f(p). If rank dpf < 2 then Jf,gt = Jf,gs = 0.

Assuming rank dfp = 2 pick a basis v1, v2 of df(TpΣ) and extend it to a basis for TqM .We let Sf be the Jacobian matrix of partial derivatives of f with respect to the bases (∂xα) and(vi). The choice of basis implies that all entries of Sf are zero apart from the upper 2×2 minor.Let Sf be that minor. Then

|Jf,gt − Jf,gs | =∣∣∣(det(STf (gt(vi, vj)(q))Sf ))1/2 − (det(STf (gs(vi, vj)(q))Sf ))1/2

∣∣∣=∣∣∣det(Sf )

∣∣∣ ∣∣∣(det(gt(vi, vj)(q)))1/2 − (det(gs(vi, vj)(q)))1/2∣∣∣

=

∣∣∣∣∣1− (det(gs(vi, vj)(q)))1/2

(det(gt(vi, vj)(q)))1/2

∣∣∣∣∣ Jf,gt ≤ ψ(s, t)Jf,gt .

(A.1.12)

Then|Areagt(f)−Areags(f)| ≤

∫Σ|Jf,gt − Jf,gs | ≤ ψ(s, t) Areagt(f)

finishing the proof.

Proof of Lemma 2.7.3. Recall that the derivative of an invertible, time dependent matrix A(t)is

dt det(A(t)) = det(A(t)) trace(A−1(t)dtA(t)) .

Let p ∈ Σ be such that Jf,gt 6= 0. Pick normal coordinates at p. LetA(t) = (gt(df(∂xα), df(∂xβ ))βα)so that Jf,gt(p) =

√detA(t). Then

dtJf,gt(p) =1

2√

detA(t)dt detA(t) =

12

√detA(t) trace(A(t)−1dtA(t))

=12Jf,gt trace(A(t)−1dtA(t)) .

(A.1.13)

Note that trace(A(t)−1dtA(t)) is the trace of the tensor dtgt ∈ T2 df(TpΣ). Hence, using com-pactness of the unit ball T1,gtM and continuity of dtgt,

trace(A(t)−1dtA(t)) ≤ 2 maxv∈T1,gtM

‖dtgt(v, v)‖2gt .

ThereforedtJf,gt(p) ≤ max

v∈T1,gtM‖dtgt(v, v)‖2gtJf,gt .

1To see this, suppose a subsequence of (det Sk)2 does not converge to 1. Use compactness of the unit sphereto choose a further subsequence, such that the basis vectors (vki ) converge. The limit vectors will constitute anorthonormal set by the choice of vki . By continuity of det this triggers a contradiction.

107


By (A.1.12), for t in a compact interval containing t0,

Jf,gt ≤ (1 + ψ(t, t0))Jf,gt0

and hence dtJf,gt close to t0 is dominated by an integrable map independent of t. Therefore,integration under the integral sign is justified and t 7→ Areagt(f) is in C1.

Proof of Lemma 2.7.4. Since gt is a Ricci flow, with notation as in the previous lemma, dtA(t) =(−2 Ricgt(df(∂xα), df(∂xβ )))βα. Since the trace is basis invariant, we have for fixed t and anorthonormal basis e1, e2 of Tf(p)M with respect to gt that

trace(A(t)−1dtA(t)) = −2 trace((Ricgt(ei, ej))ji ) = −2

∑i=1,2

Ricgt(ei, ei)

= −2

∑i=1,2

Ricgt(ei, ei) + Ricgt(nf , nf )

+ 2 Ricgt(nf , nf )

= −2Sgt + 2 Ricgt(nf , nf ) .

Hence

dtJf,gt(p) =12Jf,gt trace(A(t)−1dtA(t)) = (−Sgt + Ricgt(nf , nf ))Jf,gt (A.1.14)

and we conclude


ΣdtJf,gt = −

∫Σ

(Sgt −Ricgt(nf , nf )) Jf,gt .

