Deformation of C0 Riemannian metrics in the directionof their Ricci curvature.

Miles Simon

Abstract

The purpose of this paper is to evolve non-smooth Riemannian metric

tensors by the dual Ricci-Harmonic map flow. This flow is equivalent

(up to a diffeomorphism) to the Ricci flow. One application will be the

evolution of metrics which arise in the study of spaces whose curvature is

bounded from above and below in the sense of Aleksandrov, and whose

curvature operator (in dimension three Ricci curvature) is non-negative.

We show that such metrics may always be deformed to a smooth metric

having the same properties in a strong sense.

§ 1. Introduction and statement of results

Let (Mn, D) be an n-dimensional manifold with a smooth (C∞) structureD. We say that a tensor S on a smooth manifold (M,D) is Ck or S is inCk((M,D)) if in local co-ordinates (which come from the structure), S = Sij,is Ck. To avoid any confusion we will fix the differentiable structure D of Mand do not consider other structures (M, D). For this reason we will suppressthe D in (M,D).

When considering a Riemannian metric tensor g = gij on a compactmanifold M we often assume g is C2. This allows us to define the Riemanniancurvature tensor which is then continuous. Given a C∞ Riemannian metricg0 on a compact manifold M , we can always find a T > 0 and a 1-parameterfamily of C∞ Riemannian metrics g(t)t∈[0,T ] on M , denoted (M, g(t)), suchthat

∂

∂tg(t) = −2Ricci(g(t)), for all t ∈ [0, T ]

g(0) = g0,(1.1)

where g is C∞(M × [0, T ]) (C∞ on the manifold (M × [0, T ]) with the inducedstructure), and Ricci(g(t)) is the Ricci curvature of the Riemannian manifold(M, g(t)). Notice that (1.1) makes no sense if g is not twice differentiable inspace for all t ∈ [0, T ]. The family (M, g(t))t∈[0,T ] is called a solution to theRicci flow with initial value g0. Ricci flow was invented, and used by Hamiltonto prove that every compact three manifold which admits a C∞ Riemannianmetric g0 with Ricci(g0) > 0 also admits a metric g∞ of constant positivesectional curvature [Ha 1]. The flow was constructed in such a way that variousgeometrical conditions are preserved by the flow, and so that it is ‘nearly’ a

1

gradient flow for the Yamabe quotient

E(g) =

∫

MR(g)volg

(∫

Mvolg)

(n−2)n

,

where R(g) is the scalar curvature of (M, g) and volg is the volume form withrespect to g on M . Many metric tensors on manifolds arise from Riemannianmetric tensors which are not smooth. For example the geometric object ob-tained by cupping a two dimensional cylinder off with two hemispheres ([Pe]example 1.8) is a nice geometrical object sitting in R3. As a manifold it is sim-ply topologically S2, and we give this S2 the standard differentiable structure.It inherits a natural Riemannian metric g from the ambient space R3 (along thejoins we define g by continuity). This metric g on S2 is C1,1 with respect to thestandard differentiable structure of S2, but not C2. The curvature is definedaway from the join and is bounded from above and below, but has a disconti-nuity at the join. This manifold with metric tensor is a well known exampleof a ‘metric space with curvature bounded from above and below’ studied ini-tially by Aleksandrov [Al] in connection with his investigation of the intrinsicgeometry of convex surfaces, and later for it’s own sake by Aleksandrov andhis followers (see [BN] for an overview of the theory and a good bibliography).Here the curvature bound from below is zero.

If we take two copies of a two dimensional truncated cone imbedded inR3 and join them at their boundary we obtain a nice geometrical object (asa manifold it is topologically equivalent to the infinite cylinder R × S1). Themetric g inherited from the ambient space R3 may be defined on the join bycontinuity and is then C0,1 (Lipschitz continuous), but not C1. Note that if weapproximate this metric g by a family of metrics α

gα∈1,2,... withα

g → g as

α→ ∞ in the C0 norm, then supx∈M |Riem(α

g)| → ∞ as α→ ∞. In this senseg has infinite curvature at the join, and (M, g) is not a manifold with curvaturebounded from above and below.

The third example is the cone. Let us consider the two dimensional conesitting in R3 as a graph over R2. This cone then inherits a metric g from theambient space R3. Clearly g is C∞ with respect to the standard co-ordinates inR2 away from the point corresponding to the tip of the cone (for simplicity letthis point be 0 = (0, 0)), but g cannot be continuously extended to this point.We see this as follows. The cone C is a graph over R2, C = (x, α|x|), x ∈ R2,where α > 0 is some fixed constant, and hence using the formula for the metricof a graph, we obtain

gij = δij + α2 ∂

∂xi|x| ∂∂xj

|x| = δij + α2xixj

|x|2 .

2

Clearly xǫ = (ǫ, ǫ) → 0 as ǫ→ 0 and limǫ→0 g12(xǫ) = α2

2 . Also yǫ = (ǫ,−ǫ) → 0

as ǫ → 0, but limǫ→0 g12(yǫ) = −α2

2 . Hence there is no way to continuously

extend g to the point 0. Note however that

δij ≤ gij ≤ (1 + α2)δij , for all x ∈M − (0),

in the sense of tensors. Later we shall see that metrics which fulfill such esti-mates, with 0 < α2 ≤ ǫ(n) small, can nevertheless be flown.

We would like to have a way of evolving C0 metrics g0 by something likeRicci flow, so that for all times t bigger than zero, the solution g(t) is smooth,and as time approaches zero from above, the metric g(t) approaches g0 uniformlyon all compact subsets of M . The flow should also preserve various curvatureconditions.

Non-regular Example . Let M = S1 ×N , where N is a compact manifoldwhich admits a positive Einstein metric γ, g0 be the warped product metric onM given by g0(x, q) = h0(x) ⊕ γ(q), where h0 is a Riemannian metric on S1.Then the Ricci flow has the solution g(x, q, t) = h0(x) ⊕ (1 − 2kt)γ(q), whichhas for all times t ≥ 0 the same regularity as the regularity of h0.

This means clearly that we cannot hope that the Ricci flow will ‘smoothmetrics out’ on M with respect to the fixed differentiable structure.

It is well known that if a metric is a C2 Einstein metric then one mayintroduce Harmonic co-ordinates for which the metric is C∞ ([DK]). Such co-ordinates are only C2,α compatible with our fixed structure D on M , andso not admissible as smooth (C∞) co-ordinates for (M,D). We note that inexample one, if we introduce Harmonic co-ordinates (change the structure) themetric will never be C2 (otherwise we could apply the result of [DK] mentionedabove, and introduce Harmonic co-ordinates which make the metric C∞ whichcontradicts the fact that the scalar curvature has a discontinuity at the join).

In this paper we shall consider the dual Ricci-Harmonic Map flow (seesection 6. [Ha 3]). This leads to a more general version of the Ricci DeTurckflow, considered initially by DeTurck in [DeT]. In the paper [Bem] the authorsuse Ricci flow to smooth out C2 metrics by introducing harmonic co-ordinatesat appropriately chosen times.

We give here a short introduction to the the dual Ricci-Harmonic and theRicci DeTurck flow.

Let g(t), t ∈ [0, T ] be an arbitrary one parameter family of smooth metrics,and φt : M →M an arbitrary one parameter family of smooth diffeomorphisms.Then the metric g(t) defined by g(t) = φ∗t g(t) satisfies

∂

∂tgij(t) = (φ∗t

∂

∂tg(t))ij +

t∇iVj +t∇j Vi,

3

where (V )i = (φ∗tV )i, Vα(p) = ( ∂∂tφt(p))

βgβα, andt∇ is the co-variant deriva-

tive with respect to the metric g (see [Si], proposition 1.4). In particular if g(t),t ∈ [0, T ] is a solution to Ricci flow, then

∂

∂tgij(t) = −2Ricci(g(t)) +

t∇iVj +t∇j Vi, (1.2),

where (V )j(p, t) =∂

∂xαφi(p, t)V α((φt(p), t)gij ,

and∂

∂t(φt(p)) = V (φt(p), t).

We have now the freedom to choose the diffeomorphism φ. If g0 is alreadyEinstein, then the solution to the Ricci flow g(t) is given by g(t) = (1−2kt)g0,isalso Einstein and has the same regularity as g0. Hence g(t) = φ∗t g(t) is alsoEinstein. We want to choose φt so that g(t) will be regular for t > 0. Asmentioned before, in harmonic co-ordinates an Einstein metric is regular. Tothis end we let φ = f−1 : (M × [0, T ]) → (M × [0, T ]), where f is the solutionto the Harmonic map heat flow equation:

∂

∂tf(p, t) = (

g(t),h

∆f)(f(p, t)),

f(p, 0) = Id(p),

where h is some fixed smooth background metric. For an arbitrary functionf : (M, g) → (N,h) between two Riemannian manifolds, the Laplacian of f isthen a vector field in TN defined in co-ordinate form by

(g,h

∆f i)(y) =gαβ(x)(∂

∂xα

∂

∂xβf i(x) − g

Γη

αβ(x)∂

∂xηf i(x))

+h

Γi

jk(y)gαβ(x)∂

∂xαf j(x)

∂

∂xβfk(x),

where f(x) = y. Since V (y, t) = − ∂∂tf(x, t) = −∆f(y, t), we obtain

V (y, t) = gβδ(y, t)(g(t)

Γα

βδ −h

Γα

βδ)(y, t),

where g = f∗g, in view of the way Christoffel symbols and tensors change undera co-ordinate transformation. So we see that the system (1.2) may be written

∂

∂tgij(t) = −2Ricci(g(t)) +

t∇iVj +t∇j Vi, where

V (x, t)i = gij(x, t)gkl(x, t)(

g(t)

Γj

kl −h

Γj

kl)(x, t).

(1.3)

4

The reader is referred to [Ha 3] section 6 for further discussion of the system(1.3) which is called the dual Ricci harmonic map heat flow, or [ES], [St] forfurther information about harmonic map heat flow. It is shown in [Ha 3] section6 , that the evolution equation (1.3) for g(t) is a strictly parabolic system ofequations. In particular if we choose h = g0, then the evolution equation for g(t)in (1.3) is the Ricci-DeTurck flow, which was first introduced in [De] to provethe short time existence for Ricci flow on a compact manifold using standardparabolic techniques (short time existence for Ricci flow on a compact manifoldwas first proved by Hamilton [Ha 1] and relied upon the sophisticated machineryof the Nash-Moser Inverse function Theorem).

The evolution equation for φt in (1.2) may be written as a first orderevolution equation in terms of g. That is,

∂

∂tφα(p, t) = (φt∗V )α(φ(p, t)) =

∂

∂xiφα(p, t)gjk(

g(t)

Γi

jk − h

Γi

jk)(p, t),

φ(p, 0) = Id(p),(1.4)

in view of the derivation of V given above. If we can solve the evolution equationfor g(t), t ∈ [0, T ] in(1.3), and the solution g(t) is sufficiently regular, then wemay solve (1.4) and then define g(t) to be g(t) = (φ−1

t )∗(g(t)), which is then asolution to the Ricci flow. We say that g(t) solves the h Ricci flow or h flowof g0. Many geometric quantities that are preserved by Ricci flow will also bepreserved by h flow.

In Shi’s paper [Sh], the Ricci-DeTurck flow was written term by term toobtain the evolution equation for solutions to (1.3) in co-ordinate form. Wepresent here the evolution equation, in co-ordinate form, for metrics whichsolve (1.3) for an arbitrary smooth fixed background metric h. For the rest ofthe paper we shall be chiefly concerned with solutions of (1.3) and not solutionsof Ricci flow. For this reason we will use the notation g(t), t ∈ [0, T ] to refer to asolution of (1.3). Let g(t), t ∈ [0, T ] be a solution to (1.3). Then g(t), t ∈ [0, T ]solves the evolution equation

∂

∂tgab =gcd∇c∇dgab − gcdgapg

pqRbcqd − gcdgbpgpqRacqd

+1

2gcdgpq(∇agpc · ∇bgqd + 2∇cgap · ∇qgbd

− 2∇cgap · ∇dgbq − 4∇agpc · ∇dgbq),

g(0) =g0,

(1.5)

where Rabcd = Riem(h)abcd and ∇ is the co-variant derivative with respect toh. Note that if h is not twice differentiable, then (1.5) makes no sense, since

5

then Rabcd = Riem(h)abcd is not defined. If we choose h = g0, that is we wishto examine the Ricci DeTurck flow, and g0 is not twice differentiable, then wecannot make sense of the above equation. For this reason we will always choosea smooth h not equal to g0 (but close to g0 in some to be specified C0 sense)when examining (1.5).

