Mathematisches Institut der Ludwig-Maximilians-Universitat Munchen

ON THE LONGTIME BEHAVIOUR

OF SOLUTIONS TO THE RICCI FLOW

ON COMPACT THREE-MANIFOLDS

Diplomarbeit

von

Florian Schmidt

betreut von

Prof. Dr. Bernhard Leeb

Abgabedatum: 31.03.2006

Table of Contents

Table of Contents 1

1 Introduction 3

2 Maximum Principles 52.1 Differentiability to the Right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Parabolic Maximum Principles for Functions . . . . . . . . . . . . . . . . . . . . . . 152.3 Parabolic Maximum Principle for Sections of Vector Bundles . . . . . . . . . . . . . 16

3 General Results on Ricci Flow 183.1 The Normalized Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Evolution of Geometric Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Evolution of the Levi-Civita-Connection . . . . . . . . . . . . . . . . . . . . 223.2.2 Evolution of Orthonormal Frames . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Evolution of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.4 A ”New” Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.5 Evolution of Curvature Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Curvature Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Long Time Existence - Curvature Blow Up . . . . . . . . . . . . . . . . . . . . . . . 463.5 Limits of Ricci Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Compact Three-Manifolds with Positive Ricci Curvature 624.1 Some Special Features of Dimension Three . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Pinching the Sectional Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Estimating the Gradient of Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . 744.4 Uniform Explosion of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Nonsingular Solutions 885.1 Evolution of Scalar Curvature Under the Normalized Ricci Flow . . . . . . . . . . . 885.2 Long Time Pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Three-Manifolds with Nonnegative Ricci Curvature . . . . . . . . . . . . . . . . . . . 935.4 A Certain Singular Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Spherical Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6 Flat Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

1

2

5.7 Hyperbolic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Bibliography 107

Chapter 1

Introduction

In the early 1980’s R.S. Hamilton introduces the Ricci flow and applies it to prove that compact

three-manifolds with positive Ricci curvature are diffeomorphic to spherical space forms [H1]. Later

in the paper “four-manifolds with positive curvature operator”[H2] he shows that a compact four-

manifold with positive curvature operator is diffeomorphic to the standard sphere S4 or to the real

projective space RP4 by using his tensor maximum principles. In 1999 Hamilton publishes his work

on nonsingular solutions to of the Ricci flow on three-manifolds [H5]. There he shows that all

compact three-manifolds, which admit a nonsingular noncollapsed solution to the normalized Ricci

flow are geometrizable in the sense of Thurston. Cheeger and Gromov prove the collapsed case in

[C-G].

Many people are optimistic that Perelman has proved Thurston’s Geometrization Conjecture and

the Poincare Conjecture.

The aim of this work is to illustrate the longtime behaviour of solutions to the Ricci flow on compact

three-manifolds. Chapter 2 is about Hamilton’s tensor maximum principles which are powerful tools

in the study of the Ricci flow.

In the third chapter we introduce the Ricci flow and compute the evolution equations for several

geometric quantities without using local coordinates. Moreover, we prove the Shi estimates and use

them to prove that solutions to the Ricci flow with finite lifetime become singular. At the end of

this chapter we find a compactness theorem for solutions to the normalized Ricci flow.

In the fourth chapter we prove Hamilton’s theorem that a compact three-manifold with positive

3

4

Ricci curvature is diffeomorphic to a spherical space form. Similar to [H1] and [H2] we use the scalar

and the tensor maximum principles in the first part of the proof. In the second part, we use the

compactness theorem instead of showing exponential convergence.

The last chapter is about Hamilton’s paper “nonsingular solutions of the Ricci flow on compact

three-manifolds”[H5].

I would like to thank my supervisor Prof. Dr. Bernhard Leeb for giving me this interesting diploma

thesis. I am also thankful to Dr. Hartmut Weiß for his constant support.

Chapter 2

Maximum Principles

2.1 Differentiability to the Right

Here we derive some basic properties of maps, which are called “differentiable to the right”.

Definition 2.1.1. A map f : I ⊂ R −→ R is called differentiable to the right in t0 ∈ I, if

d

dt+

t=t0

f(t) := limh↘0

f(t0 + h)− f(t0)h

exists. f is called differentiable to the right if it is differentiable to the right in all points points

in I.

In this chapter “h↘ x0” stands for “h −→ x0, h > x0”.

There is a nice and very useful example.

Proposition 2.1.1. Let M be a compact manifold and f : M × R −→ R a smooth map. Define

f : R −→ R : t 7→ minp∈M

f(p, t)

f : R −→ R : t 7→ maxp∈M

f(p, t)

Then f and f are differentiable to the right and

d

dt+

t=t0

f(t) = min{(

∂

∂tf

)(p, t0) | f(p, t0) = f(t0)

}d

dt+

t=to

f(t) = max{(

∂

∂tf

)(p, t0) | f(p, t0) = f(t0)

}Moreover, f and f are locally lipschitz.

5

6

Proof. We proof this only for f . The proof for f is almost the same.

For t ∈ R define Ft :={p ∈M |f(p, t) = f(t)

}. First we show that f is locally lipschitz:

Let [a, b] ⊂ R and t, t′ ∈ [a, b] be arbitrary. W.l.o.g. we assume t ≤ t′.

Then for all p′ ∈ Ft′ :

f(t′)− f(t) ≥ f(p′, t′)− f(p′, t) =∫ t′

t

(∂

∂tf

)(p′, s)ds

≥(

mins∈[t,t′]

(∂

∂tf

)(p′, s)

)(t′ − t)

≥(

mins∈[a,b]

minq∈M

(∂

∂tf

)(q, s)

)(t′ − t)

and for all p ∈ Ft:

f(t′)− f(t) ≤ f(p, t′)− f(p, t) =∫ t′

t

(∂

∂tf

)(p, s)ds

≤(

maxs∈[t,t′]

(∂

∂tf

)(p, s)

)(t′ − t)

≤(

maxs∈[a,b]

maxq∈M

(∂

∂tf

)(q, s)

)(t′ − t)

where we use that M is compact and that f is smooth.

Hence,

| f(t′)− f(t) |≤ L|t′ − t|,

where L := max {K ′,K} with

K ′ :=∣∣∣∣ mins∈[a,b]

minq∈M

(∂

∂tf

)(q, s)

∣∣∣∣and

K :=∣∣∣∣ maxs∈[a,b]

maxq∈M

(∂

∂tf

)(q, s)

∣∣∣∣ .This shows that f is lipschitz on every compact time interval. Thus, f is locally lipschitz.

Now we fix t0 ∈ R. Then for all p ∈ Ft0 there exists ρp ∈ o(h), such that

f(p, t0 + h) = f(p, t0) + h

(∂

∂tf

)(p, t0) + ρp(h)

for small h, since f is smooth.

Hence,

f(t0 + h)− f(t0)h

≤ f(p, t0 + h)− f(p, t0)h

=(∂

∂tf

)(p, t0) +

ρp(h)h

7

for all p ∈ Ft0 and h > 0 small enough.

Therefore we get

lim suph↘0

f(t0 + h)− f(t0)h

≤(∂

∂tf

)(p, t0)

for all p ∈ Ft0 . Thus,

lim suph↘0

f(t0 + h)− f(t0)h

≤ minp∈Ft0

(∂

∂tf

)(p, t0)

On the other hand, for all h > 0 and p(h) ∈ Ft0+h,

f(t0 + h)− f(t0)h

≥ f(p(h), t0 + h)− f(p(h), t0)h

=1h

∫ h

0

(∂

∂tf

)(p(h), t0 + s)ds

≥ mins∈[0,h]

(∂

∂tf

)(p(h), t0 + s)

Now let hk −→ 0, hk > 0 be an arbitrary sequence. For each k we choose pk ∈ Ft0+hkand

sk ∈ [0, hk], such that (∂

∂tf

)(pk, t0 + sk) = min

s∈[0,hk]

(∂

∂tf

)(pk, t0 + s).

Since M is compact, the sequence (pk) must have at least one accumulation point in M . Let p∞ be

one of these. Then (pk, t0 + sk) −→ (p∞, t0) after passing to a subsequence. Since f is continuous

f(t0 + hk) = f(pk, t0 + hk) converges to f(t0) = f(p∞, t0), so p∞ ∈ Ft0 . Since ∂∂tf is continuous(

∂∂tf)(pk, t0 + sk) converges to

(∂∂tf)(p∞, t0). This shows

lim infk−→∞

f(t0 + hk)− f(t0)hk

≥(∂

∂tf

)(p∞, t0) ≥ min

p∈Ft0

(∂

∂tf

)(p, t0).

Since (hk) was arbitrary we get

lim infh↘0

f(t0 + h)− f(t0)h

≥ minp∈Ft0

(∂

∂tf

)(p, t0).

8

Note that proving Proposition 2.1.1 we have only used that:

• f is continuous and C1 w.r.t. t

• f |M×[a,b] is proper and uniformly bounded in absolute value for all a, b ∈ R.

Using the same notation as above, Proposition 2.1.1 generalizes to the following setup:

Proposition 2.1.2. Let M be a (not necessarily compact) manifold and f : M × R → R a map

satisfying

• f is continuous and C1 w.r.t. t

• f(p, t) ≥ ϕ(t) for all (p, t) ∈M × R, where ϕ : R −→ R is continuous

• f |M×[a,b] is proper for all a, b ∈ R, a ≤ b.

Then f is differentiable to the right with

d

dt+

t=t0

f(t) = min{(

∂

∂tf

)(p, t0)|f(p, t0) = f(t0)

}Proof. Since f |M×{t} is continuous, proper and bounded from below by ϕ(t) for all t it must

achieve its minimum, so f is defined all over R. Moreover, for all a, b ∈ R with a ≤ b, f |[a,b] is

bounded from above by some constant 0 < C <∞, otherwise f is not defined. From below it is

bounded by c := mint∈[a,b] ϕ(t). Now we get⋃t∈[a,b]

Ft ⊂((f |[a,b]

)−1 ([c, C])),

where(f |[a,b]

)−1 ([c, C]) ⊂M × [a, b] is a compact subset, because f |[a,b] was assumed to be proper.