Proof of Corollary 2.7.5. Since M is compact, the set of unit vectors T1,gtM with respect tofixed t is compact. Then since both Sgt , Ricgt and gt are smooth, we can define

C1 = maxt∈I

maxx∈M,v∈T1,gtM

|Sgt(x)− Ricgt(v, v)|

andC2 = max

t∈Imax

x∈M,v∈T1,gtM|dt (Sgt(x)− Ricgt(v, v))| .

By Lemma 2.7.4 and (A.1.14) arguing for differentiating under the integral sign as above, wehave

|dt2 Areagt(f)| Lem. 2.7.4=∣∣∣∣dt ∫

Σ(Sgt −Ricgt(nf , nf )) Jf,gt

∣∣∣∣ =∣∣∣∣∫

Σdt (Sgt −Ricgt(nf , nf )) Jf,gt

∣∣∣∣≤∫

Σ|dt (Sgt −Ricgt(nf , nf ))| Jf,gt +

∫Σ|Sgt −Ricgt(nf , nf )| |dtJf,gt |

(A.1.14)

≤ C2

∫ΣJf,gt + C1

∫Σ|Sgt −Ricgt(nf , nf )| Jf,gt

≤ (C2 + C21 ) Areagt(f) ,

108

Proof of Theorem 2.7.6. See [ET88, Theorem 4].

Proof of Proposition 2.7.7. From (2.7.5) and (2.7.3) we have

dt Areagt f(2.7.3)

= −∫

Σ

(12

Sgt +Kf(Σ) +12‖A‖2gt

)Jf,gt ≤ −

∫Σ

(12

S +KΣ

)Jf,gt

(2.7.5)

≤ −4π − 12

∫Σ

Sgt Jf,gt .

109

Appendix B

Differential Geometry

B.1 The Grassmannian bundle

Let M be a manifold and k ∈ 1, . . . ,dimM. For each x ∈ M , we let Gk(TxM) be theGrassmannian manifold, confer [Lee03, Example 1.24]. Recall that, for each V,W ∈ Gk(TxM)decomposing TxM as a direct sum W ⊕V = TxM , letting UV be the set of elements of Gk(TxM)trivially intersecting W , the map

ψ(A) = x+Ax |x ∈ V

from L(V,W ) to UV is a chart. We let GkM be the Grassmannian bundle∐x∈M Gk(TxM) with

projection π : GkM →M . For (x, V ) ∈ GkM let (U,ϕ) be a chart of M containing x. Using ϕwe can identify TxM for all x ∈ U and thereby get charts ϕ×ψ : ϕ(U)×L(V,W )→

∐x∈U UV ⊂

GkM containing (x, V ).Now, suppose M is embedded in Rm. We use the identification TxRm ∼= Rm to define 〈X,Y 〉

and ‖X − Y ‖ for all vectors X,Y ∈ TM . In the same way, for V ∈ GkM we let PV : TM →V ⊆ TM be the orthogonal projection onto V . Define a map dGkM : GkM ×GkM → R by

dGkM (V,W ) = ‖π(V )− π(W )‖+ supX∈V, ‖X‖=1

‖X − PWX‖+ supX∈W, ‖X‖=1

‖X − PVX‖ .

Proposition B.1.1. The map dGkM is a metric inducing the standard topology on GkM .

Proof. Clearly dGkM is non-negative and symmetric. Furthermore, dGkM (V,W ) = 0 if V = W ,and the converse follows easily. For V,Z,W ∈ GkM and X ∈ V with ‖X‖ = 1 we have

‖X − PWX‖ ≤ ‖X − PWPZX‖≤ ‖X − PZX‖+ ‖PZX − PWPZX‖≤ ‖X − PZX‖+ sup

Y ∈Z, ‖Y ‖=1‖Y − PWY ‖‖PZX‖

≤ ‖X − PZX‖+ supY ∈Z, ‖Y ‖=1

‖Y − PWY ‖ ,

using that ‖PZX‖ ≤ 1 and PW is orthogonal. Combined with the triangle inequality for ‖ · ‖,this shows that dGkM satisfies the triangle inequality.