The first part of this paper is concerned with finding a sensible solution tothe h flow for initial data g0 which is non-smooth. Theorem 1.1 (below) is thetarget theorem of this section.

Definition 1.1 . Let M be a complete manifold and g a C0 metric, and1 ≤ δ < ∞ a given constant. A metric h is said to be a δ fair backgroundmetric for g, or ‘δ fair to g’, if h is C∞ and there exists a constant k0 with

supx∈M

h|Riem(h)(x)| = k0 <∞, (1.6)

and1

δh(p) ≤ g(p) ≤ δh(p) for all p ∈M. (1.7)

Remark 1. By the result of Shi [Shi ], if g is Riemannian metric and h asmooth Riemannian metric satisfying (1.6) and (1.7) then there exists a smoothmetric h′ which is 2δ fair to g, and

supx∈M

h|h∇j

Riem(h)(x)| = kj <∞,

whereh∇

jis the jth covariant derivative with respect to h. We will assume

(without loss of generality) that our h always fulfills such estimates.

Remark 2. Let M be a compact manifold, and g a C0 metric on M for which(M, g) is complete. Then for every 0 < ǫ < 1 there exists a metric h(ǫ), forwhich h(ǫ) is 1 + ǫ fair to g.

Proof (of Remark 2): We may use de Rham regularisation [deR], or a locallyfinite partition of unity and Sobolev averaging (see section on mollifiers in [GT])to obtain a C∞ metric h which is C0 as close as we like to g. A bound on thecurvature follows from the compactness of M . ♦

Theorem 1.1. Let g0 be a complete metric and h a complete metric on Mwhich is 1 + ǫ(n) fair to g0, ǫ(n) as in Lemma 2.4. There exists a T = T (n, k0)

6

and a family of metrics g(t), t ∈ (0, T ] in C∞(M × (0, T ]) which solves h flowfor t ∈ (0, T ],h is (1 + 2ǫ) fair to g(·, t) ,for t ∈ (0, T ] and

limt→0

supx∈Ω′

h|g(·, t) − g0(·)| = 0,

supx∈M

|h∇i

g|2 ≤ ci(n, k0, . . . , ki)

ti, for all t ∈ (0, T ], i ∈ 1, 2, . . .,

where Ω′ is any open set satisfying Ω′ ⊂⊂ Ω, where Ω is any open set on whichg0 is continuous (see Theorem 5.2).

Remark 3. As a consequence of Theorem 1.1, we see that if the metric g0 iscontinuous except for a set I ⊆M of isolated points, then the distance functionρ(t) : M ×M → R, defined by ρ(t)(x, y) = dist(g(t))(x, y) is lipschitz, andsmooth almost everywhere, for all t > 0, and satisfies limt→0 ρ(t)(·, ·) = ρ(0)(·, ·)uniformly on any compact subset of M − I.

Remark 4. If M is not compact, g is C0 on M , and g is a ‘metric of curvaturebounded from above an below’ (see below) outside some compact set Ω, andsatisfies the global bound

supx∈M−Ω

|Riem(g)(x)|2 ≤ k0,

for some constant k0 <∞, then for every 0 < ǫ < 1 there exists an h(ǫ) so thath is 1 + ǫ fair to g.

Proof (of Remark 4): We mollify g as in the proof of remark two to obtaina metric h which is C0 as close as we like to g. One needs to check thatsupM Riem(h) < ∞. On Ω this follows by compactness. Outside of Ω this istrue because a metric with bounded curvature also has bounded curvature afterit is mollified (see Lemma 6.1).

The second section of the paper is concerned with flowing metrics g0 ofbounded curvature from above and below (initially studied by Aleksandrov [Al],see [BN] for a good overview), or locally Lipschitz metrics which satisfy (forexample) Ricci(g0) ≥ 0, to obtain a smooth metric g which satisfies Ricci(g) ≥0. The main theorem of this section is as follows. Let R(g) be the curvatureoperator of g, and G(g) : Λ2(M) ⊗ Λ2(M) → R be the operator defined by

G(g)(φ, ψ) = φijψklgikgjl, (1.8)

where Λ2(M) is the space of smooth two forms on M . I(g) will refer to theIsotropic curvature in the case that Mn = M4 (see the proof of Theorem 6.7for an overview of Isotropic curvature, and the discussion before Theorem 6.6for an overview of the curvature operator).

7

Theorem 1.2. Let M(n, k0, d, v) be the set of (Mn, g) such that Mn is ann-dimensional compact manifold and g is a metric with curvature K(M, g)bounded from above and below which satisfies

−k0 ≤ K(M, g) ≤ k0, vol(M, g) ≥ v, diam(M, g) ≤ d.

There exists an ǫ1(3, k0, d, v) > 0, ǫ2(n, k0, d, v) > 0, ǫ3(4, k0, d, v) > 0 andǫ4(n, k0, d, v) > 0 with the following properties. If (M3, g) is an element ofM(3, k0, d, v) and satisfies Ricci(g) ≥ −ǫ1g, then there exists a smooth Rieman-nian metric g′ onM3 where (M3, g′) has non-negative Ricci-curvature. If (M, g)is an element of M(n, k0, d, v) and satisfies R(g) ≥ −ǫ2G(g), then there exists asmooth Riemannian metric g′ on M where (M, g′) has non-negative curvatureoperator. If (M, g) is an element of M(4, k0, d, v) and satisfies I(g) ≥ −ǫ3,then there exists a smooth Riemannian metric g′ on M where (M, g′) has non-negative Isotropic curvature. If (M, g) is an element of M(n, k0, d, v) and sat-isfies R(g) ≥ −ǫ4, (scalar curvature), then there exists a smooth Riemannianmetric g′ on M where (M, g′) has non-negative scalar curvature (see Theorem6.8).

Remark 5. In dimension three non-negative curvature operator is equivalentto non-negative sectional curvature. In dimensions bigger than three, non-negative curvature operator implies non-negative sectional curvature.

Theorem 1.2 is proved by an application of Cheeger’s finiteness Theoremand Gromov’s compactness Theorem for metrics in M(n, k0, d, v) and a contra-diction argument, and an application of the following theorem.

Theorem 1.3. Let Mn be a manifold (compact or not compact) which admitsa complete metric g0 of bounded curvature from above and below. If R(g0) ≥ 0then Mn admits a smooth Riemannian metric g satisfying R(g) ≥ 0. If R(g0) ≥0 then Mn admits a smooth Riemannian metric g satisfying R(g) ≥ 0. If n = 3and Ricci(g0) ≥ 0 then M3 admits a smooth Riemannian metric g satisfyingRicci(g) ≥ 0. If n = 4 and I(g0) ≥ 0, then M4 admits a smooth Riemannianmetric g satisfying I(g) ≥ 0 (see Theorem 6.2, 6.6 and 6.7).

Theorem 1.3 is proved by flowing the metric g0 with the hflow from The-orem 1.1, and showing that the smooth solution g(t) also satisfies the curvaturebounds from below.

We may slightly weaken the hypotheses of theorem 1.3 in the Ricci curva-ture case. We replace the bound on the curvature from above by a Lipschitzcondition.

8

Theorem 1.4. Let M3 be a three manifold, and g0 be a complete locally Lip-schitz metric on M which satisfies Ricci(g0) ≥ 0, in the weak sense of definition(6.4). Then the solution g(x, t), t ∈ (0, T ] to h flow of g0 exists (for some smoothmetric h) and satisfies Ricci(g(x, t)) ≥ 0 for all t ∈ (0, T ] in the usual smoothRiemannian sense (see Theorem 6.5).

§ 2. A priori parabolicity.

Let g0 be C0, and h be δ fair to g0. We define the function φ0 : M → Mas follows,

φ0(p) = gij0 (p)hij(p).

We may always choose local co-ordinates around a fixed point p, so that at pwe have hij(p) = δij , and gij(p) = δijλi(p), and hence

φ0(p) =1

λ1+

1

λ2+ . . .+

1

λn

,

and hence, δ fairness implies that supp∈M φ0(p) ≤ nδ< ∞. We will use similar

techniques to those of Shi [Sh] to obtain a priori estimates for a priori smoothsolutions to the hflow with initial C∞ data g0, where h is a metric δ fair to g0(0 < δ <∞).

Lemma 2.1. Let D be a compact region in M , Let g0 be a C∞(D) metric andh a metric on M which satisfies

g0 ≥ (1 − δ)h. (2.1)

Let g(t), t ∈ [0, T ] be a C∞(D × [0, T ]) solution to the h flow with Dirichletboundary conditions g|∂D(·, t) = g0(·), g(0) = g0. For every σ > 1 there existsan S = S(n, k0, δ, σ) such that

g(t) ≥ (1 − σ)(1 − δ)h,∀t ∈ [0, S] ∩ [0, T ].

Proof : We define the function φ by

φ(x, t) = gj1i1(x, t)hi1j2gj2i2(x, t)hi2j3 . . . g

jmim(x, t)himj1 ,

and note that it satisfies

sup(x,t)∈D×0∪∂D×[0,T ]

φ(x, t) = supx∈D

φ0(x) ≤n

(1 − δ)m, (2.2)

9

due to (2.1). Using (1.5) as in [Sh] Lemma 2.2 we see that

∂

∂tφ ≤ gij∇i∇jφ+ k0φ

1+ 1m ,

and hence

∂

∂t(φ)−

1m ≥ gij∇i∇j(φ

− 1m ) − (m+ 1)φ

1m ∇iφ

− 1m ∇jφ

− 1m − 1

mk0.

This implies

(φ)−1m +

1

mkt ≥ inf

(x,t)∈D×0∪∂D×[0,T ](φ)−

1m

≥ 1

supx∈D×0 φ1m

≥ (1 − δ)

n1m

,

in view of (2.2) and the parabolic maximum principle. Rewriting the above

inequality, we obtain 1φ≥

(

(1−δ)

n1m

− 1mkt

)m

, which implies that

φ ≤( (1 − δ)

n1m

− 1

mkt

)−m

,

which may be rewritten in co-ordinate form as

1

λ1m +

1

λ2m + . . .+

1

λnm ≤

( (1 − δ)

n1m

− 1

mkt

)−m

.

Since all the terms on the left hand side of the above equation are positive, we

see that(

1λi

)m

≤(

(1−δ)

n1m

− 1mkt

)−m

, for fixed i ∈ 1, 2, . . . n which implies

that

λi(x, t) ≥( (1 − δ)

n1m

− 1

mkt

)

.

This means for any given σ > 0, we may find an S = S(k0, n, σ), such that

λi ≥ (1 − δ)(1 − σ),∀t ∈ [0, S] ∩ [0, T ].

♦We wish also to obtain bounds from above for g in terms of h.

10

Lemma 2.2. There exists an δ(n) > 0 such that the following is true. Let Mn

be an n-dimensional manifold, D be a compact region inM , and δ ≤ δ(n) and g0be a C∞(D) metric and h a metric on M which satisfies h ≤ g0 ≤ (1+ δ)h. Letg(t), t ∈ [0, T ] be a C∞(D×[0, T ]) solution to the h flow with Dirichlet boundaryconditions g|∂D(·, t) = g0(·), g(0) = g0. There exists an S = S(n, k0) > 0 suchthat

gij ≤ (1 + 2δ)hij ,

for all t ∈ [0, S] ∩ [0, T ].

Proof : Choose m(n), and α(m) so that

m ≥ 64n2 + 1 + 2c(n, h), (2.3)

and large enough so that

(2n)1m ≤ (1 + σ) (2.4),

and choose α = α(m) > 0 so small that

(1 − α)m−1 ≥ 3

4. (2.5)

By the previous theorem, there exists an S = S(n, k0, α) > 0 such that

g(·, t) ≥ h(1 − α) for all t ∈ [0, S] ∩ [0, T ]. (2.6)

Equation (2.3) implies that

4 − (m− 1)(1 − α)m−1 1

2(2n)1+1m

≤ 4 − (1 − α)m−1 64n2

8n2

= 4 − 8(1 − α)m−1,

which combined with (2.5) gives

4 − (m− 1)(1 − α)m−1 1

2(2n)1+1m

≤ −2. (2.7)

Similar to Shi [Sh], we define

G = hi1j1gj1i2hi2j2gj2i3 . . . h

imjmgjmi1 , (2.8)

11

and for (1+ δ)m − [ 12n

](G) > 0, we define F = 1(1+δ)m−[ 1

2n](G)

. See [Sh], Lemma

2.3, equation (79). In our preferred co-ordinate system,

F (x, t) =1

(1 + δ)m − [ 12n

](λm1 (x, t) + λm

2 (x, t) + . . . λmn (x, t))

. (2.9)

From (2.8) and the fact that h is 1 + δ fair to g0 we see that

(1 + δ)m − [1

2n](G(x, 0)) ≥ (1 + δ)m − [

n

2n](1 + δ)m ≥ 1

2(1 + δ)m, (2.10)

and hence F (x, 0) = 1(1+δ)m−[ 1

2n](G(x,0))

< ∞, is well defined at time zero.