The rest of the proof can be done in the same way as above. There is only one exception:

we have to use compactness of(f |[a,b]

)−1 ([c, C]) (or more precisely: compactness of its projection

onto M) for appropriate a, b, c, C ∈ R, whenever we used compactness of M before.

Remark 2.1.1. Replacing f ≥ ϕ by f ≤ ϕ we get the corresponding result for f .

Remark 2.1.2. Proposition 2.1.2 cannot be generalized to the case, where only f |M×{t} is proper

for all t ∈ R. Counterexamples can be constructed as follows:

first define

α(x) :=

{e−

1x , if x > 0

0 else.,

9

then

β(x) := e2α(1 + x)α(1− x)

and

ϕ(x) := 1− 2β(x− 2).

1. Now let

f : R× R −→ R : (x, t) 7→ ft(x) = x2ϕ(tx) + (x2 − arctan(t))β(tx− 2)

Then f : R× R −→ R is a smooth function, ft is proper for all t and

f(t) =

{− arctan(t), if t > 0

0, else

Clearly f is differentiable to the right and 1-lipschitz. But

d

dt+

t=0

f(t) = −1,

while

min{(

∂

∂tf

)(x, 0)|f(x, 0) = f(0)

}= 0.

2. If we define

g : R× R −→ R : (x, t) 7→ gt(x) = x2ϕ(tx) + (x2 −√|t|))β(tx− 2),

then g is smooth, gt is proper for all t and

g(t) = −√|t|.

Hence, g is continuous but not differentiable to the right in 0.

The following proposition may be viewed as the Fundamental Theorem of Calculus of functions

being locally lipschitz and differentiable to the right.

Proposition 2.1.3. If f : R −→ R is locally lipschitz and differentiable to the right, then for all

s, s′ ∈ R with s ≤ s′ ∫ s′

s

(d

dt+f

)(t)dt = f(s′)− f(s)

10

Proof. Since f is continuous, there is a function F : R −→ R with ddtF = f . Let L be the Lipschitz

constant of f on [s, s′ + 1]. Define fk(t) := f(t+ 1k )−f(t)1k

for all k ∈ N. Then | fk |≤ L and ddt+ f is

the point wise limit of (fk). Using dominated convergence, we get∫ s′

s

(d

dt+f

)(t)dt =

∫ s′

s

limk−→∞

fk(t)dt

= limk−→∞

∫ s′

s

fk(t)dt

= limk−→∞

{F (s′ + 1

k )− F (s′)1k

−F (s+ 1

k )− F (s)1k

}= f(s′)− f(s)

Functions which are differentiable to the right and continuous have the same monotonicity

properties as differentiable functions.

Proposition 2.1.4. If f : R → R is continuous and differentiable to the right with ddt+ f > 0, then

f is strictly monotonically increasing. If ddt+ f ≥ 0 only, then f is monotonically increasing.

Proof. We treat the case ddt+ f > 0 first:

Let a < b ,a, b ∈ R be arbitrary. We assume f(b) ≤ f(a) and lead this to a contradiction.

Since f is continuous, there exists ξ ∈ [a, b], where f |[a,b] reaches its maximal value.

If ξ = b, then f(ξ) = f(a). Thus, f ≤ f(a) on [a, b], yielding

d

dt+

t=a

f(t) = limh↘0

f(a+ h)− f(a)h

≤ 0,

which is impossible!

If ξ < b, then

d

dt+

t=ξ

f(t) = limh↘0

f(ξ + h)− f(ξ)h

≤ 0,

which is impossible either!

Now we treat the case ddt+ f ≥ 0:

Let ε > 0 and define

fε(t) := f(t) + εt

11

Then

d

dt+fε ≥ ε > 0,

so fε is strictly monotonically increasing on R. Now we are done, letting ε to zero.

The above proposition has a useful corollary.

Corollary 2.1.5. Let (gt)t∈R be a smooth family of Riemannian metrics on a compact manifold

M . If the smooth function f : M × R −→ R satisfies the PDI

∂

∂tf(p, t) ≥ ∆gt

f(p, t),

then f is (weakly) increasing. If it satisfies

∂

∂tf(p, t) ≤ ∆gt

f(p, t),

then f is (weakly) decreasing.

Proof. We proof only the first statement. Then the second follows from the first by replacing f by

−f . We know from Proposition 2.1.1 that f is differentiable to the right with

d

dt+

t=t0

f(t) = min{(

∂

∂tf

)(p, t0)|f(p, t0) = f(t0)

}for arbitrary t0 ∈ R. But

∆gt0f(p0, t0) ≥ 0

in all points p0, where the map p 7→ f(p, t0) has a minimum. Thus, we get

d

dt+

t=t0

f(t) ≥ 0

from the PDI. Since t0 was arbitrary, the result follows from Proposition 2.1.4.

The corollary from above generalizes to the following one.

Corollary 2.1.6. Let (gt)t∈R be a smooth family of Riemannian metrics on a compact manifold

M . If ϕ : R −→ R is continuous, X is a time dependent vector field on M and f satisfies(∂

∂tf

)(p, t) ≥ ∆gt

f(p, t) + gt(∇gtf,X) + ϕ(t),

12

then for all t0 ∈ R and t ≥ 0

f(t+ t0) ≥ f(t0) + φt0(t),

where φt0(t) :=∫ tt0ϕ(s)ds. Replacing “ ≥′′ by “ ≤′′ gives the corresponding result for f .

Proof. We only prove the first statement. The second follows from the first replacing f by −f .

Moreover, we only need to prove it for t0 = 0. Set φ := φ0

For all ε > 0 we consider the auxiliary function φε defined by φε(t) := φ(t)− εt.

Using Proposition 2.1.1 we get that the map

t 7→ minp∈M

(f(p, t)− φε(t)

)is differentiable to the right. We compute

d

dt+minp∈M

(f(p, t)− φε(t)

)=

d

dt+f(t)− ϕ(t) + ε

Since the gradient of a function vanishes at those points where the function reaches a minimum we

can argue as above and getd

dt+f(t) ≥ ϕ(t).

Sod

dt+minp∈M

(f(p, t)− φε(t)

)≥ 0

everywhere on R and therefore

t 7→ minp∈M

(f(p, t)− φε(t)

)increases monotonically by Proposition 2.1.4. Now we let ε tend to 0.

Proposition 2.1.7. If f : R → R is differentiable to the right in 0 with ddt+

t=0

f(t) > 0, then

there exists ε > 0 such that f |(0,ε) > f(0).

Proof. Assume that for all ε > 0 there is a xε ∈ (0, ε), such that f(xε) ≤ f(0).

Then

limh↘0

f(h)− f(0)h

= limε↘0

f(xε)− f(0)xε

≤ 0,

a contradiction.

The corollary below is a weak maximum principle for parabolic PDIs on compact manifolds.

13

Corollary 2.1.8. Let (gt)t∈R be a smooth family of Riemannian metrics on a compact manifold

M and X a time dependent vector field. If f satisfies the PDI(∂

∂tf

)(p, t) ≥ ∆gtf(p, t) + gt(∇gtf,X) + φ(f(p, t), t),

where

φ : R× R −→ R : (x, t) 7→ φ(x, t)

is continuous and weakly increasing in x, then for all s ∈ R and all solutions ϕs to the initial value

problem {ϕs(t) = φ(ϕs(t), t)

ϕs(0) = f(s)

holds

f(t+ s) ≥ ϕs(t),

whenever t ≥ 0 is in the domain of ϕs. Replacing “ ≥′′ by “ ≤′′ gives the corresponding result for f .

Proof. We only prove the first statement. The proof of the second statement is almost the same.

Moreover, we only need to prove it for s = 0. From Peanos theorem follows that the given initial

value problem has a at least one solution ϕ = ϕ0 with initial data ϕ(0) = f(0) and that this

solution exists on R. For all ε > 0 we define auxiliary functions ϕε through

ϕε(t) := ϕ(t)− εt.

Now we fix ε for a moment. Arguing as in Corollary 2.1.6 we find

d

dt+(f(t)− ϕε(t)

)≥ minp∈M

φ(f(p, t), t)− φ(ϕ(t), t) + ε.

Since φ is weakly increasing in x we get

minp∈M

φ(f(p, t), t) = φ(f(t), t).

This implies that at time 0 holds

d

dt+

t=0

(f(t)− ϕε(t)

)≥ ε > 0.

Since ϕε(0) = f(0), by Proposition 2.1.7, we find some δ = δ(ε) > 0, such that

(f − ϕε

)|(0,δ) > 0.

14

Hence,d

dt+(f(t)− ϕε(t)

)> 0

on [0, δ) and therefore f(t)− ϕε(t) increases strictly on this interval by Proposition 2.1.4. Actually,

this argument shows that f − ϕε must increase as long as it is defined. Thus, we get

(f(t)− ϕε)|[0,∞) > 0.

Letting ε to 0 we are done.

The following proposition may be called the “mean value theorem for continuous functions being

differentiable to the right”. We will not use it later. But it is a nice result and , by what we have

done yet, the proof is very easy. Furthermore it has a nice corollary clarifying the relation between

differentiability to the right and and C1-differentiability.

Proposition 2.1.9. If f : (a, b) ⊂ R → R is continuous and differentiable to the right with

ddt+

t=t0

f(t) ∈ (A,B) ⊂ R for all t0 ∈ (a, b), then for all s, s′ ∈ (a, b), s 6= s′ the difference quotient

f(s)− f(s′)s− s′

lies in (A,B) as well.

Proof. We define two auxiliary functions g and h on (a, b) by

• g(t) := f(t)−At and

• h(t) := f(t)−Bt.