Left is only to show that dGkM induces the standard topology on GkM . By the embeddingof M in Rm we only need to show that dGkM induces the subspace topology of Rm ×Gk(Rm).

111

Appendix B. Differential Geometry

This reduces to showing that V j → V in Gk(Rm) if and only if

supX∈V j , ‖X‖=1

‖X − PVX‖+ supX∈V, ‖X‖=1

‖X − PV jX‖ → 0 .

Recall that a chart on Gk(Rm) around V is given by the map

A 7→ x+Ax |x ∈ V

where A is a linear map V → V ⊥. For large enough j, we can let Aj be such a map so thatV j → V if and only if Aj → 0. Then result follows from observing that for any X ∈ V j ,Aj(PVX) = X − PVX.

Lemma B.1.2. Let Σ be a surface and u, v ∈ W 1,2(Σ,M). Then there exists C > 0 such thatfor all p ∈ Σ with Jv(p), Ju(p) > 0,

dGkM (dpv(TpΣ), dpu(TpΣ))

≤ max(√

2,C

min(Ju(p), Jv(p))1/2

)(‖v(p)− u(p)‖+ ‖dpu− dpv‖2

)1/2.

Proof. Let C = 1/cF , where cF is as defined in (2.2.1). Let p ∈ Σ with Jv(p), Ju(p) > 0 anddenote by Pdpu, Pdpv the orthogonal projections onto du(TpΣ) and dv(TpΣ) respectively. Thenfor X ∈ TpΣ such that ‖dv(X)‖ = 1 we have

‖dpv(X)− Pdpu(dpv(X))‖ ≤ ‖dpv(X)− dpu(X)‖ ≤ ‖dpv − dpu‖∞‖X‖ ≤‖dpv − dpu‖∞‖dpv‖∞

,

where ‖ · ‖∞ denotes the operator norm on L(TpΣ,Rm). By the choice of C,

‖dpv(X)− Pdpu(dpv(X))‖ ≤ C ‖dpv − dpu‖F‖dpv‖F

≤ C√2‖dpv − dpv‖F√

Jv.

In the last inequality we used that 2Jv(p) ≤ ‖dpv‖2F . Similarly

‖dpu(X)− Pdpv(dpu(X))‖ ≤ C√2‖dpu− dpv‖F√

Ju.

Note that for a, b ∈ R, Lemma C.1.4 gives a + b ≤√a2 + b2 + 2ab ≤

√2√a2 + b2. Thus, using

the above inequalities and the metric on the Grassmannian bundle,

dGkM (dpv(TpΣ), dpu(TpΣ))= ‖v(p)− u(p)‖

+ supX∈dpv(TpΣ), ‖X‖=1

‖dpv(X)− Pdpu(dpv(X))‖

+ supX∈dpu(TpΣ), ‖X‖=1

‖dpu(X)− Pdpv(dpu(X))‖

≤ ‖v(p)− u(p)‖+C√

2 min(Ju(p), Jv(p))1/2‖dpu− dpv‖

≤ max(√

2,C

min(Ju(p), Jv(p))1/2)(‖v(p)− u(p)‖2 + ‖dpu− dpv‖2

)1/2.

112

Tubular neighborhoods

B.2 Tubular neighborhoods

Let M ⊂ Rm be an embedded manifold. At each point x ∈ M the tangent space TxRm splitsinto tangent space to M , TxM , and the normal space to M ,

NxM =v ∈ TxRm | 〈v, w〉TxRm = 0 for all w ∈ TxM

.

Definition B.2.1 (Tubular neighborhood). Let δ : M → R be a positive, continuous map. A δ-tubular neighborhood Mδ is a set x+ v | (x, v) ∈ NxM, ‖v‖TxRm < δ(x) ⊂ Rm with the propertythe set (x, v) ∈ NxM | ‖v‖TxRm < δ(x) is diffeomorphic to Mδ under the map (x, v) 7→ x+ v.