Since D is compact, and g(x, t) is a priori smooth, there is some maximal T ′ ∈[0, T ]∩ [0, S], such that F (·, t) is well defined (not infinity) for all t ∈ [0, T ′), andif supD×[0,T ′) F < ∞, then [0, T ′] = [0, T ] ∩ [0, S]. Since the function F is well

defined on [0, T ′), we see that (1 + δ)m − [ 12n

](G(x, t)) > 0 for all t ∈ [0, T ′),that is

∑ni λ

mi ≤ (2n)(1+ δ)m. If we choose δ = δ(n) sufficiently small, then we

may assume that

n∑

i

λmi ≤ (4n) for all t ∈ [0, T ′). (2.11)

Then we calculate as in Shi (but remove the error stemming from (82),that is he should have there −2 λi

λαRiαiα not −2 1

λiλαRiαiα. Once corrected one

calculates as he does and no problem occurs) up to equation (89) on page 241,that

∂

∂tF ≤ gαβ∇α∇βF + F 2 m

(1 − α)

(

4 − (m− 1)(1 − α)m−1 1

(1 + δ)(2n)1+1m

)

|h∇g|2

+mn2k0

(1 − α)F 2 ∀t ∈ [0, T ′),

where here we have used (2.3),(2.6) and (2.11) to arrive at this evolution equa-tion in the same way Shi does. Substituting (1 + δ) < 2 and then (2.5) into theabove we get

∂

∂tF ≤ gαβ∇α∇βF +

mn2k0

(1 − α)F 2, ∀t ∈ [0, T ′).

From the parabolic maximum principle, and equation(2.10) we obtain

F (·, t) ≤ a(t) ≤ a0

s(t), (2.12)

12

where a(t) = sup(x,t)∈D×[0,T ′) F (x, t), a0 = a(0), s(t) = 1 − a0b0t, and b0 =mn2k0

(1−α) . Substituting (2.9) into (2.12) we see that

(1 + δ)m − 1

2nG ≥ s(t)

a0≥ s(t)

(1 + δ)m

2∀t ∈ [0, T ′),

in view of (2.10). Rewriting this equation we get

G ≤ 2n(1 + δ)m(1 − s(t)

2) = 2n(1 + δ)m(

1

2+ a0b0t),∀t ∈ [0, T ′), (2.13)

in view of the definition of s(t). Without loss of generality, we assume that S ≤1

4a0b0, which implies that a0b0t ≤ 1

4 for all t ∈ [0, T ′), which when substitutedinto (2.13) implies that

λm1 (x, t) + λm

2 (x, t) + . . .+ λmn ≤ 3n

2(1 + δ)m,∀t ∈ [0, T ′) (2.14).

Equation (2.14) then implies that

(1 + δ)m − [1

2n](G) ≥ 1

4(1 + δ)m > 0∀t ∈ [0, T ′),

and hence T ′ = min(S, T ). From equation (2.14) we see, for fixed i ∈ 1, . . . n,that

λi(x, t) ≤ (3n

2)

1m (1 + δ) for all t ∈ [0, S] ∩ [0, T ].

Substituting (2.4) into this inequality gives us λi(x, t) ≤ (1 + δ)(1 + σ) asrequired. ♦Theorem 2.3. Let D be a compact region in M , and g0 be a C∞(D) metricand h a metric on M which satisfies 1

δ1h ≤ g0 ≤ δ2h, where 1+(1/2)δ(n) ≥ δi ≥

1 for i ∈ 1, 2. Let g(t), t ∈ [0, T ] be a C∞(D×[0, T ]) smooth solution to h flowsatisfying the Dirichlet boundary conditions g|∂D(·, t) = g0(·), g(·, 0) = g0(·).Then there exists an S = S(n, k0) > 0 such that

1

2δ1hij ≤ gij ≤ 2δ2

for all t ∈ [0, S] ∩ [0, T ].

Proof : The proof relies on some simple scaling arguments. First note that ifg(·, t) is a solution to hflow, then so is cg(·, 1

ct), with initial data cg(·, 0). Let

g(·, t) = δ1g(·, 1δ1t). Then g0(·) = g(·, 0) = δ1g0(·) satisfies h ≤ g(·, 0) ≤ δ1δ2h.

From the previous two lemmas, we may find an S = S(n, k0) so that hij12 ≤

gij ≤ 2δ1δ2hij , for all t ∈ [0, S] ∩ [0, T ]. Multiplying the above equation by 1δ1

,we obtain the result. ♦

13

Lemma 2.4 . Let g(t), t ∈ [0, S] be a C∞(D × [0, S]) solution to the h flow,for some h which is 1 + ǫ(n) fair to g(t), for all t ∈ [0, S] (ǫ(n) to be specifiedin the proof below) with Dirichlet boundary conditions g(·, t)|∂D = h(·)|∂D,g(·, 0) = g0(·). Then

supx∈D

|h∇g(x, t)|2 ≤ c(n, h, δ,D, supD

|h∇g0|2) for all t ∈ [0, S].

Proof :

Let

φ(x, t) = gj1i1(x, t)hi1j2gj2i2(x, t)h

i2j3 . . . gjmim(x, t)himj1 . (2.15)

We may always choose co-ordinates at a point so that hij(p, t) = δij , gij(p, t) =λiδij , and then φ(p, t) = (λ1)

m + . . .+ (λn)m. Notice that since (1 − ǫ)h(x) ≤g(x, t) ≤ (1 + ǫ)h, we get

1 − ǫ ≤ λi ≤ (1 + ǫ) (2.16)

in our preferred co-ordinate system. We will calculate the evolution equation

for the function ψ = (φ(x, t)+a(n))|h∇g(x, t)|2. Calculating as in Shi, and using

the fact that h is a priori (1 + ǫ) fair to g(x, t), we see, as in Shi([Sh],§ 4 , page250, estimate (33)) that the function ψ satisfies

∂

∂tψ ≤ gαβ∇α∇βψ − 1

2ψ2 + c0(n, k0, k1),

as long as ǫ(n) > 0 is chosen small enough (as is the case in [Sh],§ 4, (33)) . Fornow we only need the fact that this implies that

∂

∂t(ψ − c0t) ≤ gαβ∇α∇β(ψ − c0t),

although the term −12ψ

2 shall be important later. From the maximum principlewe obtain that

supD×[0,S]

(ψ − c0t) ≤ sup∂D×[0,S]∪D×0

ψ. (2.17)

Applying Lemma 3.1, VI, § 3 [LSU] to the evolution equation (1.5) for g−h, we

get sup∂D×[0,S] |h∇g| ≤ c(n, δ, ∂D), in view of the apriori parabolicity (Theorem

2.3). Upon substituting this inequality into (2.17) we obtain the result. ♦

14

Lemma 2.5. Let g(t), t ∈ [0, S] be a C∞(Br(x0)× [0, S]) solution to the h flowwhich is 1 + ǫ(n) -fair to h, ǫ(n) as in Lemma 2.5, where Br(x0) is a ball ofradius r and centre x0 with respect to the metric h. Then

supx∈B r

2(x0)

|(h∇m

)g(x, t)|2 ≤ c(n, h,m, g0|Br) for all t ∈ [0, S].

Proof : We proceed as in the proofs of Lemmas 4.1 and 4.2 of Shi [Sh]. ♦§ 3. Solution to the Dirichlet problem.

Theorem 3.1. Let g0 be a C∞(D) metric and h a metric which is 1+ ǫ(n) fairto g0 on D, and g0|∂D = h|∂D where D ⊂ M is a compact domain in M (ǫ(n)as in Lemma 2.4). There exists an S = S(n, k0, δ) > 0 and a family of metricsg(t), t ∈ [0, S] which solves h flow, h is 1 + 2ǫ(n) fair to g(t) for all t ∈ [0, S],and g|∂D(·, t) = g0(·), g(0) = g0.

Proof : We consider the family of evolution problems

sL(sg(x, t)) = 0, for (x, t) ∈ D × [0, T ],sg(·, 0) = (1 − s)g0(·) + sh(·),sg(x, t) = (1 − s)g0(x) + sh(·)(x) = h(x) for all x ∈ ∂D, for all t ∈ [0, T ],

(3.1)where

sL(v)kl =∂

∂tvkl − vij∇i∇jvkl + vcdvkph

pqRlcqd + vcdvlphpqRkcqd

− 1

2vcdvpq(∇avpc · ∇bvqd + 2∇cvap · ∇qvbd

− 2∇cvap · ∇dvbq − 4∇avpc · ∇dvbq),

where Rabcd = Riem(h)abcd and ∇ =h∇, v(·, t) = (1 − s)v(·, t) + sh(·) and

s ∈ [0, 1] (strictly speaking the notation v should be sv since the operator ˆdepends on s). Then if sg(x, t), t ∈ [0, T ] is a classical solution to (3.2), onemay verify that v(x, t) = sg(x, t), t ∈ [0, T ] solves

∂

∂tvkl = vij∇i∇j vkl − (1 − s)vcdvkph

pqRlcqd − (1 − s)vcdvlphpqRkcqd

− 1

2vcdvpq(∇avpc · ∇bvqd + . . .),

v(·, 0) = (1 − s)g0 + sh,

v(x, t) = (1 − s)g0(x) + sh(x), for all x ∈ ∂D, for all t ∈ [0, T ].

15

This is essentially equation (1.5), and we may use the same techniques we usedthere to obtain Lemma 2.3, 2.4 and 2.5 for all classical solutions sg(·, t), t ∈ [0, S]of Ls(

sg) = 0 (independently of s) for all t in [0, S], where S = S(k0, n, 1+ ǫ) isas in Theorem 2.3. Then h is a priori 1+2ǫ fair to sg(t) for all t ∈ [0, S] becauseof Theorem 2.3, and the fact that h is 1 + ǫ fair to sg(0). Also using Lemma

2.4,and Lemma 2.5 we get that supD |(h∇m

)sg(t)| ≤ c(D,m, n, g0, h, 1 + ǫ).

We may now use the same argument as in [LSU], Theorem 7.1, ChapterVII to show that a smooth solution exists on a time interval [0, S]. (the argu-ment used there is based on the Leray-Schauder fixed point argument of [LSU],Theorem 6.1 of Chapter V.

Notice that in this argument, all derivatives (time like and spatial) of thesolutions sg(·, t), t ∈ [0, T ), and hence of the solution g(x, t), t ∈ [0, T ) to (1.5),are bounded by constants depending on g0,h,1 + ǫ, n and D , as long as T ≤S(n, k0, δ, δ), S as in thm. 2.3. ♦.

§ 4. A priori interior estimates for the gradientand higher order mixed derivatives of g.

To obtain interior estimates for the first derivative of g(x, t) we may modifythe argument used in Shi.

Lemma 4.1 . Let g(t), t ∈ [0, S] be a C∞(D × [0, S]) solution to the h flow,for some h which is 1+ ǫ(n) fair to g(t),for all t ∈ [0, S] (ǫ(n) as in Lemma 2.4).Then

supB(h)(x0,r)

|h∇g(x, t)|2 ≤ c(n, h,1

r)1

t, for all t ∈ [0, S],

where B(h)(x0, r) denotes a ball of radius r with centre x0 calculated withrespect to the metric h.

Proof : In Lemma 2.4 we saw that the functionψ = (φ(x, t) + a(n))(|h∇g(x, t)|2, satisfies

∂

∂tψ ≤ gαβ∇α∇βψ − 1

2ψ2 + c0(n, ǫ(n), k0, k1) for all (x, t) ∈M × [0, T ].

We have been careful to include the dependence of the constant c0 here, andnote that it does not depend upon g0 or D. Using this inequality we see thatthe function f(x, t) = ψ(x, t)t satisfies

∂

∂tf ≤ gαβ∇α∇βf − 1

2tf2 + c0t+

f

tfor all (x, t) ∈M × [0, T ] (4.2).

16

Let x0 be fixed in M , and Ω = B(h)(x0, r) a ball of radius r in M , where herewe have used the notation B(h)(x0, r) to make clear that the ball B(h)(x0, r)is calculated in terms of the metric h. That is

Ω = B(h)(x0, r) = x ∈M : dist(h, x, x0) ≤ r.