Then ddt+ g > 0 and d

dt+h < 0 on (a, b), so g increases strictly and h decreases strictly monotone on

this interval by Proposition 2.1.4. Now let s, s′ ∈ (a, b) with s 6= s′ be arbitrary. Note that

interchanging s and s′ within the difference quotient doesn’t change its value, so we can w.l.o.g.

assume s < s′. Then on the one hand

0 <g(s′)− g(s)s′ − s

=f(s′)− f(s)

s′ − s−A

and

0 >h(s′)− h(s)

s′ − s=f(s′)− f(s)

s′ − s−B

on the other.

15

Corollary 2.1.10. If f : R → R is continuous and differentiable to the right, such that ddt+ f is

continuous in t0 ∈ R, then f is continuous differentiable in t0 and ddt+

t=t0

f(t) = ddt

t=t0

f(t). In

addition, if ddt+ f is continuous on all over R, then f is C1 and d

dt+ f ≡ddtf .

Proof. Clear by the latter proposition.

2.2 Parabolic Maximum Principles for Functions

In this section we state the weak and the strong parabolic maximum principle without proving

them. For the proofs we refer to [P-W].

Let U ⊂ Rn an open, bounded and connected set and

E =∑i,j

aij∂2

∂xi∂xj+∑k

bk∂

∂xk

a time dependent elliptic operator on U × [0, T ) with uniformly bounded coefficients.

Theorem 2.2.1 (Weak parabolic maximum principle). Let u : U × [0, T ) −→ R is a C2 function

satisfying the PDI u ≥ Eu. Then u( ·, 0) ≥ 0 implies u ≥ 0.

Theorem 2.2.2 (Strong parabolic maximum principle). If u : U × [0, T ) −→ [0,∞) is a C2

function satisfying the PDI

u ≥ Eu.

Then u(x0, 0) > 0 for some x0 ∈ U implies u( ·, t) > 0 for all 0 < t < T .

One can apply these results to get the following corollaries

Corollary 2.2.3 (Weak parabolic maximum principle on manifolds). Let M be a manifold and

(gt)t∈[0,T ) a smooth family of Riemannian metrics on M . If u : M × [0, T ) −→ R is a C2 function

satisfying the PDI

u ≥ ∆gtu.

Then u( ·, 0) ≥ 0 at time 0 implies u ≥ 0.

16

Corollary 2.2.4 (Strong parabolic maximum principle on manifolds). Let M be a connected

manifold and (gt)t∈[0,T ) a smooth family of Riemannian metrics on M . If u : M × [0, T ) −→ [0,∞)

is a C2 function satisfying the PDI

u ≥ ∆gtu.

Then u(p0, 0) > 0 for some p0 ∈M implies u( ·, t) > 0 for all 0 < t < T

2.3 Parabolic Maximum Principle for Sections of Vector

Bundles

In this section we state the weak and the strong parabolic maximum principle for sections of vector

bundles without proving them. Let M be a compact manifold and (gt)t∈[0,T ) a smooth family of

Riemannian metrics on M . Furthermore, let (E, 〈·, ·〉) −→M × [0, T ) be a Euclidiean vector

bundle equipped with a metric connection ∇.

We consider the PDE

∇ ∂∂tu = ∆u+ φ(u)

for sections u of E where φ is a vertical vector field on E. We ask ourselves if we can get control

over solutions u to the PDE if we can control solutions to the associated vertical ODE, which is

defined by

u = φ(u).

We get the following result, which is also called the weak (parabolic) tensor maximum

principle. It is due to R.S.Hamilton [H2].

Theorem 2.3.1. Suppose that C ⊂ E is a subbundle of convex sets C(x,t) ⊂ E(x,t) which is parallel

in spatial direction and preserved by the flow of the vector field φ+ ∂∂t ∈ Γ(TE). Then, if u solves

the PDE and takes values in C at time 0, then u takes values in C everywhere on M × [0, T ).

There is also a strong version of Theorem 2.3.1. Let M be a connected manifold. Adopting the

notation from the previous section we get the so called strong (parabolic) tensor maximum

principle which is also due to Hamilton [H2].

17

Theorem 2.3.2. Suppose that C ⊂ E is a parallel family of closed convex sets C(x,t) ⊂ E(x,t)

which is preserved by the flow of φ+ ∂∂t .Then, if t0 > 0 and u : M × [0, t0] −→ E is a solution of

the PDE and if u(x0, t0) ∈ ∂C for some x0 ∈M , then u takes all values in ∂C.

Now we give an example showing how this may be applied. Again, this is due to Hamilton [H2]

Theorem 2.3.3. Consider the bundle Sym(E) −→M × [0, T ) where E is as above. Sym(E) is

defined to be the set of all self adjoint endomorphisms of E. Suppose that F ∈ Γ (Sym(E)) satisfies

the evolution equation

∇ ∂∂tF = ∆F + φ(F )

where φ is a vertical vector field with φ(G) ≥ 0 whenever G ≥ 0 is an element of Sym(E).

Then F ≥ 0 to start implies F ≥ 0 in the future. Moreover, there exists a nonempty interval (0, δ)

on which the rank of F is constant and the kernel of F is parallel and also lies in the kernel on

φ(F ).

Chapter 3

General Results on Ricci Flow

In this chapter we introduce the Ricci flow, compute the evolution equations for several geometric

quantities and discuss some general results which we use throughout the rest of the text.

Let M be a smooth manifold and (gt)t∈[0,T ) a smooth family of Riemannian metrics on M . Then

we call this family a solution to the Ricci flow if it satisfies the evolution equation

d

dtg = −2Ric,

i.e. ifd

dtgt(X,Y ) = −2Rict(X,Y )

holds for all vector fields X and Y at all times t ∈ [0, T ). Hamilton proved in [H1] that this

equation can be solved uniquely with any prescribed initial metric provided that the underlying

manifold is compact. More precisely we have the following theorem.

Theorem 3.0.4. Given an n-dimensional compact Riemannian manifold (M, g0) there is a unique

solution g to the Ricci flow with initial metric g0 which is defined on a maximal time interval [0, T ),

where T > 0.

Hamilton’s proof involves the Nash-Moser Theorem. De Turck gave an alternative proof of this

theorem in [DeT] uses only the classical inverse function theorem. A version of De Turck’s Trick is

presented in [H3].

18

19

3.1 The Normalized Ricci Flow

Suppose that (gt)t∈[0,T ) is a solution to the Ricci flow on a compact n-dimensional manifold M .

Then we can construct a new family g of Riemannian metrics from the given one. As we shall see

below this can be done in a way, such that all members of the new family have the same volume

vol. Moreover, g will satisfy the following evolution equation

d

dtget =

2nr(t)get − 2Ricet ,

where

r(t) :=1

vol(t)

∫M

scal( · , t)dvolet

denotes the average scalar curvature of (M, get).

Let r(t) denote the average scalar curvature of gt. First, we want to find a smooth family (ψt)t∈[0,T )

of scale factors, such that ddtvol (M,ψ(t)gt) = 0. Now, the time derivative of vol(t) := vol(M, gt) is

given by ddtvol(t) = −r(t)vol(t) as we will proof in the following section. By separation of variables

we find that

vol(t) = e−R t0 r(s)dsvol(0).

This implies

d

dtvol(M,ψ(t)gt) =

n

2ψ(t)

n−22 ψ(t)vol(t)− ψ(t)

n2 r(t)vol(t) =

(n

2ψ(t)ψ(t)

− r(t)

)vol(M,ψ(t)gt)

Thus, if we want to have constant volume, then ψ must necessarily satisfy ψ = 2nrψ. Hence it is

given by

ψ(t) = e2n

R t0 r(s)ds,

when we choose ψ(0) = 1. It is clear from the calculations above that if we define ψ in that way,

then vol (M,ψg) is constant.

Unfortunately, the family (ψg)t∈[0,T ) doesn’t satisfy the desired evolution equation. We get

d

dt(ψg) = ψ

(2nrg − 2Ric

)instead. How can we repair this without destroying the fine properties of ψg? We answer this

question by scaling time as well:

20

If we scale time by t 7→ t := τ(t) then we can write dt = dτdt. Thus, defining τ(t) :=∫ t0ψ(s)ds and

get := ψ(t)gt will lead tod

dtg =

2nrg − 2Ric.

The corollary to the following lemma assures us that the right hand side equals

2nrg − 2Ric.

Lemma 3.1.1. Let (M, g) be a Riemannian manifold and f : M −→ R a smooth function with

f > 0. If we define a Riemannian metric through a conformal change g := fg, then the corresponding

Levi-Civita-connections ∇ and ∇ are related as follows

∇ = ∇+ Sf ,

where we define Sf by

Sf (X,Y ) :=12f

((Xf)Y + (Y f)X − g(X,Y )∇f)

and ∇f is the gradient of f with respect to g. In particular, if f is constant, then ∇ = ∇ for .

Proof. This follows from the Koscul formula. Let X,Y, Z be arbitrary vector fields. Then

2fg(∇XY, Z

)= X (fg (Y, Z)) + Y (fg (Z,X))− Z (fg (X,Y ))

+ f (g ([X,Y ], Z)− g ([Y,Z], X) + g ([Z,X], Y ))

= g ((Xf)Y, Z) + g (Z, (Y f)X)− g(∇f, Z) + 2fg(∇XY, Z)

= g (2f∇XY + 2fSf (X,Y ) , Z)

Corollary 3.1.2. Let (M, g) be a Riemannian manifold and λ > 0 a real number. If we define

g := λg, then the corresponding curvature quantities are related as follows

• R = R

• Ric = Ric for the (2,0) Ricci tensor but

21

• Ric = 1λ Ric for the (1,1) Ricci tensor

• scal = 1λscal

• r = 1λr

Proof. This is immediate from the previous lemma.

We need a name for this new family of Riemannian metrics.