For further details see [Lee03, Page 255]. Note that M ⊂Mδ and Mδ is open.

Theorem B.2.2. Every embedded manifold M ⊂ Rm has a δ-tubular neighborhood.

Proof. See [Lee03, Theorem 10.19].

Note that if M is compact and Mδ a tubular neighborhood, we can let δ = minx∈M δ(x).Considering δ as a constant map, Mδ is then a tubular neighborhood.

Proposition B.2.3. Let M ⊂ Rm be a compact, embedded manifold. Then there exists δ > 0such that M has a tubular neighborhood Mδ and a smooth retraction r : Mδ → M , called thenearest point retraction, satisfying the following properties:

(i) no point in M is closer to x than r(x);

‖r(x)− x‖Rm = infy∈M‖y − x‖Rm ,

(ii) the map x 7→ dxr is Lipschitz on Mδ,

(iii) there exists Cδ > 0 such that for each x ∈M ,

‖dxr‖∞ ≤ 1 + Cδ‖x− r(x)‖Rm ≤√

2 .

Note that, in particular, Property (iii) implies ‖dxr(V )‖ ≤ ‖V ‖ for all x ∈M .

Proof. By Theorem B.2.2 there exists a δ > 0 such that Mδ is a tubular neighborhood of M . Itis the result of [Lee03, Problem 10-2] that we can shrink δ such that a retraction r : Mδ → Mexists satisfying (i). Recall that Mδ is diffeomorphic to the set

Mδ = (x, v) ∈ NxM | ‖v‖TxRm < δ

Let π : NM →M be the natural projection (x, v) 7→ x. Then r is just the map π|Mδ. Since the

map x 7→ dxr is smooth, we can shrink δ such that it is Lipschitz on Mδ.Considering (iii). We claim that ‖dxr‖∞ = 1 for x ∈ M . To see this, let V ∈ TxRm and

write V = VT + VN , VT ∈ TxM and VN ∈ NxM . Since r|M is the identity, dxr(VT ) = VT . Inparticular ‖dxr‖∞ ≥ 1. Define a path γ(t) = x+ tVN so that γ′(0) = VN . Since

r γ(t) = π γ(t) = π(x, tVN ) = x ,

113

Appendix B. Differential Geometry

dxr(VN ) = 0. Thus‖dxr(V )‖ = ‖VT ‖ ≤ ‖V ‖TxRm

using that VT ⊥ VN in the last inequality. Thus ‖dxr‖∞ ≤ 1.Because x 7→ dxr is Lipschitz, there exists Cδ > 0 such that∣∣‖dxr‖∞ − ‖dr(x)r‖∞

∣∣ ≤ ‖dxr − dr(x)r‖∞ ≤ Cδ‖x− r(x)‖Rm .

Thus‖dxr‖∞ ≤ ‖dr(x)r‖∞ + Cδ‖x− r(x)‖Rm = 1 + Cδ‖x− r(x)‖Rm .

Replacing δ by min(δ, (√

2− 1)/Cδ) we get

‖dxr‖∞ ≤ 1 + Cδ‖x− r(x)‖Rm ≤ 1 + Cδ

√2− 1Cδ

≤√

2

proving (iii).

114

Appendix C

Miscellaneous

Lemma C.1.4 (Absorbing inequalities). For any a, b, x, y ∈ R we have

2abxy ≤ a2x2 + b2y2 .

In particular,

(a+ b)2 = a2 + b2 + 2ab ≤ 2a2 + 2b2 ,

2xy ≤ x2/2 + 2y2 ,

and for t ∈ [0, 1],2t(t− 1)xy ≤ (t− t2)x2 + ((1− t)− (1− t)2)y2 .

Proof. The first inequality follows immediately from

0 ≤ (ax− by)2 = a2x2 + b2y2 − 2abxy .

The second and third follows just by plugging in, and the fourth by setting a =√t− t2, b =√

(1− t)− (1− t)2 =√t− t2 and noticing that ab =

√t− t2

√t− t2 = t(t− 1).