For fixed x0, we use the background metric h and the Hessian comparisonprinciple to construct a time independent cut off function η satisfying

η(x) = 1∀x ∈ B(h)(x0, r), (d1)

η(x) = 0∀x ∈M −B(h)(x0, 2r), (d2)

0 ≤ η(x) ≤ 1∀x ∈M, (d3)

h|∇η|2 ≤ c1(1

r)η, (d4)

∇α∇βη ≥ −c2(supM

˜Riem,1

r)hαβ = c2(k0,

1

r). (d5)

(see [Sh] Theorem 1.1). Note that the constants c1 and c2 decrease (get better)as r increases. Note also that the function is C∞ almost everywhere, andLipschitz everywhere. If we mollify the function η then we obtain a C∞ functionsatisfying the same properties, but for slightly different balls (B(h)(x0, r − ǫ)and B(h)(x0, 2r + ǫ)) and slightly different constants (c1 + ǫ , c2 + ǫ).

Using (d3) in equation (4.2) we get

∂

∂t(fη) ≤gαβ∇α∇β(fη) − 1

16tf2η − 2gαβ∇αf∇βη − fgαβ∇α∇βη

+ c(n, r, h) +fη

t, for all x ∈ Ω, t ∈ [0, S]

(4.3)

In this proof a large number of constants depending on n, h, r appear. Tosimplify the proof we use a small c to denote a constant c = c(n, h, r). We oftenrewrite algebraic expressions involving c and other constants simply as c. Forexample n

2 c2 + c4 would be turned into c.

Let us assume that (x0, t0) is an interior point of B(h)(x0, 2r)×[0, T ] wherethe supremum of (fη)(x, t), for (x, t) ∈ B(h)(x0, 2r) × [0, T ] is obtained. Since(fη)(x0, t0) is a maximum of the function fη(·, t0), we get

−2gαβ∇αf∇βη = −2gαβ 1

η∇α(fη)∇βη + 2gαβ f

η|∇η|2

= 2gαβ f

η|∇η|2,

17

at the point (x0, t0), which implies

−2gαβ∇αf∇βη ≤ cf, (4.4)

at (x0, t0), in view of (d4). Substituting inequality (4.4) and (d5) into (4.3),we obtain ∂

∂t(fη)(x0, t0) ≤ gαβ∇α∇β(fη) − 1

16t0f2η + cf + c+ fη

t0, at (x0, t0),

which implies that

0 ≤ ∂

∂t(fη)(x0, t0) ≤ − 1

16t0f2η + cf + c+

fη

t0, (4.5)

in view of the fact that ∂∂t

(fη)(x0, t0) ≥ 0, and gαβ∇α∇β(fη)(x0, t0) ≤ 0, atthe maximal point (x0, t0). Multiplying equation (4.5) by η(x0) we get

1

16t0(fη)2(x0, t0) − c(fη)(x0, t0) −

(fη)(x0, t0)

t0≤ c,

which implies

(fη)(x0, t0)( 1

16t0(fη)(x0, t0) − c− 1

t0

)

≤ c,

that is (fη)(x0, t0) ≤ c(n, h), in view of the fact that t0 ≤ S = S(n, k0).This implies that supB(h)(x0,r) f(x, t) ≤ max(supB(h)(x0,2r) f(x, 0), c) = c, sincef(x, 0) = 0. Using 1+ ǫ(n) fairness and the definition of f we obtain the result.♦.

We have now obtained the important a priori parabolic estimates and thea priori interior gradient estimate. To obtain further interior estimates we mayapply the above techniques and those of Shi.

Lemma 4.2. Let g(t), t ∈ [0, S] be a C∞(D)× [0, S] solution to the h flow, forsome h which is 1 + ǫ(n) fair to g(t), for all t ∈ [0, S], ǫ(n) as in Lemma 2.4Then

supB(h)(x0,r)

|h∇i

g|2 ≤ c(n, i, 1r, k0, . . . , ki+1)

tp(i,n)for all t ∈ [0, S], i ∈ 1, 2, . . .,

where p(i, n) is an integer and B(h)(x0, r) denotes a ball of radius r containedin D.

Proof : Whenever we write |T | for some tensor T in the following calculation,

we shall meanh|: the modulus of the tensor taken with respect to the metric

18

h. We calculate similar to Shi ([Sh], Lemma 4.1, equation (4),(5),...) using theevolution equation (1.5) for h flow, that

∂

∂t

h|h∇m

g|2 = gijh∇i

h∇j |h∇

m

g|2 − 2gijh∇i(h∇

m

g)h∇j(

h∇m

g)

+ 2∑

i+j+k=m,i,j,k,l≤m

h∇i

g−1 ∗ h∇j

g ∗ h∇k

Riem(h) ∗ h∇m

g

+ 2∑

i+j+k+l=m+2,i,j,k,l≤m+1

h∇i

g−1 ∗ h∇j

g−1 ∗ h∇k

g ∗ h∇l

g ∗ h∇m

g,

for all x ∈ Ω, for all t ∈ [0, S]

where here T ∗ S, (T and S are tensors), refers to some trace with respect tothe metric h which results in a tensor of the appropriate type (in the aboveformula the tensor product should result in a function). Using the fact that his C∞ and 1 + ǫ fair to g(x, t), we get

∂

∂t

h|h∇m

g|2 ≤ gijh∇i

h∇j |h∇

m

g|2 − 2gijh∇i(h∇

m

g)h∇j(

h∇m

g)

+2c(n, h)∑

i+j+k=m,i,j,k,l≤m

|h∇i

g||h∇j

g||h∇m

g|

+2∑

i+j+k+l=m+2,i,j,k,l≤m+1

(1 + ǫ)4|h∇i

g||h∇j

g||h∇k

g||h∇l

g||h∇m

g|.

for all x ∈ Ω, for all t ∈ [0, S](4.6)

We will prove interior gradient bounds by induction in m. Assume that weknow already that

h|h∇i

g|2 ≤ c(n, h,m)

tp(i,n)for all x ∈ Ω, t ∈ [0, T ], i ∈ 1, 2, . . . ,m− 1,

where p(i, n) is an integer. We show that this implies a similar bound for

|h∇mg|2. We will write c(n, h,m) simply as c, to simplify readability of the

proof (as in the proof of Theorem 4.1). Combining the evolution equation (4.6)with our induction hypothesis, we obtain

∂

∂t

h|h∇m

g|2 ≤ gijh∇i

h∇j |h∇

m

g|2 − 2gijh∇i(h∇

m

g)h∇j(

h∇m

g)

+c

tq|h∇

m

g| + c|h∇m

g|2

+c∑

i+j+k+l=m+2,m≤i,j,k,l≤m+1

|h∇i

g||h∇j

g||h∇k

g||h∇l

g||h∇m

g|,

19

which implies

∂

∂t|h∇

m

g|2 ≤ gijh∇i

h∇j |h∇

m

g|2 − 2gijh∇i(h∇

m

g)h∇j(

h∇m

g)

+c

tq|h∇

m

g| + c|h∇m

g|2 + c|h∇m

g|2(|∇2g| + |∇g|2)

+c|h∇m+1

g||h∇m

g||∇g|,

where q = q(n, p,m) is some integer. In what follows we shall freely replacepowers of q by q. For example 2q2 will be replaced by q. Since m ≥ 2, we mayuse our induction hypothesis on the gradient terms of order one and two, andthe Cauchy-Schwarz inequality to obtain,

∂

∂t|h∇

m

g|2 ≤ gijh∇i

h∇j |h∇

m

g|2 − 2gijh∇i(h∇

m

g)h∇j(

h∇m

g)

+c

tq|h∇

m

g|3 +(1 + ǫ)

2|h∇

m+1g|2 +

c

tq.

(4.7)

Finally, substituting

2gijh∇i(h∇

m

g)h∇j(

h∇m

g) ≥ 1

(1 + ǫ)

h|h∇m+1

g|2

into (4.7), we get

∂

∂t|h∇

m

g|2 ≤gijh∇i

h∇j |h∇

m

g|2 − 1

(1 + ǫ)

h|h∇m+1

g|2

+c

tq|h∇

m

g|3 +c

tqfor all x ∈ Ω, for all t ∈ [0, S].

Similarly

∂

∂t|h∇

m−1g|2 ≤ gijh∇i

h∇j |h∇

m−1g|2 − 1

(1 + ǫ)

h|h∇m

g|2 +c

tq,

in view of the induction hypothesis. Following Shi ([Sh], Lemma 4.2 equation(80)) we define

ψ(x, t) = (a+h|h∇

m−1g|2)h|h∇

m

g|2,

20

where a is a constant to be chosen later. In view of the previous two evolutionequations we get

∂

∂tψ(x, t) ≤gijh∇i

h∇jψ + (− 1

(1 + ǫ)

h|h∇m

g|4 +c

tqh|h∇

m

g|2)

+ (a+h|h∇

m−1g|2)(− 1

(1 + ǫ)

h|h∇m+1

g|2 +c

tq|h∇

m

g|3 +c

tq)

− 2gijh∇i

h|h∇m−1

g|2h∇j

h|h∇m

g|2

≤− 1

2(1 + ǫ)

h|h∇m

g|4 − a

2(1 + ǫ)

h|h∇m+1

g|2 +c(1 + a4)

tq

− 2gijh∇i

h|h∇m−1

g|2h∇j

h|h∇m

g|2, (4.8)

where here it is clear that we have used our inductive assumption (and not thatof Shi) and the Cauchy-Schwartz inequality. The last term satisfies

−2gijh∇i

h|h∇m−1

g|2h∇j

h|h∇m

g|2 ≤ 2(1 + ǫ)h|h∇

m

g|2h|h∇m−1

g|h|h∇m+1

g|

≤ 1

4(1 + ǫ)

h|h∇m

g|4 +c

tqh|h∇

m+1g|2,

where we have used our inductive assumption on the termh|h∇

m−1g| and the

Cauchy-Schwartz inequality. Substituting this inequality into (4.8) we get

∂

∂tψ ≤ gijh∇i

h∇jψ+ (c

tq− a

2(1 + ǫ))

h|h∇m+1

g|2 − 1

4(1 + ǫ)

h|h∇m

g|4 +c(1 + a4)

tq,

(4.9)where here it is clear that we have used our inductive assumption (and not thatof Shi), and (1 + ǫ) fairness to obtain upper and lower bounds for the metricg(x, t) in terms of h (as in Shi). We now modify ψ to our purposes. We considerthe new function w defined by

w = tq+1(2c(1 + ǫ)t−q + |h∇m−1

g|2)|h∇m

g|2,

where q and c are the constants appearing in equation (4.9) (the constant q =q(n, p,m) is now fixed!). That is we have chosen a to be a constant dependingon t (who’s time derivative we must therefore calculate), and multiplied thewhole function by a power of t. Note that this function is zero at time zero andhence must attain a maximum at some time bigger than zero. Without loss of

generality we may assumeh|h∇

ig|2 ≤ a(t), for all i ∈ 1, . . . ,m. Also note that

21

in (4.9) we then get that ( ctq − a

2(1+ǫ) ) ≤ 0 and c(1+a4)tq ≤ c

t5q . In view of these

two inequalites and (4.9), we get

∂

∂tw ≤ gijh∇i

h∇jw − 1

4(1 + ǫ)tq+1|h∇

m

g|4 +(q + 1)

tw − qc|h∇

m

g|2 +c

t4q,

= gijh∇i

h∇jw − 1

4(1 + ǫ)

w2

tq+1(a(t) + |h∇m−1g|2)2

+(q + 1)

tw − c|h∇

m

g|2 +c

t4q,

where the last equality follows from the definition of w. Hence

∂

∂tw ≤ gijh∇i

h∇jw − 1

48(1 + ǫ)

w2

tq+1a2(t)+

(q + 1)

tw +

c

t4q

= gijh∇i

h∇jw − 1

c2tq−1w2 +

(q + 1)

tw + cw +

c

t4q,

(4.10)

in view of the fact that a(t) = ct−q, andh|h∇

m−1g|2 ≤ a(t). Now, as in the

estimate of the first derivative of g, we multiply w by a cut off function η andcalculate the evolution equation of wη. Using (4.10) and d1 – d5 as in theestimate of the first derivative of g, we get

∂

∂t(wη) ≤ gijh∇i

h∇j(wη) −1

c2tq−1w2η +

(q + 1)

t(ηw) + cw +

c

t4q.