Definition 3.1.1. A smooth family (get)et∈[0,eT) of Riemannian metrics on a compact n-dimensional

manifold M is said to be a solution to the normalized Ricci flow, if it satisfies

d

dtg =

2nrg − 2Ric.

Now we are able to state the result from above in the following way.

Theorem 3.1.3. Given an n-dimensional compact Riemannian manifold (M, g0) there is a unique

solution g to the normalized Ricci flow with initial metric g0 which is defined on a maximal time

interval [0, T), where T > 0.

If we are given a solution (get)et∈[0,eT) to the normalized Ricci flow on a compact manifold M , then it

is clear that we can scale space and time to get a solution (gt)t∈[0,T ) to the Ricci flow with initial

metric g0 = g0. We simply define

ψ(t) := e−2n

Ret0 er(es)des,

t = τ(t) :=∫ et

0

ψ(s)ds

and finally

gt := ψ(t)get .

Then g is the desired solution to the Ricci flow. It is defined on M × [0, T ), where

T = limet−→eT τ(t).

22

3.2 Evolution of Geometric Quantities

In this section we will derive evolution equations for several geometric quantities from the

evolution equation of the metric. Let (gt)t∈[0,T ) be a solution to the Ricci flow on a compact

manifold M . We start with the easiest examples:

Lemma 3.2.1. The volume vol(t) satisfies ddtvol = −rvol.

Proof. Locally, we may write the volume form as dvol =√detgijdx. We compute

d

dtdvol =

(ddet)gij

ddtgij

2√detgij

dx =−(detgij)gijRicij√

detgijdx = −scaldvol

which implies ddtvol = −

∫Mscaldvol = −rvol.

Proposition 3.2.2. The Gradient operator ∇ satisfies the evolution equation

∇ = 2Ric ◦ ∇

Proof. Pick a function f and a vector field X. then

0 =d

dtdf(X) =

d

dtg(∇f,X) = −2Ric(∇f,X) + g(

d

dt(∇f), X) = g(−2Ric∇f + ∇f,X)

3.2.1 Evolution of the Levi-Civita-Connection

Definition 3.2.1. If (M,g) is a Riemannian manifold and {Ei(p)} is an orthonormal basis of TpM ,

we can extend it locally to an orthonormal frame via parallel translation along radial geodesics

through p. Each Ei has the important property, that it is parallel in p. Such a frame is called a

Fermi frame with origin p.

Proposition 3.2.3. The Levi-Civita-connection ∇ evolves by the formula

∇ = divR−∇Ric,

where we obtain divR by contracting ∇R against its first two variables.

23

Proof. Let X, Y and Z be vector fields on M and p ∈ M arbitrary. Because ∇ is a 1-form with

values in End(TM), (∇XY )(p) depends only on X(p) and Y(p). Now let t in [0,T). Extend X, Y

and Z around p so that the covariant derivatives of X, Y and Z vanish in p at time t. We compute

in p at time t:

Note that ∇X = ∇Y = ∇Z = 0 in p at time t implies [X,Y ] = [Y,Z] = [Z,X] = 0 in p at time t,

because the Levi-Civita-connection is torsion free. But then we get [X,Y ] = [Y, Z] = [Z,X] = 0 in

p for all times. So from the Koszul formula and the evolution equation of g we get

d

dtg(∇XY, Z) =

12d

dt(Xg(Y, Z) + Y g(Z,X)− Zg(X,Y ))

= −XRic(Y,Z)− Y Ric(Z,X) + ZRic(X,Y )

= −g((∇XRic)(Y ), Z)− g((∇YRic)(Z), X) + g((∇ZRic)(X), Y )

= −g((∇XRic)(Y ), Z)− g((∇YRic)(Z), X) + g((∇ZRic)(Y ), X).

Using the second Bianchi identity one gets

(∇YRic)(Z) =∑i

(∇YR)(Z,Ei)Ei

=∑i

−(∇ZR)(Ei, Y )Ei − (∇EiR)(Y, Z)Ei

= (∇ZRic)(Y )−∑i

(∇EiR)(Y,Z)Ei

where {Ei} is a gt-Fermi frame with origin p. By the symmetries of the Riemannian curvature

tensor, which descend to symmetries of its covariant derivative, it follows that

d

dtg(∇XY, Z) = −g((∇XRic)(Y ), Z) +

∑i

g((∇EiR)(Ei, X)Y,Z)

= g((divR)(X,Y )− (∇XRic)(Y ), Z)

On the other hand

d

dtg(∇XY,Z) = −2Ric(∇XY,Z) + g(∇XY,Z) = g(∇XY,Z)

in p at time t. So

g(∇XY, Z) = g((divR)(X,Y )− (∇XRic)(Y ), Z).

Now the claim follows, since X,Y,Z,p and t were arbitrary and g is fiber wise non-degenerate.

24

Lemma 3.2.4. ∇ has the following properties:

1. ∇ is symmetric

2. If {Ei} is a gt-orthonormal basis of TpM , then∑i ∇t

EiEi = 0

Proof. 1) This follows from the fact that the difference of two torsion free connections on TM is a

(2,1) tensor which is symmetric in the first two variables together with

∇t = limh−→0

∇t+h −∇t

h

2) Extending {Ei} to a Fermi frame with origin p, we see that in p at time t holds

∑i

(divR)(Ei, Ei) =∑i,j

(∇EjR)(Ej , Ei)Ei =

∑j

(∇EjRic)Ej .

The claim follows.

3.2.2 Evolution of Orthonormal Frames

Proposition 3.2.5. Given X0 ∈ Γ(TM) there is a unique solution X : M × [0, T ) −→ TM to the

vertical ODE X = RicX with initial condition X(·, 0) = X0, where T is maximal. In particular, we

have X(·, t) ∈ Γ(TM) for all t. If Y is another solution to the ODE from above, then

d

dtg(X,Y ) = 0

Proof. Ric is a smooth and fiber wise linear operator. Therefore the vertical ODE has a unique

solution Xp : [0, T ) −→ TpM with initial data (X0)p within each fiber. Since X0 is smooth the fiber

wise solutions fit together to form the desired time dependent vector field X.

To proof the last statement compute

d

dtg(X,Y ) = −2Ric(X,Y ) + g(X,X) + g(X, Y ) = −2Ric(X,Y ) + g(RicX, Y ) + g(X,RicY ) = 0

Corollary 3.2.6. If we take a (local) g0-orthonormal frame {(ei)0} as initial condition for a (local)

fundamental system of solutions {ei} to the vertical ODE from above, then for all t ∈ [0, T ) {ei(t)} is

25

a (local) gt-othonormal frame. Such a fundamental system will be referred to as a (local) evolving

orthonormal frame

Proof.d

dtg(ei, ej) = −2Ric(ei, ej) + g(Ricei, ej) + g(ei, Ricej) = 0

3.2.3 Evolution of Curvature

Before entering the subject we recall some general facts and fix some notation. Given a vector

bundle p : E −→ N with connection ∇ over a Riemannian manifold, there is a natural and well

known way to extend ∇ to a connection on T k0 N ⊗ E, using the Levi-Civita Connection of N,

which will also be denoted by ∇. Inductively, for k ≥ 0, we define the (k+1)-th iterated

covariant derivative by

∇k+1Xk+1,...,X1

s := (∇Xk+1∇ks)(Xk, ..., X1).

The iterated covariant derivatives have a very important and useful but also natural property,

which will be stated in the following lemma.

Lemma 3.2.7. For all k, l ≥ 0 holds

∇k∇ls = ∇k+ls.

Proof. Fix an l ≥ 0. The lemma follows by induction over k.

For all vector fields Xl+1, ..., X1 we get

(∇∇ls)(Xl+1, ..., X1) = (∇Xl+1∇ls)(Xl, ..., X1) = ∇l+1

Xl+1,...,X1s

by definition.

Assuming the formula to be true for k ≥ 0 we choose vector fields Xk+1+l, ..., X1 and compute:

(∇k+1∇ls)(Xk+1+l, ..., X1) = (∇Xk+1+l∇k∇ls)(Xk+l, ..., X1)

= (∇Xk+1+l∇k+ls)(Xk+l, ..., X1)

= ∇k+1+lXk+1+l,...,X1

s.

26

In the first and the last step we have used the definition - note that we have used it on different

bundles. In the second we used the induction hypothesis.

From the fact that the Levi-Civita-connection on N has vanishing torsion we get that

R(X,Y )s = ∇2X,Y s−∇2

Y,Xs.

Furthermore, we have the following lemma, which we will use very often in later calculations.

Lemma 3.2.8. For all sections T ∈ Γ(T r0N⊗E) and all vector fields X,Y, Z1, ..., Zr ∈ Γ(TN) holds

(R(X,Y )T )(Z1, ..., Zr) =R(X,Y )(T (Z1, ..., Zr))−∑i

T (Z1, ..., Zi−1, R(X,Y )Zi, Zi+1, ..., Zr)

Proof. Pick X,Y, the Zi’s and a point p. Since R(·, ·)T is tensorial in all of its arguments we can

w.l.o.g. assume that the covariant derivatives of the chosen vector fields vanish in p. Then we get

there

(∇2X,Y T )(Z1, ..., Zr) = ∇X((∇Y T )(Z1, ..., Zr))

= ∇X(∇Y T (Z1, ..., Zr))−∑i

∇XT (Z1, ..., Zi−1,∇Y Zi, Zi+1, ..., Zr)

= ∇2X,Y T (Z1, ..., Zr)−

∑i

(∇XT )(Z1, ..., Zi−1,∇Y Zi, Zi+1, ..., Zr)

+∑i

T (Z1, ..., Zi−1,∇X∇Y Zi, Zi+1, ..., Zr)

= ∇2X,Y T (Z1, ..., Zr) +

∑i

T (Z1, ..., Zi−1,∇2X,Y Zi, Zi+1, ..., Zr)

Now the claim follows from R(X,Y )T = ∇2X,Y T −∇2

Y,XT

Now we are ready to compute the evolution equations for several curvature quantities. We start

with the (3, 1) curvature tensor.