Lemma C.1.5. For 2× n matrices S, T , n ∈ N we have∣∣det(STS)− det(T TT )∣∣ ≤ 2‖S − T‖F max(‖S‖3F , ‖T‖3F ) .

Proof. The Frobenius norm is induced by the inner product 〈S, T 〉 = tr (STT ). Thus, |tr (STT )| =| 〈S, T 〉 | ≤ ‖S‖F ‖T‖F by Cauchy-Schwartz. In addition, by submultiplicativity of ‖ · ‖F ,‖ST‖F ≤ ‖S‖F ‖T‖F . For any 2× 2 matrix A, ‖A‖F = ‖ adj(A)‖F . For t ∈ [0, 1] define

A(t) = (S + t(T − S))T (S + t(T − S))

and recall that ∂t det(A(t)) = trace(adj(A(t))∂tA(t)). By the mean value theorem, we havet0 ∈ [0, 1] such that ∂t det(A(t))|t0 = det(STS)− det(T TT ). Thus∣∣det(STS)− det(T TT )

∣∣ = |trace(adj(A(t0))∂tA(t)|t0)| ≤ ‖A(t0)‖F ‖∂tA(t)|t0‖F .

115

Appendix C. Miscellaneous

By convexity of the norm, ‖A(t0)‖F ≤ ‖S + t0(T − S)‖2F ≤ max(‖S‖2F , ‖T‖2F ). Also, lettingS = (sij) and T = (tij), we have

∂tA(t)|t0 = ∂t

(∑k

(ski + t(tki − ski))(skj + t(tkj − skj))

)ij

=

(∑k

(tki − ski)(skj + t(tkj − skj)) +∑k

(skj + t(tkj − skj))(tki − ski)

)ij

= (T − S)T (S − t(T − S)) + (S − t(T − S))T (T − S)

so that ‖∂tA(t)|t0‖F ≤ 2‖T − S‖F max(‖S‖F , ‖T‖F ) proving the result.

Lemma C.1.6. Let I ⊂ R be a closed interval and suppose I = I1, . . . , Ir is a finite collectionsof closed intervals covering I. Then there exists a subcollection I ′ ⊂ I such that I ′ covers I,each Ij ∈ I ′ intersects at most two intervals Ij1 , I

j2 of I ′ \ Ij and those intervals are disjoint;

Ij1 ∩ Ij2 = ∅.

Proof. We construct I ′ inductively. Let I ′1 = I1 and suppose for j ∈ 2, . . . , r that I ′j−1

covers ∪k∈1,...,j−1Ik and each interval in I ′j−1 intersects at most two other intervals of I ′j−1,

which are disjoint. We aim at constructing I ′j such that it covers ∪k∈1,...,jIk and maintainsthe above properties.

We construct I ′j in several steps. If Ij is contained in the union of some of the intervals inI ′j−1 we can just let I ′j = I ′j−1, so suppose not. Then let I ′j = I ′j−1 ∪ Ij. Clearly I ′j thencovers ∪k∈1,...,jIk. Now discard from I ′j all the intervals contained in Ij and replace I ′j bythe resulting set. Among the intervals intersecting Ij are only those left having one endpointin Ij and one endpoint outside Ij ; no interval is contained in Ij and no interval contains Ij .Thus, among those left intersecting Ij , those intersecting each other have the same (left/right)endpoint in Ij ; otherwise they would contain Ij . But more than two might still intersect Ij . Wecorrect this by excluding all those intersecting Ij having the right endpoint in Ij but the onehaving the left endpoint furthest away from Ij . Similarly, we discard all the ones having the leftendpoint in Ij except for the one having the right endpoint furthest way from Ij . Now clearlythe intersection property is fulfilled. Note that we have only discarded intervals contained ineither Ij or the union of Ij and one of the intervals of I ′j−1. Thus I ′j still covers ∪k∈1,...,jIk.This concludes the inductive step.

We conclude the proof by letting I ′ be the collection I ′r. Since I ′r covers ∪k∈1,...,rIk whichcovers I, the result follows.