At an interior point (x0, t0) of Ω× [0, T ] which is a maximum of w we argue asin the proof of Lemma 4.1 to get

(ηw)(x0, t0)(1

c2t5q−10 (ηw)(x0, t0) − (q + 1)) ≤ c,

from which we obtain (ηw)(x0, t0) ≤ c

t5q

0

. Using the definition of w and the

above inequality, we get

tq+1(4ct−q + |h∇m−1

g|2)|h∇m

g|2 ≤ c

t5q, for all x ∈ Br(x0), t ∈ [0, T ],

which implies the desired result ♦.

Theorem 4.3. Let g(t), t ∈ [0, S], h be as in Theorem 4.2. Then

supx∈M

|h∇i

g(x, t)|2 ≤ c(n, i, Ri)

tifor all t ∈ [0, T ], i ∈ 1, 2, . . .,

22

where R2i = max(k0, k1, k2, . . . , ki), kj as in definition 1.1.

Proof : We derive this corollary from Theorem 4.2 and a scaling argument.Let g(·, t) = 1

Rg(x,Rt) for some constant R > 0, and h(·) = 1

Rh(·). Then h is

1+ ǫ fair to g(x, t) and g(·, t),t ∈ [0, SR

] solves hf low. Without loss of generality

we assume that S ≤ 1. For a given t0 ∈ [0, S], let R = t0 ≤ 1. Then the ki ≤ ki,

where ki = suph

M|h∇iRiem(h)|2. Hence by Lemma 4.2 we get

h|∇ig|2|(x,1) ≤ c(n, k0, . . . , ki) ≤ c(n, k0, . . . , ki).

But

h|∇ig|2|(x,1) = (hj1k1 hj2k2 . . . hjiki hmnhpq∇j1∇j2 . . . ∇jigmp∇k1∇k2 . . . ∇ki

gnq)(x, 1)

= Ri|h∇i

g|2(x,R),

from which the result follows. ♦§ 5. Existence of entire solutions

Lemma 5.1. Let g0 be a C∞(M) metric and h a metric on M which is 1+ǫ(n)fair to g0, ǫ(n) as in Lemma 2.4. There exists a T = T (n, k0) and a family ofmetrics g(t), t ∈ [0, T ] in C∞(M × [0, T ]) which solves h flow, h is (1 + 2ǫ) fairto g(·, t) for t ∈ [0, T ], and

|h∇i

g|2 ≤ ci(n, k0, . . . , ki)

ti, for all t ∈ (0, T ], i ∈ 1, 2, . . ..

Proof : IfM is compact, then we obtain the result using Theorems 3.1, Lemmas2.4,2.5 and 4.3. Let Di, i ∈ 1, 2, . . . ,∞ be a family of compact sets whichexhaust M , Di = B(h)(x0, i), where B(h)(x0, i) is the ball of radius i forsome fixed arbitrary x0, with respect the metric given by h. We set ig0 =ηig0 + (1 − ηi)h where ηi : Mn → R is smooth ηi = 0 on Di −Di−1, ηi = 1 on

Di−2, 0 ≤ ηi ≤ 1 and |h∇m

(ηi)|2 ≤ c(h, i, n,m). Note that ig0 is also 1 + ǫ(n)fair to h.

Leti

g(·, t), t ∈ [0, T ] be the solutions obtained to the Dirichlet problemon Di with boundary data h from Theorem 3.1. Using the interior estimates(Lemma 2.4,2.5) and Arzela-Ascolii Theorem, we may let i → ∞ and take adiagonal subsequence to obtain a limit g(·, t), t ∈ [0, T ] which solves h flow andhas initial data g0. The estimates are satisfied in view of Theorem 4.3. ♦.

23

Theorem 5.2. Let g0 be a complete metric and h a complete metric on Mwhich is 1 + ǫ(n) fair to g0, ǫ(n) as in Lemma 2.4. There exists a T = T (n, k0)and a family of metrics g(t), t ∈ (0, T ] in C∞(M × (0, T ]) which solves h flowfor t ∈ (0, T ],h is (1 + 2ǫ) fair to g(t) for t ∈ (0, T ], and

limt→0

supx∈Ω′

h|g(·, t) − g0(·)| = 0,

supx∈M

|h∇i

g|2 ≤ ci(n, k0, . . . , ki)

ti, for all t ∈ (0, T ], i ∈ 1, 2, . . .,

where Ω′ is any open set satisfying Ω′ ⊂⊂ Ω, where Ω is any open set on whichg0 is continuous.

Remark. In particular, if M is compact, and g0 is continuous then g(·, t) →g0(·) uniformly on M as t→ 0.

Proof : Let α

g0α∈N be a sequence of smooth metrics which satisfylimα→∞α

g0 = g0, where the limit is uniform in the C0 norm. It follows thenthat h is (1+ ǫ

2 ) fair toα

g0 for all α ≥ N for some N ∈ N. We flow each metricα

g0 by h flow (using Lemma 5.1) to obtain a family of metricsα

g(·, t), t ∈ [0, T ],T = T (n, k0) independent of α which satisfy

|h∇j

(α

g(·, t))|2 ≤ cjtj, for all t ∈ (0, T ],

independently of α, for all α ≥ N . We then obtain a limiting solution g(x, t) ,t ∈ (0, T ) via g(x, t) = limα→∞

α

g(x, t), which is defined for all t ∈ (0, T ). Thislimit is obtained using the Theorem of Arzela-Ascolii (is uniform on compactsubsets of M), and it may be necessary to pass to a sub-sequence to obtainthe limit. It remains to show that the metrics g(·, t)|Ω′ uniformly approachesg0(·)|Ω′ as t approaches zero. As a first step we obtain estimates on the rate atwhich g(·, t) → g0(·, t) as t→ 0 if g0(·) is smooth.

Let ǫ > 0 be given. Arguing as in [Sh] Lemma 2.2, we see from (66) and(68) in the proof of Lemma 2.2, and using (1 + ǫ(n)) fairness that gij satisfies

∂

∂tgij ≤ gαβ∇α∇βg

ij + c(h, n)gij − Sij ,

where Sij is a positive tensor obtained from the square of ∇g (the last term in[Sh] Lemma 2.2 , equation (68)). Since Sij is positive, we get

∂

∂t(gij − lij) ≤ gαβ∇α∇β(gij − lij)+c(h, n)(gij − lij)+gαβ∇α∇βl

ij +c(h, n)lij ,

(5.1)

24

for any time independent tensor lij . Fix x0 in Ω′, and fix a co-ordinate chartaround x0, ψ : U →M , x0 ∈ U ⊂⊂ Ω. Define the (0, 2) tensor l by

l(V,W )(x) = Vi(x)Wj(x)hqp(x0)gqi0 (x0)h

pj(x),

where on the right hand side we have used our fixed co-ordinate chart to help usdefine this tensor. That this is a well defined tensor (for example linearity) fol-lows from the definition. Notice that the right hand side in the above definition*is dependent* on the coordinate chart. That is we have used our fixed co-ordinate chart to help us define this tensor. Also notice that lij(x0) = gij

0 (x0).By definition of l , we get

h|gij0 (x) − lij(x)| ≤ h|gij

0 (x) − gij0 (x0)| +

h|gij0 (x0) − hqp(x0)g

qi0 (x0)h

pj(x)∣

∣

≤ ǫ

2, (5.2)

for all x ∈ B(h)(x0, r) ⊆ U for some small r = r(g0, h, ǫ) > 0, where the lastinequality follows from the continuity of gij

0 and the continuity of hij . Thisgives us that

(1 − 2ǫ)h ≤ l ≤ (1 + 2ǫ)h, (5.3)

for all x ∈ B(h)(x0, r), in view of (5.2) and the fact that h is (1 + ǫ) fair to g0.

We also have that

supB(h)(x0,r)

h|h∇h∇l| ≤ c(h, n, U), (5.4)

as a consequence of the definition of l, the inequality (5.3), and the fact thatU ⊆ Ω is some fixed compact set. Substituting (5.4) into (5.1) and using (5.3)we get

∂

∂t(gij − lij) ≤ gαβ∇α∇β(gij − lij) + c(h, n, U)(gij − lij) + c(h, n, U)hij ,

and hence

∂

∂t(e−ct(gij − lij) − cthij) ≤ gαβ∇α∇β(e−ct(gij − lij) − cthij),

for all x ∈ B(h)(x0, r). Define the tensor f to be f ij = e−ct((gij − lij)− cthij).We construct a cut off function η, as in the proof of Lemma 4.1, for the ball

25

B(h)(x0, r), with η ≡ 1 on B(h)(x0,r2 ) and η ≡ 0 on ∂B(h)(x0, r). Using the

properties of η, as in the proof of Lemma 4.1, we see that

∂

∂t(ηf ij) ≤ gαβ∇α∇β(ηf ij) + c1(ηf

ij) − 2∇αη∇β(f ijη) + c1fij ,

which combined with the fact that f is bounded gives

∂

∂t(ηf ij − c1h

ijt) ≤ gαβ∇α∇β(ηf ij − c1hijt),

where c1 = c1(1r, n, h, U). Using the maximum principle and the fact that

ηf ij−c1hijt = −c1hijt ≤ 0 on ∂B(h)(x0, r), we get that η(·)f ij(·, t)−c1hij(·)t ≤ηf ij(·, 0) ≤ ǫ

2hij for all x ∈ B(h)(x0, r), where the last inequality follows from

(5.2), and the definition of f . This implies that f ij(·, t) ≤ (c1t + ǫ2 )hij for

all x ∈ B(h)(x0,r2 ) and for all t ∈ [0, S] in view of the fact that η is equal

to one on B(h)(x0,r2 ). Substituting t ≤ ǫ

2c1into the above inequality, we get

f ij(·, t) ≤ ǫhij for x ∈ B(h)(x0,r2 ), for all t ≤ ǫ

2c1. Substituting the definition of

f into this inequality, we see that (gij − lij) ≤ ect(ǫ+ ct)hij , which implies that(gij − lij) ≤ 2ǫhij , for all x in B(h)(x0,

r2 ), for all t ≤ T (c, c1, ǫ). Substituting

the inequality (5.2) into the above inequality, we get that

gij − g0ij = (gij − lij) + (lij − g0

ij) ≤ 3ǫhij , (5.5)

for all x in B(h)(x0,r2 ), for all t ≤ T (c, c1, ǫ).

Notice that this argument applies to each solution αg(·, t) defined at thebeginning of the proof. That is, αgij ≤ αg0

ij + 3ǫhij for all x ∈ B(h)(x0,rα

2 ),for all t ≤ T (n,U, h, 1

rα, ǫ), where here we write rα, as rα may possibly depend

on α. In the estimate (5.2) we see that rα > 0 is chosen so that

h|αgij0 (x) − αlij(x)| ≤ ǫ

2,

for all x ∈ B(h)(x0, rα). But then for x ∈ B(h)(x0, rα), β > α we get

h|βgij

0 (x) − βlij

(x)| ≤ h|βgij

0 (x) − αgij0 (x)| + h|αgij

0 (x) − αlij(x)| + h|αlij(x) − βlij

(x)|≤ 3ǫ,

if α, β are chosen large enough, due to the continuity of h, the definition of land the fact that αg0 → g0 in Ω as α→ ∞. So we see that we may choose r > 0independent of α. Hence we obtain (5.5) for the metric g(·, t) = limα→∞ αg(·, t).

26

Let φ be the function defined in (2.15). Arguing similarly to Shi, we seethat φ satisfies

∂

∂tφ ≤ gαβ∇α∇βφ+ c0(h, n) − m2

8|h∇g|2,

as shown in [Sh] (§ 4, equation (19)), where we use (1 + ǫ(n)) fairness as Shidoes. Arguing as above, but for φ instead of gij , we see that there exists aS = S(n, h,Ω′, g0, ǫ) > 0 such that

φ(·, t) ≤ φ(·, 0) + 3ǫ, (5.6)

for all t ∈ [0, S], for all x ∈ Ω′. Using the inequalities (5.5) and (5.6) we see

that supΩ′

h|g0(·) − g(·, t)| ≤ c(n)ǫ1

m(n) , for all t ∈ [0, S], for all x ∈ Ω′. ♦§ 6. Applications to metrics with

‘curvature bounded from above and below’.

Assume that our initial metric g0 is a ‘metric with bounded curvature’on a compact manifold M , in the sense of Aleksandrov ([Al], see [BN] fora good overview). Such metrics are locally C1,α, and using a Theorem ofNikolaev, we may approximate g0 by a family of smooth Riemannian metricswhose sectional curvatures are bounded from above and below by constantswhich approximate the bounds for g0. Furthermore the bounds from above andbelow for Ricci curvature and curvature operator of the approximating metricsare approximately the same as those for g0. We state this more precisely.