Proposition 3.2.9. The curvature tensor satisfies

• R(X,Y )Z = (∇X∇)(Y,Z)− (∇Y ∇)(X,Z)

• R(X,Y )Z = (∆R)(X,Y )Z−(R(X,Y )Ric)Z+∑i(R(X,Ei)R)(Ei, Y )Z−(R(Y,Ei)R)(Ei, X)Z

where {Ei} is a suitable orthonormal frame.

27

Proof. Pick vector fields X,Y and Z and assume that their covariant derivatives all vanish in p at

time t. Then in p at time t

(d

dt∇2)X,Y Z =

d

dt(∇X∇Y Z)− d

dt(∇∇XY

Z)

= ∇X∇Y Z +∇X∇Y Z − ∇∇XYZ −∇∇XY

Z

= (∇X∇)(Y, Z).

The first claim follows. Now pick an gt-Fermi frame with origin p. Then in p at time t holds

R(X,Y )Z = (∇X∇)(Y,Z)− (∇Y ∇)(X,Z)

= ∇X∇Y Z −∇Y ∇XZ

= −(R(X,Y )Ric)(Z) +∑i

(∇2X,Ei

R)(Ei, Y )Z − (∇2Y,Ei

R)(Ei, X)Z,

where we use the evolution equation of ∇ in the last step. From the second Bianchi identity we get

∑i

(∇2X,Ei

R)(Ei, Y )Z =∑i

(R(X,Ei)R)(Ei, Y )Z + (∇2Ei,XR)(Ei, Y )Z

=∑i

(R(X,Ei)R)(Ei, Y )Z − (∇2Ei,Ei

R)(Y,X)Z − (∇2Ei,YR)(X,Ei)Z

= (∆R)(X,Y )Z

+∑i

((R(X,Ei)R)(Ei, Y )Z − (R(Y,Ei)R)(Ei, X)Z)

+∑i

(∇2Y,Ei

R)(Ei, X)Z,

and the claim follows.

Proposition 3.2.10. The (1,1) Ricci tensor satisfies

RicX = (∆Ric)(X) + 2∑i

R(X, ei)(Ricei)

Proof. Choose a vector field X and a point p. Assume (∇X)(p) = 0 at time t. Then we choose an

evolving orthonormal frame {ei}, which is a Fermi frame with origin p at time t. Now in p at time

28

t we get

Ric(X) =∑i

(R(X, ei)ei)·

=∑i

R(X, ei)ei +R(X,Ricei)ei +R(X, ei)(Ricei)

=∑i

R(X, ei)ei + 2R(X, ei)(Ricei),

where we used the first Bianchi identity and∑iR(Ricei, ei)X = 0. The latter is true, since for all

U, V ∑i

g(R(Ricei, ei)U, V ) = trace(W 7→ (R(U, V )Ric)(W )),

so this expression doesn’t depend on the choice of the orthonormal frame and computing in an

eigenbasis of Ric gives the result.

It remains to show that ∑i

R(X, ei)ei = (∆Ric)(X).

From Lemma 3.2.4 we get that in p at time t holds

∑i

R(X, ei)ei = ∇X∇eiei −∇ei

∇Xei

= −∑i

(∇ei∇)(ei, X)

=∑i

(∇2ei,ei

Ric)(X)−∑i,k

(∇2ei,ek

R)(ek, ei)X

= (∆Ric)(X) +12

∑i,k

(R(ek, ei)R)(ek, ei)X.

But using Lemma 3.2.8 gives

∑i,k

(R(ek, ei)R)(ek, ei)X =∑i,k

[R(ek, ei), R(ek, ei)]X +R(Ricei, ei)X +R(Ricek, ek)X

= 0

and we are done

Proposition 3.2.11. The scalar curvature evolves by the formula

˙scal = ∆scal + 2 ‖Ric‖2

29

Proof. Pick a point p and an evolving orthonormal frame {ei} over it. Then in p at any time

˙scal =

(∑i

g(Ricei, ei)

)·=∑i

−2Ric(Ricei, ei) + g(Ricei, ei) + g(Ricei, ei) + g(Ricei, ei)

=∑i

(g(∆Ric)ei, ei) + 2

∑k

g(R(ei, ek)(Ricek), ei)

)= trace(∆Ric) + 2 ‖Ric‖2

= ∆scal + 2 ‖Ric‖2

To see the last step, observe that whenever F is an endomorphism field of a Euclidean vector bundle

with a metric connection D, then

d(traceF )X = trace(DXF ).

Now fix t and extend {ei} to a gt-Fermi frame with origin p. Then in p at time t

∆scal =∑i

(∇ei

dscal)(ei)

=∑i

ei(trace(∇ei

Ric))

=∑i

trace(∇2ei,ei

Ric)

= trace(∆Ric)

3.2.4 A ”New” Setup

In order to apply the Maximum Principles to the evolution equations of curvature, we have to

translate them into the language we have used in the previous chapter. To do so, we have to

construct the corresponding Euclidean bundles over M × [0, T ) together with metric connections:

let π : M × [0, T ) −→M be the canonical projection. Then (π∗TM, g) becomes a Euclidean vector

bundle over M × [0, T ) if we define g(p,t) := (gt)p for all (p, t) ∈M × [0, T ).

30

On each time slice M × {t} we have the corresponding Levi-Civita-connection ∇t. Defining ∇π

through

(∇πX+f ∂

∂tY )(p, t) := (∇t

XY )(p, t) + fY (p, t)

for X + f ∂∂t ∈ Γ(T (M × [0, T ))) and Y ∈ Γ(π∗TM), clearly yields a connection. However, it is not

metric in time direction if the metric changes under the Ricci flow. We repair this by letting

∇ := ∇π − dt⊗Ric

Proposition 3.2.12. ∇ is a metric connection on (π∗TM, g)

Proof. It is clear that ∇ is a connection and that it is metric in spatial direction. For arbitrary

X,Y ∈ Γ(π∗TM) we compute:

∂

∂tg(X,Y ) = −2Ric(X,Y ) + g(X, Y ) + g(X, Y )

= g(X −RicX, Y ) + g(X, Y −RicY )

= g(∇ ∂∂tX,Y ) + g(X,∇ ∂

∂tY )

Remark 3.2.1. We note that

• X ∈ Γ(π∗TM) solves X = RicX if and only if ∇ ∂∂t

X = 0.

• ∇ induces connections on all other tensor bundles over M × [0, T ) in the usual way. These

connections will be compatible with the corresponding induced metrics.

3.2.5 Evolution of Curvature Revisited

Proposition 3.2.13. In π∗End(TM) the (1,1) Ricci tensor satisfies

(∇ ∂∂tRic)X = (∆Ric)X + 2

∑i

R(X, ei)Ricei

Proof. The metric connection on π∗End(TM) is given by the formula

∇ = ∇π − dt⊗ [Ric, ·] .

31

Therefore we get

∇ ∂∂tRic =

(Ric− [Ric,Ric]

)= Ric

and the claim follows from Proposition 3.2.10.

Proposition 3.2.14. In π∗T 31M the (3,1) Curvature tensor satisfies

(∇ ∂∂tR)(X,Y )Z = (∆R)(X,Y )Z +

∑i

2[R(X, ei), R(ei, Y )]Z +R(R(X,Y )ei, ei)Z

Proof. Let X,Y and Z be arbitrary time dependent vector fields. As usual, we pick a point (p, t)

and assume that the covariant derivatives of the chosen vector fields vanish in (p, t). Then in (p, t)

holds

(∇ ∂

∂tR)

(X,Y )Z = ∇ ∂∂tR(X,Y )Z

= R(X,Y )Z +R(X, Y )Z +R(X, Y )Z +R(X,Y )Z −Ric(R(X,Y )Z)

= R(X,Y )Z +R(RicX, Y )Z +R(X,RicY )Z +R(X,Y )(RicZ)−Ric(R(X,Y )Z),

which equals

R(X,Y )Z +R(RicX, Y )Z +R(X,RicY )Z + (R(X,Y )Ric)(Z)

by Lemma 3.2.8.

From Proposition 3.2.9 we know that

R(X,Y )Z = (∆R)(X,Y, )Z − (R(X,Y )Ric)Z +∑i

(R(X, ei)R)(ei, Y )Z − (R(Y, ei)R)(ei, X)Z︸ ︷︷ ︸(I)

.

Using Lemma 3.2.8 once more, we see that

∑i

(R(X, ei)R)(ei, Y )Z =∑i

R(X, ei)R(ei, Y )Z −R(R(X, ei)ei, Y )Z −R(ei, R(X, ei)Y )Z

−R(Y, ei)R(ei, X)Z

= −R(RicX, Y )Z +∑i

[R(X, ei), R(ei, Y )]Z −R(ei, R(X, ei)Y )Z

and

∑i

(R(Y, ei)R)(ei, X)Z = −R(RicY,X)Z +∑i

[R(Y, ei), R(ei, X)]Z −R(ei, R(Y, ei)X)Z,

32

so the difference (I) of the terms on the left is given by

−R(RicX, Y )Z −R(X,RicY )Z +∑i

2 [R(X, ei), R(ei, Y )]−R(ei, R(X, ei)Y −R(Y, ei)X︸ ︷︷ ︸(II)

)Z.

By the first Bianchi identity (II) equals

R(X,Y )ei

and hence (I) becomes

−R(RicX, Y )Z −R(X,RicY )Z +∑i

2 [R(X, ei), R(ei, Y )] +R(R(X,Y )ei, ei)Z.

Now all the “bad” terms cancel and the proposition follows.