C.2 Convergence in measure

Definition C.2.1. A sequence of measurable maps (f j), f j : N → R from a manifold N issaid to converge in measure to a measurable map f : N → R if for all δ > 0 the measure ofx ∈ N | |f j(x)− f(x)| > δ

goes to zero as i→∞.

Convergence in measure is weaker than L1 convergence:

116

Convergence in measure

Lemma C.2.2. If (f j), f j ∈ L1(N,R) and f jL1(N,R)→ f , f ∈ L1(N,R), then f j

measure→ f .

Proof. Follows from∫N|f j − f | ≥

∫x∈N | |fj(x)−f(x)|>δ

> δ

∫x∈N | |fj(x)−f(x)|>δ

1

and∫N |f

j − f | → 0.

We will need two minor results regarding convergence in measure, and a version of theLebesque Dominated Convergence Theorem.

Lemma C.2.3. Let N be an manifold and suppose (f j), (gj), f j , gj : N → R are sequences ofmeasurable maps with f j

measure→ f , gj measure→ g, and ‖f j‖∞, ‖g‖∞ < C for some C ≥ 0. Thenf jgj

measure→ fg.

Proof. Follows immediately from∣∣f jgj(x)− fg(x)∣∣ ≤ ∣∣(f j − f)g(x)

∣∣+∣∣f j(gj − g)(x)

∣∣ ≤ ∣∣f j − f ∣∣C + C∣∣gj − g∣∣ .

Lemma C.2.4. Let N be a manifold and S a metric space with metric dS. Suppose (f j) is asequence of maps N → S such that, for an f : N → S and each j, dS(f j , f) is measurable. Ifh : S → R is a uniformly continuous function and dS(f j , f) measure→ 0 then h f j measure→ h f .

Proof. Given δ, ε > 0 use uniform continuity of h to pick ν > 0 such that

∀x, y ∈ S : d(x, y) < ν ⇒ |h(x)− h(y)| < δ .

Use convergence in measure of ds(f j , f) to pick K such that for all i ≥ K the measure ofx ∈M | dS(f j(x), f(x)) > ν

is less than ε. Then clearly

x ∈M | |h(f j(x))− h(f(x))| > δ⊆x ∈M | d(f j(x), f(x)) > ν

implying that the measure of

x ∈M | |h(f j(x))− h(f(x))| > δ

is less than ε.

Theorem C.2.5 (Lebesgue Dominated Convergence Theorem for convergence in measure).

Suppose f j , f, gj , g : N → R. If f j measure→ f , gjL1(N,R)→ g and |f j | ≤ gj for all i ∈ N, then∫

N fj →

∫N f .

Proof. See [Roy88, Page 96].

117

Appendix C. Miscellaneous

C.3 Generalized difference quotients

Lemma C.3.1. Let f, g : R → R be functions, and suppose one of them is continuous. If thesum D+

x f(x) +D+x g(x) is not of the form ∞−∞ then

D+x (fg)(x) ≤ (D+

x f(x))g(x) + (D+x g(x))f(x) .

Proof. By interchanging f and g, we can suppose g is continuous. Since

f(x+ h)g(x+ h)− f(x)g(x) = f(x) (g(x+ h)− g(x)) + g(x+ h) (f(x+ h)− f(x))

and for any sequences (an), (bn) with lim supn an + lim supn bn 6=∞−∞,

lim supn

(an + bn) ≤ lim supn

an + lim supn

bn

(see [Rud76, Page 78]) we get

D+x (fg)(x) = lim sup

h→0+

f(x+ h)g(x+ h)− f(x)g(x)h

= lim suph→0+

f(x) (g(x+ h)− g(x)) + g(x+ h) (f(x+ h)− f(x))h

≤ lim suph→0+

f(x) (g(x+ h)− g(x))h

+ lim suph→0+

g(x+ h) (f(x+ h)− f(x))h

.