Lemma 6.1. Let g be a ‘metric with bounded curvature’ on a manifold M ,with curvature K(g)

C ′ ≤ K(g) ≤ C

in the sense of Aleksandrov. We may approximate g by smooth Riemannianmetrics, α

gα∈N so that

C ′ − 1

α≤ K(

α

g) ≤ C +1

α,

andlim

α→∞|αg − g|C1,β(Ω) → 0, lim

α→∞

g0|αg − g|C0(M) → 0 (6.1)

for open Ω ⊆M whose closure is compact. Furthermore if the curvature satisfies

B′g ≤Ricci(g) ≤ Bg,

(B′G ≤R(g) ≤ BG, )

27

then

(B′ − 1

α)

α

g ≤Ricci(α

g) ≤ (B +1

α)

α

g, (6.2)

((B′ − 1

α)

α

G ≤R(α

g) ≤ (B +1

α)

α

G), (6.3)

Proof : The approximation is achieved by mollifying or regularising g . Herewe use Sobolev averaging and a partition of unity (Nikolaev used De Rhamregularisation to obtain the estimates for the sectional curvatures: see[Re]).Let Us be a locally finite cover by co-ordinate neighborhoods of M , andUs ⊆ Us′ for some co-ordinate chart Us′ . For x ∈ Us ⊆ Us′ define

s,α

gij(x) =

∫

|z|≤1

ρ(z)gij(x− 1

αz)dz,

where here 1α

is small enough so that x − 1αz ∈ Us′ , for all z ∈ B1(0) (which

then means that gij(x − 1αz) is well defined for this fixed co-ordinate system,

(Us′ , ψ)). From work of Berestovskij [Be] we know that (M, g) is actually amanifold (and not just a metric space) and that g is continuous. Nikolaev[Ni] then used these facts to prove that locally g ∈ W 2,p. It then follows (seeBerestovskij, Nikolaev[BN]) that g has a second derivative except on a set ofmeasure zero Σ1 ⊆M . Hence we have the formula

∂

∂xk

∂

∂xl

s,α

gij(p) =s,α

(∂

∂xk

∂

∂xlgij)(p), for all p ∈M (6.4)

where here we have used g ∈ W 2,p in order to make sense of the right handside. The local formula for the Riemannian curvature tensor of a metric g isgiven by

R(g)ijkl =∂

∂xj

∂

∂xkgil +

∂

∂xi

∂

∂xlgjk − ∂

∂xj

∂

∂xlgik − ∂

∂xi

∂

∂xkgjl

+ (g−1 ∗ ∂g ∗ ∂g)ijkl, (6.5)

where the last term is a product of two first derivatives of g and the inverse ofg. Since the first derivatives of g are continuous (as is g itself) we obtain, inview of (6.4) and (6.5)

R(s,α

g)ijkl(p) =s,α

(R(g)ijkl)(p) ± ǫijkl(p), for all p ∈M,

28

where ǫijkl is a tensor, |ǫ| goes to zero as α→ ∞. Using the partition of unityUs, ηs, we construct our approximating metric

α

g = ηs,α

s g. From construction(6.1) is true, and

R(α

g)ijkl =α

(R(g)ijkl) ± cǫijkl, (6.6)

where the constant c = c(η1, . . . , ηN ) comes from taking first and second deriva-tives of the unity functions ηs. The estimates (6.2) and (6.3) then follow simplyby taking traces with respect to

α

g of (6.6), in view of (6.1). ♦If the dimension of X is three, and (X, g0) is a space with curvature

bounded from above and below with Ricci(g0) ≥ 0, then we may use the hflowto flow g0 and so obtain a family g(t), t ∈ (0, T ) of smooth metrics all of whichsatisfy Ricci(g(t)) ≥ 0.

Theorem 6.2. Let g0 be a complete metric with bounded curvature on amanifold M3 of dimension three, −k2

0 ≤ K ≤ k20, such that

Ricci(g0) ≥ 0

in the Aleksandrov sense. Then there exists a metric h which is 1+ǫ(n) fair to g0(ǫ(n) as in Lemma 2.4), a T (n, h, k0) > 0, and a family of smooth Riemannianmetrics g(x, t), t ∈ [0, T ] such that g(x, t), t ∈ (0, T ] solves h flow, g(·, t) → g0(·)uniformly on compact subsets of M as tց 0, and

0 ≤ Ricci(g(x, t)) ≤ c2(k0, n, δ, h) for all t ∈ (0, T ].

Proof :Letα

g0, be the approximating metrics for g0 (obtained from Lemma

6.1), and let h beN

g0 a metric which is 1 + ǫ(3) fair to allα

g0 for α bigenough.Also let

α

g(x, t), t ∈ [0, T ] denote the corresponding solutions to thehflow, and g(x, t), t ∈ (0, T ] the limit (as α→ ∞) solution. Note that each

α

g0

satisfies supM

h|h∇(α

g0)| ≤ c1, from (6.1), and hence we see, arguing as in theproof of Lemma 4.1 (but without multiplying our test function by time t) that

supM×[0,T )

h|h∇(α

g)| ≤ c1. Calculating the evolution equation of the function

t(a+h|h∇(

α

g)|2)h|h∇2(

α

g)|2 as in the proof of Lemma 4.2, and arguing as in the

proof of Lemma 4.2, we get that supM×[0,T ]

h|h∇2(

α

g)| ≤ c2√t, in view of the fact

that |h∇(α

g)|2 is bounded.This then implies that the tensor V (

α

g) =α

gij(Γijk (

α

g) − Γijk (h)) satisfies

supM×[0,T ]

h|V (g)| ≤ c3,

supM×[0,T ]

h|Riem(g)| + h|h∇V (g)| ≤ c4√t.

(6.7)

29

We wish to calculate the evolution of the curvature tensor of the metricsα

gFor a fixed point p, let φ : Bǫ(p) × [0, ǫ] → M be a time dependent localdiffeomorphism satisfying the equation

∂

∂tφi(p, t) = (φt ∗ V )(φt(p), t) = V (φt(p), t),

V (p, t) = (g(t)

Γi

jk − h

Γi

jk)(p, t),

φ(p, 0) = Id(p),

and define g(t) = (φt∗g)(t). As explained in the introduction, g(t) satisfies the

Ricci flow equation (1.1). Also

Ricciij(g(t))(p) = Ricciij(φt∗g(t))(p) = Ricciαβ(g(t))(φt(p))∂

∂xiφα ∂

∂xjφβ ,

which gives us that

∂

∂tRicciij(g(t))(p)

=( ∂

∂tRicciαβ(g(t))(φt(p)) +

∂

∂xηRicciαβ(g(t))(φt(p))

∂

∂tφt(p)

η) ∂

∂xiφα ∂

∂xjφβ

+ Ricciαβ(g(t))(φt(p))∂

∂xi

∂

∂tφα ∂

∂xjφβ + Ricciαβ(g(t))(φt(p))

∂

∂xiφα ∂

∂xj

∂

∂tφβ

=∆Ricci(g)ij + θ(Ricci(g))ij + (∇sRicciij)Vs

+ Ricciljg∇iV

l + Ricciilg∇jV

l, (6.8)

where in order to obtain the last equality we have used the fact that g(t) satisfiesthe Ricci flow equation (1.1) (as explained in the introduction) and here θ isa quadratic term coming from the curvature evolution equation. In dimensionthree, the evolution of the Ricci curvature for a family of metrics g evolving byRicci flow is given by

∂

∂tRicci(g) =

g

∆Ricci(g) + θ(Ricci(g)),

where θ(Ricci) is a quadratic in the Ricci curvature (See [Ha 1]). More specifi-cally, if we choose co-ordinates around x0 for given t0 so that Ricciij(x0, t0) isdiagonal, with values Ricci11 = λ ≤ Ricci22 = µ ≤ Ricci33 = ν, then

θ(Ricci)11 = (µ− ν)2 + λ(µ+ ν − 2λ) ≥ RR11 − 3gklR1kR1l, (6.9)

30

and similarly

θ(Ricci)ij ≥ RijR− 3gklRikRjl. (6.10)

Now we would like to apply Corollary 7.4 to this tensor. But first wehave to check that Ricci(g(t)) ≥ −c holds for some well defined time interval( c some constant). This is done as follows: let η be a cut off function foran arbitrary point x0 with η|B1(x0) = 1, η|M−B2(x0) = 0, 0 ≤ η ≤ 1 and and

|h∇η|2η

≤ 1, h|h∇η| ≤ c(k0, n) and h|∇h∇hη| ≤ c(k0, n): this always exists since

supM |Riem(h)| < ∞. Here Br(x0) is a ball of radius r and centre point x0

w.r.t h. Now use the triangle inequality to get

c(n, k0) ≥h|h∇h∇η| ≥ h|g∇g∇η| − h|h∇h∇η − g∇g∇η|,

which implies

h|g∇g∇η| ≤ c(n, h, c0) +h|(Γ(g) − Γ(h)) ∗ h∇η| ≤ c(c0, n, h),

in view of the fact that Γ(g(t)) − Γ(h) andh∇η are uniformly bounded. Com-

bining this inequality with (6.8), (6.10) and |Riem(g(t))| ≤ c(n,k)√t

we get:

∂

∂t(ηRicci(g(t)) + (1 +m

√t)g)ij

≥ ∆(ηRicci(g) + (1 +m√t)g)ij + η(RijR− 3gklRikRjl) + (∇s(ηRicciij)V

s)

+ ηRicciijg∇iV

l + ηRicciilg∇jV

l − 2

ηgsl∇sη∇l(ηRicci)ij

+ 2|∇η|2η

(Ricci)ij +m

2√t− 2(1 +m

√t)Ricci(g)ij (6.11)

for m = m(n, k) large enough. If there is a first point and time (p0, t0) anddirection W where (ηRicci(g(t))+(1+m

√t)g)(W,W ) = 0 then this must occur

in a compact region (w.l.og. —W— = 1). Assume that m√t ≤ 1. Using the

inequalities |g∇V | ≤ c(n,k)t

, m√t ≤ 1, |Riem(g(t))| ≤ c(n,k0)√

t, |V | ≤ c(n, k0) and

ηRicci(g(t))(W,W ) = −(1 +m√t) in (6.11) we get

0 ≥ ∂

∂t(ηRicci(g(t)) + (1 +m

√t)g)(W,W )

> ∆(ηRicci(g(t)) + (1 +m√t)g)(W,W ) +

m

4√t> 0

31

Clearly θ satisfies the conditions of Theorem 7.3 and so, in view of (6.7) andthe initial conditions and (6.9), we may apply the Corollary 7.4 to the tensorN = Ricci(αg(t)) whose evolution equation is given by (6.8), and satisfies (6.7)to obtain

Ricci(α

g(x, t)) ≥ − 4

α, for all t ∈ [0, T ′′],

where T ′′ = T ′′(3, k0) (note that N = Ricci ≥ −m(n,K0) is satisfied as we havejust shown, so the corollary is apllicable).

We may argue similarly with the function R (scalar curvature), which

satisfies ∂∂tR ≤ ∆R + c(n)R2 to obtain supM×[0,T ′′]

h|Riem(α

g)|2 < c(3, k0, h)(we use the cut off function above and the argument above which was used forηRicci). Letting α go to infinity gives us the result.♦

Hence, if M is compact, we may apply the result of Hamilton ([Ha 2]) toobtain M is diffeomorphic to a quotient of one of the spaces S3 or S2 × R1, orR3 by a group of fixed point free isometries in the standard metric.

When the dimension of M is two we obtain a similar result by examiningscalar curvature and arguing as in the theorem above. Note that in dimensiontwo, the scalar curvature evolves according to the equation ∂

∂tR = ∆R+R2.

Theorem 6.3. Let g0 be a complete metric on a manifold M2,

−k0 ≤K(g0) ≤ 0

(0 ≤K(g0) ≤ k0)

in the sense of Aleksandrov. Then there exists a metric h which is 1+ǫ(n) fair tog0 (ǫ(n) as in Lemma 2.4), a T (n, k0) > 0, and a family of smooth Riemannianmetrics g(x, t), t ∈ [0, T ] such that g(x, t), t ∈ (0, T ] solves h flow, g(·, t) → g0(·)uniformly on compact subsets of M as tց 0, and

−c2(k0, h) ≤R(g(x, t)) ≤ 0 for all t ∈ (0, T ]

(0 ≤R(g(x, t)) ≤ c2(k0, h) for all t ∈ (0, T ]).