The Curvature Operator

The metric connection ∇ on π∗TM from above induces metric connections on π∗(∧2

TM) and

π∗End(∧2

TM) which will also be denoted by ∇. We want to compute the evolution equation of

the curvature operator. Before writing down the final formula, we have to introduce the #-product:

We work fiberwise. Pick (p, t) ∈M × [0, T ) and let (V, g) := (TpM, g(p,t)). The map

ρ = ρp :∧

2V −→ so(V, g) : X ∧ Y 7→ g(X, ·)⊗ Y − g(Y, ·)⊗X,

which is an isomorphism of vector spaces, allows us to define a Lie bracket on∧2

V , declaring

[X ∧ Y,U ∧ V ] := ρ−1[ρ(X ∧ Y ), ρ(U ∧ V )].

Writing (X ∧ Y )∗ := g(X ∧ Y, ·) we define

[(X ∧ Y )∗, (U ∧ V )∗] := [X ∧ Y,U ∧ V ]∗

on two forms of V. Finally we define the #-Product # = #p on End(V ∧ V ) by

(α⊗ v)#(β ⊗ w) :=12[α, β]⊗ [v, w],

where α, β ∈ (∧2

V )∗ and v, w ∈∧2

V .

33

Remark 3.2.2. The maps from above have only been defined on generating subsets of the corre-

sponding vector spaces. One has to extend these maps linearly and bilinearly to the whole spaces.

We have worked fiberwise. Viewing ρ, the Lie brackets [·, ·] and # as global objects we find that

they are all tensorial in all of their arguments. In particular, # is symmetric.

Proposition 3.2.15. In π∗End(∧2

TM) the Curvature Operator satisfies

∇ ∂∂t

R = ∆R + 2R#R + 2R2,

Proof. Pick some arbitrary (time dependent) section X ∧ Y and a point (p, t) ∈M × [0, T ). Then

R(X ∧ Y ) =∑i<j

g(R(X,Y )ej , ei)ei ∧ ej ,

where the ei form an evolving orthonormal frame in a neighborhood of p. W.l.o.g. we can assume

that the covariant derivative of X ∧ Y vanishes in (p, t). There we get:

(∇ ∂∂t

R)(X ∧ Y ) =∑i<j

g((∇ ∂∂tR)(X,Y )ej , ei)ei ∧ ej

=∑i<j

g((∆R)(X,Y )ej , ei)ei ∧ ej︸ ︷︷ ︸(I)

+2∑i<j

∑k

g([R(X, ek), R(ek, Y )]ej , ei)ei ∧ ej︸ ︷︷ ︸(II)

+∑i<j

∑k

g(R(R(X,Y )ek, ek)ej , ei)ei ∧ ej︸ ︷︷ ︸(III)

,

using the evolution equation for the Curvature tensor, namely Proposition 3.2.14. We will treat the

three terms one by one. But before that, we do some auxiliary calculations concerning ρ, [·, ·] and

# :

Lemma 3.2.16. 1. g(ρ(X ∧ Y )U, V ) = g(X ∧ Y, U ∧ V ) for all vector fields X,Y, U, V

2.∑k(ρ(X ∧ Y )ek) ∧ ek = −2X ∧ Y for all vector fields X and Y

3. R(X,Y )U = −ρ (R(X ∧ Y )) (U)

4. ∇ρ = 0 and hence ∇kρ = 0 for all k ≥ 1

34

5. [α, β] =∑k(ιek

α) ∧ (ιekβ) for all two forms α and β, where ιXω denotes the contraction of a

differential form ω with a vector field X.

6. (L#L′)(X ∧ Y ) = 12

∑m ([L(X ∧ em),L′(Y ∧ em)] + [L′(X ∧ em),L(Y ∧ em)]) for all vector

fields X and Y .

Proof. Pick time dependent vector fields X,Y, U and V on M .

1. Recall that g(X ∧ Y, U ∧ V ) = det(g(X,U) g(X,V )g(Y,U) g(Y,V )

)by definition. So

g(ρ(X ∧ Y )U, V ) = g(g(X,U)Y − g(Y, U)X,V )

= g(X,U)g(Y, V )− g(X,V )g(Y,U)

= g(X ∧ Y,U ∧ V )

2. Just compute

∑k

(ρ(X ∧ Y )ek) ∧ ek =∑k

(Y ∧ g(X, ek)ek −X ∧ g(Y, ek)ek

)= −2X ∧ Y.

3. This is immediate from the definition of R and 1.

4. ρ is a smooth section of π∗(∧2

T ∗M ⊗ so(TM)). Given a point (p, t), we may assume the

covariant derivatives of X,Y, U and V to vanish in (p, t). Then ∇(X ∧ Y ) = 0 in (p, t) as well. We

compute in (p, t)

(∇Uρ) (X ∧ Y )(V ) = (∇Uρ(X ∧ Y )) (V )

= ∇Ug(X,V )Y −∇Ug(Y, V )X

= (Ug(X,V ))Y − (Ug(Y, V ))X

= (g(∇UX,V ) + g(X,∇UV ))Y − (g(∇UY, V ) + g(Y,∇UV ))X

= 0

A similar calculation shows that ∇ ∂∂t

ρ = 0

35

5. It is sufficient to prove this for α = (X ∧ Y )∗ and β = (U ∧ V )∗. On the one hand

∑k

ιekα ∧ ιek

β =∑k

((X∗ ∧ Y ∗)(ek, ·) ∧ (U∗ ∧ V ∗)(ek, ·)

)=∑k

((g(X, ek)g(Y, ·)− g(Y, ek)g(X, ·)) ∧ (g(U, ek)g(V, ·)− g(V, ek)g(U, ·))

)= g(X,U)g(Y ∧ V, · )− g(Y, U)g(X ∧ V, · )

− g(X,V )g(Y ∧ U, · ) + g(Y, V )g(X ∧ U, · )

and on the other

[α, β] = g([X ∧ Y, U ∧ V ], · ) = g([ρ(X ∧ Y ), ρ(U ∧ V )]·, ·)

= g((g(X,U)ρ(Y ∧ V ) + g(Y, V )ρ(X ∧ U)− g(X,V )ρ(y ∧ U)g(Y, U)ρ(X ∧ V ))·, ·)

= g(X,U)g(Y ∧ V, · ) + g(Y, V )g(X ∧ U, · )− g(X,V )g(Y ∧ U, · )− g(Y,U)g(X ∧ V, · )

6. We show

(L#L)(X ∧ Y ) =∑m

[L(X ∧ em),L(Y ∧ em)].

Then the claim follows by the symmetry of # and the polarization formula. Pick a Point p and an

orthonormal frame {ei} of TpM . Then

(L#L)(X ∧ Y ) =∑i<jk<l

(g(L( · ), ei ∧ ej)⊗ ei ∧ ej

)#(g(L( · ), ek ∧ el)⊗ ek ∧ el

)(X ∧ Y )

=12

∑i<j

k<l,m

(ιemg(L( · ), ei ∧ ej)) ∧ (ιemg(L( · ), ek ∧ el))(X ∧ Y )[ei ∧ ej , ek ∧ el]

=12

∑i<j

k<l,m

g(L(em ∧X), ei ∧ ej)g(L(em ∧ Y ), ek ∧ el)[ei ∧ ej , ek ∧ el]

− 12

∑i<j

k<l,m

g(L(em ∧ Y ), ei ∧ ej)g(L(em ∧X), ek ∧ el)[ei ∧ ej , ek ∧ el]

=∑m

∑i<j

g(L(em ∧X), ei ∧ ej)ei ∧ ej ,∑k<l

g(L(em ∧ Y ), ek ∧ el)ek ∧ el

=∑m

[L(X ∧ em),L(Y ∧ em)]

36

Now we are ready to treat term (I):

3. and 4. of the Lemma imply

g((∆R)(X,Y )U, V ) = −g(ρ((∆R)(X ∧ Y ))U, V ),

which equals

−g((∆R)(X ∧ Y ), U ∧ V )

by 1. Hence ∑i<j

g((∆R)(X,Y )ej , ei)ei ∧ ej = (∆R)(X ∧ Y )

Term (II):

Applying 3., the definition of [·, ·] and 1. gives∑i<j

∑k

g([R(X, ek), R(ek, Y )]ej , ei)ei ∧ ej =∑i<j

∑k

g([−ρ(R(X ∧ ek)),−ρ(R(ek ∧ Y ))]ej , ei)ei ∧ ej

=∑i<j

∑k

g(ρ[R(X ∧ ek),R(ek ∧ Y )]ej , ei)ei ∧ ej

=∑i<j

∑k

g([R(X ∧ ek),R(Y ∧ ek)], ei ∧ ej)ei ∧ ej

=∑k

[R(X ∧ ek),R(Y ∧ ek)].

By 6., this equals

R#R(X ∧ Y ).

Term (III):

Using the symmetries of the (4, 0) Curvature Tensor and 3., 1., 2. of the lemma we calculate∑i<j

∑k

g(R(R(X,Y )ek, ek)ej , ei)ei ∧ ej =∑i<j

∑k

g(R(X,Y )ek, R(ei, ej)ek)ei ∧ ej

=∑i<j

∑k

g(ρ(R(X ∧ Y ))ek, ρ(R(ei ∧ ej))ek)ei ∧ ej

=∑i<j

∑k

g(R(X ∧ Y ), ek ∧ ρ(R(ei ∧ ej))ek)ei ∧ ej

= 2∑i<j

g(R(X ∧ Y ),R(ei ∧ ej))ei ∧ ej

= 2R2(X ∧ Y )

37

Putting it all together gives the result.

3.3 Curvature Estimates

Let M be a compact n-dimensional manifold and (gt)t∈[0,T ) a solution to the Ricci flow on M .

We will show that uniform curvature bounds at time 0 give rise to uniform curvature bounds up

to a certain time depending only on the initial metric. In addition we will show that a uniform

curvature bound on M × [t0, T0), where 0 ≤ t0 < T0 ≤ T , leads to uniform bounds on all derivatives

of curvature on M × [t0, T0). These estimates are known as the Shi-Estimates. They have very

important consequences concerning the lifetime of solutions to the Ricci flow. These consequences

will be discussed in the next section.