By continuity of g we get

D+x (fg)(x) ≤ (D+

x f(x))g(x) + (D+x g(x))f(x) .

Lemma C.3.2. If f : [a, b] → R is continuous and D+x f(x) < 0 for all x ∈ [a, b] then f is

(strictly) decreasing on [a, b].

Proof. Suppose there exist x, y ∈ [a, b] with x < y and f(x) ≤ f(y). Use continuity of f to lets ∈ [x, y] be such that f(s) = min f |[x,y]. If f(s) = f(x) we let s = x. Then s < y. For any hsuch that s+ h ∈ [x, y], f(s) ≤ f(s+ h) by the choice of s. Hence

lim suph→0+

f(s+ h)− f(s)h

≥ 0

contradicting that, by assumption, D+x f(s) < 0.

Lemma C.3.3. Let f : [a, b] → R be increasing. Then f is differentiable almost everywhere.The derivative is integrable and ∫ b

af ′(x)dx ≤ f(b)− f(a) .

Proof. See [Roy88, P. 100].

118

Generalized difference quotients

Corollary C.3.4. If f : [a, b]→ R is continuous and D+x f(x) < 0 for all x ∈ [a, b] then

f(b)− f(a) ≤∫ b

aD+x f(x)dx .

Proof. By Lemma C.3.2, f is decreasing. Thus −f is increasing. By Lemma C.3.3, −f isdifferentiable almost everywhere and hence D+

x (−f) = −f ′ almost everywhere. In addition,∫ b

a(−f)′(x) ≤ (−f)(b)− (−f)(a) .

Thus,

f(b)− f(a) ≤∫ b

af ′(x)dx =

∫ b

aD+x f(x)dx .

119

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123

Index

C0 ∩W 1,2, 11C0 ∩W 1,2

loc , 11E∆(s, γ, ε), 58Eδ(u,B), 45Eδ(u,B), 45Lp(N,M), 9Lploc(N,M), 9W ([σ]), 22W 1,2

0 (N,Rm), 9ΩM , 21W 1,2(N,M), 9W 1,2(N,Rm), 9MB, 44W 1,2loc (N,Rm), 9

ε0, 76εC , 84εH , 32εL, 15εmin, 23εSU , 16cl (U), 5R, 8ec(si), 73ε-almost harmonic sequences, 69ε-locally converging to harmonic maps, 67int (U), 5Ric, 8MB, 34S, 8B, 44

almost conformal sequences, 82area, 12

conformal invariance, 13

ballgeodesic, 6geodesic radius, 7

projected, 6boundary

equal on the boundary, 14branched

immersion, 18minimal surface, 18

bubble convergence, 72

circle on S2, 99compactness

of almost harmonic maps, 70of harmonic maps, 69unit ball, 10

comparison map, 35conformal

chart, 6map, 6structure, 6

convergencein measure, 116strong, 10strongly except on S, 11weak, 10

dilation, 6

energy, 12conformal invariance, 13convexity, 13decrease, 45maximal improvement, 58minimizing, 14triangle inequality, 13

energy concentration map, 73exhaust in the limit, 11

Fundamental Theorem of Calculus, 11

glueing Lemma, 11

125

Index

Grassmannian bundle, 111

harmonic maps, 14convexity, 15regularity, 15unique continuation, 15

high energy sets, 69homotopy

in ΩM , 22

low energy sets, 69

manifold, 5metric

smooth family, 8minimal surface, 16

normon C0 ∩W 1,2, 11Sobolev, 9

oriented manifold, 9

parameter space, 21

Riccicurvature, 8flow, 8

Riemannian manifold, 5

scalar curvature, 8second fundamental form, 16singular points, 69Sobolev space, 9sphere

equator, 6p+ (north pole), 6S2

+ (northern hemisphere), 6p− (south pole), 6S2− (southern hemisphere), 6

S2 (2-sphere), 6stereographic projection, 6summation convention, 5surface, 5sweepout, 22

slice, 22sweepout tightening map, 32

tubular neighborhood, 113

width, 22

126

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