We can actually slightly weaken the hypothesis of ‘curvature bounded fromabove’ for Theorem 6.2 to a uniform Lipschitz condition on the initial sequenceof metrics.

Definition 6.4. Let M be a three dimensional manifold, and g be a locallyLipschitz complete metric on M . We say that Ricci(g) ≥ 0, if there exists afamily αgα∈1,2,... of smooth metrics on M which satisfy supM |Riem(gα)| <∞, Ricci(αg) ≥ − 1

α, and limα→∞ supM

g|αg − g| = 0, andg|Γ(αg) − Γ(g)| ≤

32

k for all α ∈ 1, 2, . . ., where k is some constant which does not depend on α,and Γ(g) refers to the Christoffel symbols of g.

Theorem 6.5. Let M3 be a three dimensional manifold and g0 be a completelocally Lipschitz metric on M which satisfies Ricci(g0) ≥ 0, in the sense ofdefinition 6.4. Then there is a solution g(x, t), t ∈ (0, T ] to h flow of g0 forsome smooth metric h and some T = T (k0), and it satisfies Ricci(g(x, t)) ≥0 for all t ∈ (0, T ] in the usual smooth sense.

Proof :The proof is the same as for the case of bounded curvature from above

(theorem 6.2), except that we use the family αg0 coming from the definition6.4, and not the family αg0 constructed in the proof of theorem 6.2 (whichcome from Lemma 6.1).

♦We now examine the evolution equation of the curvature operator R. In [Ha

3], Hamilton uses time dependent isomorphisms u(t) : (TM, g0) → (TM, g(t))to examine the evolution of the curvature operator. In particular if (M, gij(t)is a solution to the Ricci flow, then the pull back of the curvature operator is

R(t)(φ, ψ) = R(t)abcdφabψcd,

where R(t)abcd = Riem(g(t))ijkluiau

jbu

kcu

la, and the pull back of the metric is

gab = uia(t)uj

b(t)gij(t), and the isomorphisms u(t) are chosen so that gab haszero time derivative, and hence gab is independent of t. That is

∂

∂tui

a = gijRjkuka.

The evolution of R is then derived in [Ha 3] to be

∂

∂tR = ∆R + R2 + R#R,

where R2 is the square of the curvature operator, # is the operator givenby T#Nαβ = cγη

α cδθβ TγδNηθ, and cαγη are the structure constants given by

cαβη =< φα, [φβ , φη] >.

Theorem 6.6. Let g0 be a metric with bounded curvature on a manifoldM , −k2

0 ≤ K ≤ k20, such that R(g0) ≥ 0 in the Aleksandrov sense. Then

there exists a metric h which is 1 + ǫ(n) fair to g0 (ǫ(n) as in Lemma 2.4), aT (n, k0) > 0, and a family of smooth Riemannian metrics g(x, t), t ∈ [0, T ] such

33

that g(x, t), t ∈ (0, T ] solves h flow, g(·, t) → g0(·) uniformly on compact subsetsof M as tց 0, and

0 ≤ R(g(x, t)) ≤ c2(k0, n, h) for all t ∈ (0, T ].

The same result is achieved if we replace the curvature operator (in the abovehypotheses) by the scalar curvature.

Proof : We argue as in Theorem 6.2. Notice that we obtain initially thatsupM×[0,T ] |Riem(

α

g)| ≤ c(k, n) which implies that supM×[0,T ] λi(x, t) ≤ c′(k, n),

for all i ∈ 1, 2, . . . , n(n−1)2 where λi are the eigenvalues of the curvature oper-

ator R. Then the evolution equation for R fulfills the conditions of corollary7.5. In particular the tensor a appearing in Theorem 7.1 (which is needed forcorollary 7.5) will be a matrix coming from the eigen values of R and thensupM×[0,T ]

g|a| ≤ c(k, n). We may apply corollary 7.5 to the family of solutionsα

g(x, t), t ∈ [0, T ], and then take the limit as α→ ∞ to obtain the result. ♦For completeness we mention the following results which are proved using

the same techniques as above. Let M4 be a four manifold and I denote theisotropic curvature on this manifold (see [Ha 5]).

Theorem 6.7. Let g0 be a metric with bounded curvature on a compact realfour manifoldM4 , −k2

0 ≤ K ≤ k20, such that I(g0) ≥ 0 in the weak Aleksandrov

sense. Then there exists a metric h which is 1+ǫ(4) fair to g0, a T (k0) > 0, anda family of smooth Riemannian metrics g(x, t), t ∈ (0, T ] such that g(x, t), t ∈(0, T ] solves h flow, g(·, t) → g0(·) uniformly on compact subsets of M as tց 0,and

0 ≤ I(g(x, t)) ≤ c2(k0, h) for all t ∈ (0, T ].

Proof : In four dimensions one can decompose the real two forms Λ2 into thedirect sum of Λ2

+ and Λ2−. Then the curvature operator defined on Λ2 ⊗ Λ2

decomposes as a block matrix

R =

(

A BBt C

)

.

The manifold has non-negative isotropic curvature if and only if a1 + a2 ≥ 0and c1 + c2 ≥ 0, where a1 and a2 are the two smallest eigenvalues of A andc1, c2 are the two smallest eigenvalues of C. The ordinary differential equationfor the evolution of a (c is the same) under Ricci flow is (see [Ha 5], proof ofTheorem 1.2)

∂

∂t(a1 + a2) ≥ a2

1 + a22 + 2(a1 + a2)a3 + b21 + b22 a.e. t ∈ [0, T ], a.e. x ∈M,

34

where b1 and b2 are the two smallest eigenvalues of B (we ignore the lapcaianterm for the moment). We consider the function f(x, t) = a1(x, t) + a2(x, t)and note that it satisfies the ODE

∂

∂tf ≥ 2a3f a.e. ,

where supM×[0,T ] |a3| ≤ k <∞. We may argue then as in Theorem 6.2 to showthat if f(x, 0) ≥ 0 in the Aleksandrov sense then g0 can be evolved by hflowfor some h to obtain g(x, t), t ∈ [0, T ], and f(x, t) ≥ 0 for all t ∈ [0, T ], wheref(x, t) = a1(x, t)+a2(x, t) and a1(x, t), a2(x, t) are the two smallest eigen valuesof A(x, t), where A(x, t),B(x, t),C(x, t) are the curvature operator matrices (asabove) for the metric g(x, t). Similarly we obtain c1 + c2 ≥ 0. ♦

We now show that the theorems of Hamilton for manifolds with non-negative Ricci curvature in three dimensions, and non-negative curvature oper-ator in n-dimensions can be epsilon improved. To do this we argue by contradic-tion and apply Cheeger’s finiteness Theorem([Ch]), and Gromov’s compactnessTheorem (see [Pe] for an exposition), The argument here was inspired by theargument given in Berger [Ber 1], where it is shown that there as a δ < 1

4such that any compact, even dimensional manifold which is δ pinched is eitherhomeomorphic to an n-dimensional sphere or is isometric to one of the sym-metric spaces of rank one ( the complex projective space CPn, the quaternionprojective space HPn, or the Cayley projective space CaP 2). This is an epsilonimprovement on the 1

4 pinched rigidity Theorem (which is the same result for14 pinched manifolds). See Berger [Ber 2], Klingenberg [Kl] and Rauch [Ra] forthe Sphere Theorem, rigidity Theorems and their generalisations.

Theorem 6.8. Let M(n, k0, d, v) be the set of (Mn, g) such that Mn is ann-dimensional compact manifold and g is a metric with curvature K(M, g)bounded from above and below which satisfies

−k0 ≤ K(M, g) ≤ k0, vol(M, g) ≥ v, diam(M, g) ≤ d.

There exists an ǫ1(3, k0, d, v) > 0, ǫ2(n, k0, d, v) > 0, and ǫ3(4, k0, d, v) > 0with the following properties. If (M3, g) is an element of M(3, k0, d, v) andsatisfies Ricci(g) ≥ −ǫ1g, then there exists a smooth Riemannian metric g′ onM3 where (M3, g′) has non-negative Ricci-curvature. If (M, g) is an elementof M(n, k0, d, v) and satisfies R(g) ≥ −ǫ2G(g), then there exists a smoothRiemannian metric g′ on M where (M, g′) has non-negative curvature operator.If (M4, g) is an element of M(4, k0, d, v) and satisfies I(g) ≥ −ǫ3, then thereexists a smooth Riemannian metric g′ on M4 where (M4, g′) has non-negativeIsotropic curvature. If (M, g) is an element of M(n, k0, d, v) and satisfies R(g) ≥

35

−ǫ4, (scalar curvature), then there exists a smooth Riemannian metric g′ on Mwhere (M, g′) has non-negative scalar curvature.

Proof : All of these results are proved in the same way using Gromov’s com-pactness result and Cheeger’s finiteness Theorem for manifolds in M(n, k0, d, v).We prove the Ricci curvature result here. Fix k0, d and v. Assume, to the con-trary that there is no such ǫ1 > 0. Then we have for i ∈ 1, 2, . . ., manifoldsMi with metrics gi such that (Mi, gi) ∈ M(3, k0, d, v), and Ricci(gi) ≥ −1

i, but

there is *no* smooth gi′ on Mi such that gi

′ has non-negative Ricci curvature.By Cheeger’s finiteness Theorem, after taking a sub-sequence if necessary, wemay assume that Mi = M . By Gromov’s compactness Theorem, gi → g in C1,α

for some g ∈ M(3, k0, d, v) which satisfies Ricci(g) ≥ 0 on M in the sense ofAleksandrov. We may flow this metric g using hflow (Theorem 6.2) to obtaina metric g′ on M which is smooth and has non-negative Ricci curvature: acontradiction. ♦

§ 7. Non-compact tensor maximum principles

For scalar parabolic equations on non-compact manifolds there exist al-ready versions of the maximum principle. For example Ecker and Huisken [EH]prove a maximum principle for a scalar function on a non-compact manifoldwhich is evolving by a very general heat flow like equation (with a back groundmetric which may depend on time), as long as the function satisfies a priorivarious spatial growth conditions and the metric satisfies a priori various spa-tial and temporal growth conditions. It is well known that for non-compactmanifolds the maximum principle may be violated if at some fixed time thefunction has very large (bigger than exponential) growth in space. Here weprove a maximum principle for tensors which evolve parabolically (that is weconsider a system of equations) on non-compact manifolds.

For the proof below we introduce the notation Σ2 to be the set of two bytwo symmetric matrices.

Theorem 7.1 Non-compact Tensor Maximum Principle . Let (Mn, g)be a smooth complete Riemannian manifold (non-compact or compact),and N(t), t ∈ [0, T ] be a family of smooth symmetric two tensors on M evolvingaccording to the evolution equation

∂

∂tNij(·, t) =

g(t)

∆N(·, t)ij + ∇sNij(·, t))W s + fNij + FikNkj + φ(Nij),

Nij(·, 0) = N0(·)ij ,(7.1)

36

where N0 is a covariant two tensor satisfying N0 ≥ 0, W is a covariant threetensor, and φ : Σ2 → Σ2 satisfies φ(P )ij ≥ agklPikPlj , where a is some smoothfunction (no sign restriction). Assume also that

supM×[0,T ]

Ricci(g(t)) ≥ −k, (a)

supM×[0,T ]

g|a| + g| ∂∂tg| + g|N | + |f | + g|F | + |W | ≤ k, (b)

where k is a constant. Then the solution N(t), t ∈ [0, T ], to the equation (7.1)satisfies N(t) ≥ 0.