Theorem 3.3.1. Let ϕ(t) = ϕC(t) =(

22−6

√Ct

)2

C for some fixed C > 0 and t ∈ [0, 13√C

). Then

‖R‖2 ≤ C = ϕ(0) at time 0 implies ‖R‖2 ≤ ϕ on M × [0,min{T, 1

3√C

}).

The proof requires the following lemmata.

Lemma 3.3.2. For all F,G ∈ π∗End(∧2

TM) holds ‖F#G‖ ≤ 12 ‖F‖ ‖G‖ .

Proof. Let (p, t) ∈M × [0, T ) and a gt-orthonormal basis of TpM . Then for all i, j we have

[ei∧ej , ek∧el] = er∧es for some suitable r, s. This follows from the proof of 5. in Lemma 3.2.16. So

‖[ei ∧ ej , ek ∧ el]‖ ∈ {0, 1}. The same is true for [(ei∧ ej)∗, (ek ∧ el)∗]. Hence, for all i, j, k, l, r, s, u, v

holds ∥∥∥∥((ei ∧ ej)∗ ⊗ ek ∧ el)

#(

(er ∧ es)∗ ⊗ eu ∧ ev)∥∥∥∥ ≤ 1

2

Writing

F =∑i<jk<l

Fijkl(ei ∧ ej)∗ ⊗ ek ∧ el

and

G =∑r<su<v

Grsuv(eu ∧ ev)∗ ⊗ er ∧ es

38

gives

‖F#G‖2 =∑i<jk<l

∑r<su<v

(Fijkl)2(Grs

uv)2 ‖(ei ∧ ej)∗ ⊗ ek ∧ el#(eu ∧ ev)∗ ⊗ er ∧ es‖2

≤ 14

∑i<jk<l

∑r<su<v

(Fijkl)2(Grs

uv)2

=14‖F‖2 ‖G‖2

Lemma 3.3.3. If (E, 〈·, ·〉) −→ M is a Euclidean vector bundle with metric connection ∇ over a

Riemannian manifold (M, g), then for all sections s ∈ Γ(E) the following formula is valid:

∆ ‖s‖2 = 2〈∆s, s〉+ 2 ‖∇s‖2.

Proof. Pick a point p ∈M and a Fermi frame with origin p. Then in p holds

∆ 〈s, s〉 =∑i

eiei 〈s, s〉 = 2∑i

ei⟨∇ei

s, s⟩

= 2∑i

⟨∇2ei,ei

s, s⟩

+⟨∇ei

s,∇eis⟩

= 2 〈∆s, s〉+ 2trace((∇s)∗∇s)

= 2 〈∆s, s〉+ 2 ‖∇s‖2

Proof of Theorem 3.3.1. Using the evolution equation of the curvature operator and the lemmata

from above we compute

∂

∂t‖R‖2 = 2g(∇ ∂

∂tR,R)

= 2g(∆R,R) + 4g(R#R,R) + 4g(R2,R)

= ∆ ‖R‖2 − 2 ‖∇R‖2 + 4g(R#R,R) + 4g(R2,R)

≤ ∆R + 6 ‖R‖3

A simple computation yields

ϕ = 6ϕ32 .

39

Note that the map x 7→ 6x32 increases on [0,∞). Thus, from ‖R‖2 ≤ C = ϕ at time 0, we get

‖R‖2 ≤ ϕ

in the future by Corollary 2.1.8.

Theorem 3.3.4. If ‖R‖2 ≤ C on M × [t0, T0) for some 0 ≤ t0 < T0 < T , then there exist constants

Ck, such that ∥∥∇kR∥∥2 ≤ Ck

on M × [t0, T0).

The proof of this theorem requires a little more work. We want to apply the maximum princi-

ple. Therefore, we need evolution equations for the higher derivatives of the curvature operator.

First we will give an explicit formula for ∇, where ∇ means the connection on π∗(∧2

TM)

or

π∗End(∧2

TM). Then we give a general formula for the commutator of the metric covariant

derivative and the induced Laplacian on sections of a Euclidean vector bundle over a Riemannian

manifold. This information will enable us to write down the desired evolution equations explicitly.

Lemma 3.3.5. The metric connections on π∗(∧2

TM)

and π∗End(∧2

TM)

are given by

∇ = ∇π − dt⊗Ric ∧ id

and

∇ = ∇π − dt⊗ [Ric ∧ id, · ]

respectively, where L ∧ L′(X ∧ Y ) := LX ∧ L′Y − LY ∧ L′X for endomorphism fields on π∗TM .

Proof. The statements are clear for the spatial direction. For a section X ∧ Y ∈ Γ(∧2

π∗TM)

we

get

∇ ∂∂tX ∧ Y =

(∇ ∂

∂tX)∧ Y +X ∧

(∇ ∂

∂tY)

=(X −RicX

)∧ Y +X ∧

(Y −RicY

)=

d

dt(X ∧ Y )− (Ric ∧ id) (X ∧ Y ) .

40

The first claim follows. To proof the second claim, choose an endomorphism L ∈ Γ(∧2

π∗TM)

and

compute(∇ ∂

∂tL)

(X ∧ Y ) = ∇ ∂∂t

L(X ∧ Y )− L(∇ ∂

∂tX ∧ Y

)=

d

dtL(X ∧ Y )− (Ric ∧ id) (L(X ∧ Y ))− L

(d

dt(X ∧ Y )− (Ric ∧ id)(X ∧ Y )

)=(d

dtL− [Ric ∧ id,L]

)(X ∧ Y ).

Lemma 3.3.6. We have

∇X (U ∧ V ) =(∇X( · ) ∧ id

)(U ∧ V ) =

(divR(X, · ) ∧ id−∇X(Ric ∧ id)

)(U ∧ V )

and

∇XL = [∇X( · ) ∧ id,L] = [divR(X, · ) ∧ id−∇X(Ric ∧ id),L] (U ∧ V ).

Proof. This follows immediately from the previous lemma and the evolution equation of the Levi-

Civita-connection ∇ = divR−∇Ric.

Lemma 3.3.7. If (E, 〈·, ·〉) −→ M is a Euclidean vector bundle with metric connection ∇ over a

Riemannian manifold (M, g), then for all sections s ∈ Γ(E) the following formula is valid:

∆∇s−∇∆s = divR(·, s)− 2∑i

R(·, ei)∇eis+ (∇s) ◦Ric

Proof. We will use Lemma 3.2.7 and Lemma 3.2.8 for several times. Pick a vector field X, a point

p and a Fermi frame {ei} with origin p. Assume that ∇X = 0 in p. There we compute

(∆∇s)(X) =∑i

(∇2ei,ei

∇s)(X) =

∑i

(∇ei

∇2s)(ei, X)

=∑i

∇ei

(∇2ei,Xs

)=∑i

∇ei

(R(ei, X)s+∇2

X,eis)

= divR(X, s) +∑i

(R(ei, X)∇ei

s+(∇2ei,X∇s

)(ei))

= divR(X, s) +∑i

(R(ei, X)∇ei

s+ (R(ei, X)∇s) (ei) +(∇2X,ei

∇s)(ei))

= divR(X, s) +∑i

(2R(ei, X)∇ei

s−∇R(ei,X)eis+

(∇X∇2s

)(ei, ei)

)= divR(X, s)− 2

∑i

R(ei, X)∇eis+∇RicXs+∇X∆s.

41

The lemma follows.

Proposition 3.3.8. If L ∈ End(∧2

π∗TM) evolves by the formula

∇ ∂∂t

L = ∆L + φ(L),

where φ is a vertical vector field, then ∇L evolves by the formula

∇ ∂∂t∇L = ∆∇L +∇φ(L) + 2

∑i

R(·, ei)∇eiL

Proof. Pick a time dependent vector field X. We will compute in a point (p, t). We assume ∇X = 0

in (p, t). In particular, this implies X(p, t) = (RicX)(p, t). By Lemma 3.3.5 and Lemma 3.3.6 we

get (∇ ∂

∂t∇L)

(X) = ∇ ∂∂t

(∇XL) =d

dt(∇XL)− [Ric ∧ id,∇XL]

= ∇XL +∇X

L +∇X L− [Ric ∧ id,∇XL]

=[∇X( · ) ∧ id,L

]+∇RicXL +∇X∇ ∂

∂tL +∇X [Ric ∧ id,L]

− [Ric ∧ id,∇XL]

= [divR(X, · ) ∧ id,L]− [∇X(Ric ∧ id),L] +∇RicXL +∇X∇ ∂∂t

L

+ [∇X(Ric ∧ id),L]

= [divR(X, · ) ∧ id,L] +∇RicXL +∇X∇ ∂∂t

L

From the evolution equation of L together with Lemma 3.3.7 we get

∇X∇ ∂∂t

L = (∆∇L) (X) +∇Xφ(L)− divR(X,L) + 2∑i

R(X, ei)∇eiL−∇RicXL.

Thus, if we can show that

divR(X,L) = [divR(X, · ) ∧ id,L],

then we are done. This follows from the following lemma

Lemma 3.3.9. If (E, 〈·, ·〉) −→ M is a Euclidean vector bundle with metric connection ∇ over a

Riemannian manifold (M, g), then for all sections F ∈ Γ(End(E ∧ E)) and all vector fields X,Y

the following formulas are valid:

• R(X,Y )F = [R(X,Y ) ∧ id, F ]

42

• divR(X,F ) = [divR(X, · ), F ]

Proof. From Lemma 3.2.8 we get

(R(X,Y )F ) (s ∧ s′) = R(X,Y )F (s ∧ s′)− F (R(X,Y )(s ∧ s′)) .