Remark. It is often the case that if in condition (b) we replacesupM×[0,T ] |N | ≤ k with supM |N0| ≤ k then the smoothing properties of flowswill ensure that this bound exists for all t ∈ [0, T ].Remark. There is an earlier weaker version of the non-compact maximumprinciple for tensors in [Ha 6].Proof : We prove the result initially for the simpler evolution equation∂∂tN ≥g

∆N , as the more general case is merely a minor adjustment of thisargument.Step 1. All metrics are equivalent. Condition (b) implies that

−kg ≥ ∂

∂tg ≥ −kg, (7.2)

which implies that all metrics g(t), t ∈ [0, T ] are equivalent:1

c(k,n,T )g0 ≤ g(t) ≤ c(k, n, T )g0. Similarly if γ : [a, b] →M is a smooth curve in

M and l(t)(γ) is the length of γ with respect to g(t), then

∂

∂tl(t)(γ) =

∂

∂t

∫

γ

√

g(∂

∂sγ,

∂

∂sγ)ds ≥ −kl(t)(γ) for all t ∈ [0, T ]

∂

∂tl(t)(γ) ≤ kl(t)(γ) for all t ∈ [0, T ]

(7.3)

We define ρ(x, t) = dist(g(t))(x, x0), ρ0(x) = dist(g0)(x, x0) where x0 is somefixed arbitrary point in M . Then ρ2(p, t) is Lipschitz continuous in t for p /∈Cut(g(t))(x0), and we get

e−2ktρ20 ≤ ρ2 ≤ ρ2

0e2kt for all t ∈ [0, T ]

∂

∂t(ρ2) ≥ −knρ2 for a.e. t ∈ [0, T ]

(7.4)

37

in view of (7.3). Without loss of generality we may assume T = 1.As manyconstants appear in this proof, we shall often use a small c to denote a constantdepending on k, n. For example it is understood that 5c(k, n)+ c2(k, n) may bereplaced by c(k, n) without any harm. This implies that 1

c2(k,n)ρ0(·) ≤ ρ(·, t) ≤c2(k, n)ρ0(·).Step 2. Compactification of the problem. Let

Nij(·, t) = Nij(·, t) + ǫEij(·, t), (7.5)

where E is defined by Eij = eb(x,t)gij , where b(x, t) = (1+βt)(1+ρ2(x, t)), andwhere β = β(k, n) is a constant to be determined later. In view of the fact thatE ≥ 0 we get N0 = N0 + ǫE0 > 0. Since supM |N | ≤ k, we have

−cg ≤ N ≤ cg, for all t ∈ [0, T ]. (7.6)

Choose R = R(ǫ, k, n) so large that eρ2 ≥ 2cǫ

for all x ∈ M − B(g0)R(x0), for

all t ∈ [0, T ]. Substituting this inequality and (7.6) into the definition of N , weget

Nij(·, t) =Nij(·, t) + ǫeb(x,t)gij(·, t)

≥ −cgij + ǫ2c

ǫρ2(x, t)gij

> 0 for all t ∈ [0, T ], for all x ∈M −BR(g0)(x0).

Step 3. Evolution of N . From the definition of N we get

∂

∂tN =

∂

∂tN + ǫβ(1 + ρ2)eb(x,t)g + ǫ2

∂

∂tρρ(1 + βt)eb(x,t)g + ǫeb(x,t) ∂

∂tg

≥ ∂

∂tN + ǫeb(x,t)(−c+ β)(1 + ρ2)g, for a.e. t ∈ [0,

1

β2]

(7.7),in view of (7.2) and (7.4) and the fact that 1 + βt ≤ 2 for all t ∈ [0, 1

β2 ]. Also

from the definition of N we get

∆N = ∆N + ǫ(∆eb(x,t))g

= ∆N + ǫ((1 + βt)∆(1 + ρ2) + (1 + βt)2|∇ρ2|2)g= ∆N + ǫeb(x,t)((1 + βt)2ρ∆ρ+ 2(1 + βt)|∇ρ|2 + (1 + βt)24ρ2|∇ρ|2)g.

We use the following facts from Geometry: (i)g|g∇ρ|2 = 1 for all x ∈ M −

Cut(t)(x0), (ii)g

∆ρ ≤ (n − 1)k (1+ρ2)ρ

, for all x ∈ M − Cut(t)(x0), where (i)

38

is true for any smooth complete Riemannian manifold (M, g), and (ii) is trueunder the extra assumption that Ricci(g) ≥ −kg. Substituting (i) and (ii) intothe calculation of the Laplacian of N we get

∆N ≤ ∆N + ǫc(k, n)(1 + ρ2)eb(x,t)g for all x ∈M − Cut(t)(x0),

for all t ∈ [0,1

β2].

(7.8)

Subtracting (7.8) from (7.7) we get

(∂

∂t− ∆)N ≥ (

∂

∂t− ∆)N + ǫ(1 + ρ2)eb(x,t)(−2c+ β)g.

Choose β > 2c. Assume that at some first time t0 > 0, N(t0) >/0. Then due tothe compactification of the problem (see step 1), there exists some p0 ∈ BR(x0)and a vector vp0 such that N(t0)(vp0 , vp0) = 0. We see that if p0 /∈ M −Cut(t)(x0) then we may argue as in the proof of the compact maximum principlefor tensors ([Ha 1], Theorem 9.1) to obtain a contradiction. If p0 ∈ M −Cut(t)(x0) then we must use a trick of Calabi ([Ca]). Let γ : [0, s] be a geodesicwith (respect to the metric g(t0) ) going from x0 to p0 ∈ Cut(t0)(x0), and let q =γ(r) for some very small r ∈ (0, s). Then q is not a point in the cut locus of x0

with respect to the metric g(t) for every t ∈ [t0−ǫ′, t0+ǫ′] for ǫ′ > 0 small. Thenwe define a new function

q

ρ(x, t) = dist(g(t))(x0, q) + dist(g(t))(q, x). Noticethat

q

ρ(x, t) ≥ ρ(x, t), in view of the triangle inequality and hence, definingq

N = N + ǫe(1+βt)(1+qρ2)g, we get

q

N ≥ N > 0 for all t ∈ [0, t0), and alsoq

N(p0, t0)(vp0 , vp0) = 0 due to the definition ofq

N . Using the same argument

we used for N , we also get

(∂

∂t− ∆)

q

N(·, t) > 0, for all t ∈ [t0 − ǫ′, t0 + ǫ′],

in a small neighbourhood of p0 ∈M in view of the fact thatq

ρ(x, t)−ρ(x, t) ≤ 2rfor t near t0, and x in a small neighbourhood of p0 ∈ M , and the fact thatr ∈ (0, s) was chosen small. But the tensor

q

N(·, ·) is smooth in space and timein a small neighbourhood of (x0, t0) ∈M× [0, T ], and so we may argue as in theproof of the compact maximum principle for tensors to obtain a contradiction.Hence N(·, t) > 0 for all t ∈ [0, 1

b2], for all x ∈ M . Letting ǫ → 0 gives us that

N(·, t) ≥ 0 for all t ∈ [0, 1b2

], for all x ∈ M . Iterating this argument we obtainN(·, t) ≥ 0 for all t ∈ [0, T ].Step. 4 The general case. For the general case we argue as above to obtain

(∂

∂t− g(t)

∆)Nij ≥ǫ(β − c(k, n))(1 + ρ2)eb(x,t)gij + ∇sNij(·, t) ·W s

+ fNij + F ki Njk + φ(N)ij ,

39

which we then rewrite as

(∂

∂t− g(t)

∆)Nij = ǫ(β − c(k, n))(1 + ρ2)eb(x,t)gij + ∇k(N + ǫE)ijWk + f(N + ǫE)ij

+F ki (N + ǫE)jk − gkl(Nik + ǫEik)Njl + akl

i apqj NkpNlq

− ǫ∇kEijWk − ǫfEij − ǫEikNjlg

kl − ǫF ki Ejk,

≥ǫ(β − c)(1 + ρ2)eb(x,t)gij + ∇kNijWk + fNij

+ F ki Njk + akl

i apqj NkpNlq, for all t ∈ [0,

1

β2], (7.9)

in view of (b) and the definition of N and E. Let vp0 be an arbitrary non-zerovector of length one. Then in orthonormal co-ordinates at p0 for which g andN are diagonal, we have,

(akli a

pqj NkpNlq)(v

i, vj) = akl(v)apq(v)NkpNlq = (akl(v))2NkkNll.

For fixed k, l we see that either (case 1) (a(v)kl)2NkkNll ≥ 0 or (wlog) (case 2)Nkk < 0, Nll ≥ 0. In case 2, we get

(akl(v))2NkkNll = (akl(v))2(Nkk + ǫEkk)Nll − ǫ(akl(v))2EkkNll

= (akl(v))2NkkNll − ǫ(akl(v))2EkkNll

≥ −ǫ(akl(v))2EkkNll

≥ −ǫc(k, n)(1 + ρ2)eb(x,t)gij for all t ∈ [0, t0],

in view of (b) and the fact that N ≥ 0 for all t ∈ [0, t0]. Taking the sum overall k and l, and substituting this inequality into (7.9) we get

(∂

∂t− g(t)

∆)Nij ≥ ǫ(β − c)(1 + ρ2)eb(x,t)gij + ∇kNijWk + fNij + F k

i Njk,

for all t ∈ [0, t0]. The result follows using the argument at the end of step 3.♦Corollary 7.2. Let (Mn, g) be a non-compact smooth Riemannian manifold,

and N(t), t ∈ [0, T ] be a family of smooth symmetric two tensors on M evolvingaccording to the evolution equation (7.1) in the Theorem above. Assume thatN0 satisfies N0 ≥ −ǫg0ij . Assume that all the conditions of the theorem aboveare satisfied, (except for N0 ≥ 0). Then

Nij(t) ≥ −2ǫe(1+βt)gij(t), for all t ∈ [0, S(k, n)], (7.10)

40

where β = β(n, k) and S(k, n) > 0.

Proof : The argument is essentially contained in the proof above. Fix anarbitrary x0 ∈ M , and define N = N + 2ǫE as above. Then N0 > 0 sinceN0 + ǫg0ij > 0. we argue as before to obtain N > 0 for all t ∈ [0, T ]. Inparticular at x = x0 where ρ(x) = 0 we get (7.10). As x0 was arbitrary, theproof is finished. ♦Theorem 7.3. Let N(t), t ∈ [0, T ] be as in thm. 7.1, with all the conditionsof the theorem being satisfied. Assume that in place of condition (a) and (b)we have

supM×[0,T ]

Ricci(g(t)) ≥ − k√t, (a′)

supM×[0,T ]

g|a|√t

+g| ∂∂tg| + g|N | + |f | + |F | + |W | ≤ k√

t. (b′)

supM×[0,T ]

N ≥ −k (c′)

Then the solution N(t), t ∈ [0, T ] satisfies N(t) ≥ 0.

Proof : The proof is the same as the proof of Theorem 7.1, with some smallchanges. Wherever in the proof we use (a) or (b) to estimate we must now use(a′) and (b′). To compensate, we define a modified E,

Eij = eb(x,t)gij , where

b(x, t) = (1 + β√t)(1 + ρ2(x, t)),

where β is as in the theorem above, and set N = N+ǫE. Then all the estimatescarry through. Note that condition (c’) guarantees that the first time where Nis zero is well defined, and must occur (if it occurs) in some compact subset ofM (not at spatial inifinity).

♦Corollary 7.4. Let N(t), t ∈ [0, T ] be as in Corollary 7.2. Assume that inplace of (a),(b) we have (a’),(b’) and (c’). Then

N(t)ij ≥ −2ǫe(1+β√

t)g(t)ij , for all t ∈ [0, S(k, n)],

where β = β(n, k), and S = S(k, n) > 0.

Proof : The corollary follows by making the modifications to the proof ofcorollary 7.2 mentioned in the proof of Theorem 7.3.♦

41

Corollary 7.5. Let (M, g(t)) be as in Theorem 7.1, and R(t) : Λ2(M) ⊗Λ2(M) → R, satisfy

∂

∂tR(·, t) =

G(t)

∆R(·, t) + 〈∇R(·, t)),W 〉 + fR + φ(R),

R(·, 0) = R0(·),

where f,W, φ are as above, and supM

G|R| ≤ k√t, and R ≥ −k for all (x, t) ∈

M × [0, T ], where G(t) is the operator defined as in (1.8). Then R(·, 0)αβ ≥−ǫG0αβ , implies that

R(·, t)αβ ≥ −ǫe(1+β√

t)G(t)αβ , for all t ∈ [0, S(n, k)],

where β = c(n, k) and S = S(n, k) > 0. ♦Proof : The proof is as in Corollary 7.2 with some minor changes. In the proofwe replace E by E = eb(x,t)G(x, t), where b(x, t) is as in Corollary 7.2. Note

that as the metric G(t) is compatible with g(t),G(t)

∆ρ2 =g(t)

∆ρ2, and so on. ♦Acknowledgements

The author would like to thank Gerhard Huisken, for his continuous interest in and

support of this work, and Frank Duzaar and Ernst Kuwert for their useful comments

and support. Thanks to Wilderich Tuschmann for introducing me to spaces of bounded

curvature (in the sense of Aleksandrov). We would also like to thank the Humboldt

Universitaet (DFG-Graduiertenkolleg ”Geometrie und Nichtlineare Analysis”) and the

Albert-Ludwigs-Universitaet Freiburg, (DFG-Graduiertenkolleg ”Nichtlineare Differen-

tialgleichungen”) for their hospitality and financial support.

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