Now

∇2X,Y (s ∧ s′) =

(∇2X,Y s

)∧ s′ + (∇Xs) ∧ (∇Y s

′) + (∇Y s) ∧ (∇Xs′) + s ∧

(∇2X,Y s

′)which implies

R(X,Y )(s ∧ s′) = (R(X,Y )s) ∧ s′ + s ∧ (R(X,Y )s′) = (R(X,Y ) ∧ id) (s ∧ s′),

so the first formula is proved.

To prove the second formula note that divR(X,F ) is tensorial in X and F , so we are allowed to

compute point wise. Pick a point p and a Fermi frame with origin p. Moreover, assume that the

covariant derivatives of X and F vanish in p. There we compute

divR(X,F ) =∑i

(∇ei

R)(ei, X)F =

∑i

∇eiR(ei, X)F

=∑i

∇ei[(R(ei, X) ∧ id), F ]

=∑i

[∇ei

(R(ei, X) ∧ id), F]

= [divR(X, · ) ∧ id), F ] .

Corollary 3.3.10. If L is as above, then for all k ∈ N ∇kL evolves by the formula

∇ ∂∂t∇kL = ∆∇kL + φk(L),

where

φk(L)(Xk, ...,X1) := ∇kXk,...,X1

φ(L)

+ 2k−1∑l=0

∑i

(∇k−1−lXk,...,Xl+2

(R(·, ei)∇ei

∇lL))

(Xl+1, Xl, ..., X1)

43

Proof. The case k = 1 is already done. This is simply Proposition 3.3.8. Now we let k ≥ 1 and

assume that the formula has been proved for k instead of k + 1. We pick time dependent vector

fields Xk+1, ..., X1 and a point (p, t). Without loss of generality we assume the covariant derivatives

of the Xi to vanish in (p, t). There we compute(∇ ∂

∂t∇k+1L

)(Xk+1, ..., X1) =

(∇ ∂

∂t∇∇kL

)(Xk+1, ..., X1)

=(∇ ∂

∂t∇∇k

Xk,...,X1L)

(Xk+1)

=(∆∇∇k

Xk,...,X1L)(Xk+1) + (∇(φk(L)(Xk, ..., X1)) (Xk+1)

+ 2∑i

(R( · , ei)∇ei

∇kXk,...,X1

L)(Xk+1)

=(∆∇∇kL

)(Xk+1, ..., X1) + (∇φk(L)) (Xk+1, ..., X1)

+ 2∑i

(R( · , ei)∇ei

∇kL)(Xk+1, ..., X1)

=(∆∇k+1L

)(Xk+1, ..., X1) +

(∇k+1φ(L)

)(Xk+1, ..., X1)

+ 2k−1∑l=0

∑i

(∇∇k−1−l (R(·, ei)∇ei

∇lL))

(Xk+1, ..., X1)

+ 2∑i

(R( · , ei)∇ei

∇kL)(Xk+1, ..., X1)

=(∆∇k+1L

)(Xk+1, ..., X1) +∇k+1

Xk+1,...,X1φ(L)

+ 2k∑l=0

∑i

(∇(k+1)−1−lXk+1,...,Xl+2

(R(·, ei)∇ei

∇lL))

(Xl+1, Xl, ..., X1)

=(∆∇k+1L

)(Xk+1, ..., X1) + φk(L)(Xk+1, ..., X1)

Corollary 3.3.11.

∂

∂t

∥∥∇kL∥∥2

= ∆∥∥∇kL

∥∥2 − 2∥∥∇k+1L

∥∥2+ 2g

(φk(L),∇kL

)Proof. Use Corollary 3.3.10 and Lemma 3.3.3

Proof of Theorem 3.3.4. For k = 0 we have a uniform bound ‖R‖2 ≤ C0 on M × [t0, T0) by

assumption. Now let k ≥ 0 and assume that the claim has been proved for k instead of k + 1.

Writing

φ (R) := 2R#R + 2R2

44

the evolution equation of R reads

∇ ∂∂t

R = ∆R + φ (R) .

Corollary 3.3.11 then gives

∂

∂t

∥∥∇k+1R∥∥2 ≤ ∆

∥∥∇k+1L∥∥2

+ 2g(φk+1 (R) ,∇k+1R

).

Using the induction hypothesis we can bound

‖φk+1 (R)‖

by

C ′k+1

∥∥∇k+1R∥∥+ C ′′k+1

on M × [t0, T0) with some constants C ′k+1, C′′k+1 ≥ 0. Therefore we get

g(φk+1 (R) ,∇k+1R

)≤ C ′k+1

∥∥∇k+1R∥∥2

+ C ′′k+1

∥∥∇k+1R∥∥

≤ C ′′′k+1

∥∥∇k+1R∥∥2

+ C ′′′′k+1.

for some constants C ′′′k+1, C′′′′k+1 ≥ 0. Since the map x 7→ x increases on R and, by compactness of

M ,∥∥∇k+1R

∥∥2 is uniformly bounded at time t0, we can apply Corollary 2.1.8 and find that

maxp∈M

∥∥∇k+1R∥∥2

grows at most exponentially on [t0, T0). Hence, there exists a constant Ck+1 such that on M×[t0, T0)

holds ∥∥∇k+1R∥∥2 ≤ Ck+1.

Remark 3.3.1. Corollary 3.3.11 tells us that all covariant time derivatives of ∇kR can be expressed

in terms of spatial covariant derivatives of the curvature operator. As a consequence , we get uniform

bounds on all covariant spacetime derivatives of the curvature operator on M × [t0, T0), provided

that we have a uniform bound on the curvature operator itself on this set.

Furthermore, Theorem 3.3.4 implies that the space and time derivatives of Ric and scal are bounded

uniformly either.

We use this to get similar estimates for the normalized Ricci flow. In view of Lemma 3.1.1 and

45

its corollary we only have to get control over the time derivatives of the scale factor ψ. Note that

Lemma 3.1.1 implies that the iterated covariant derivatives satisfy ∇k ◦ τ = ∇k, because each scale

factor ψ(t) is constant.

Lemma 3.3.12. If ‖R‖2 ≤ C on M × [0, T ), then there are constants ck such that∣∣∣∣ dkdtk r∣∣∣∣ ≤ ck

holds on [0, T ) for all k ∈ N.

Proof. Theorem 3.3.4 provides constants C ′k, such that∣∣∣ dk

dtkscal ≤ C ′k

∣∣∣ holds on M × [0, T ) for all k.

As we have seen in the proof of Lemma 3.2.1 the volume forms dvolt satisfy

d

dtdvol|(p,t) = −scal(p, t)dvol|(p,t).

This leads to

dvol|(p,t) = e−R t0 scal(p,s)dsdvol|(p,0).

On the other hand we have shown ddtvol = −rvol, so

vol(t) = e−R t0 r(s)dsvol(0).

This gives

r(t) =1

vol(t)

∫M

scal(·, t)dvol( ·, t)

=1

vol(0)eR t0 r(s)ds

∫M

scal(·, t)e−R t0 scal(·,s)dsdvol0

=1

vol(0)

∫M

scal(·, t)eR t0 (r(s)−scal(·,s))dsdvol0

We will finish the proof by induction over k:

Let k = 0. Then we have |r| ≤ c0 for a constant c0, because |scal| ≤ C ′0 implies

|r| = 1vol

∫M

|scal| dvol ≤ 1vol

∫M

C ′0dvol = C ′0.

Now we let k > 0 and assume that there exist constants ci, i = 0, ..., k − 1 such that∣∣∣ di

dti r∣∣∣ ≤ ci

holds on [0, T ) for all i = 0, ..., k − 1. Using the formula from above we get∣∣∣∣ dkdtk r∣∣∣∣ ≤ 1

vol(0)

∫M

∣∣∣∣ dkdtk (scal(·, t)eR t0 (r(s)−scal(·,s))ds

)∣∣∣∣ dvol0.

46

Hence we have to find a uniform bound on the integrand∣∣∣ dk

dtk

(scal(·, t)e

R t0 (r(s)−scal(·,s))ds

)∣∣∣. We

compute∣∣∣∣ dkdtk (scal(·, t)eR t0 (r(s)−scal(·,s))ds

)∣∣∣∣ =∣∣∣∣∣k∑l=0

(k

l

)(dl

dtlscal(·, t)

)(dk−l

dtk−leR t0 (r(s)−scal(·,s))ds

)∣∣∣∣∣≤

k∑l=0

(k

l

) ∣∣∣∣ dldtl scal(·, t)∣∣∣∣ ∣∣∣∣ dk−ldtk−l

eR t0 (r(s)−scal(·,s))ds

∣∣∣∣Since we have uniform bounds on the time derivatives of scalar curvature, we are left proving that the

second factor is uniformly bounded. To see this note that for all l = 0, ..., k dk−l

dtk−l eR t0 (r(s)−scal(·,s))ds

equals

eR t0 (r(s)−scal(·,s))dsPk−l

(dk−l

dtk−l

∫ t

0

(r(s)− scal(·, s)) ds, ..., ddt

∫ t

0

(r(s)− scal(·, s)) ds)

=

where Pk−l is a polynomial in k − l variables and with degree k − l. But, as

di+1

dti+1

∫ t0

(r(s)− scal(·, s)) ds = di

dti (r(t)− scal(·, t)) holds for all i ∈ N, each argument of Pk−l turns

out to be uniformly bounded on M × [0, T ). This follows from the hypothesis of induction and the

fact that all time derivatives of scalar curvature are uniformly bounded on M × [0, T ). This proves

the lemma.

3.4 Long Time Existence - Curvature Blow Up

The aim of this section is to show that if a solution to the Ricci flow has finite lifetime T, then its

curvature necessarily blows up when time reaches T. This has important consequences. E.g., this

result together with Theorem 3.3.1 allows us to give a lower bound on the lifetime of a solution to

the Ricci flow in terms of a uniform curvature bound at time 0. Note that such a uniform curvature

bound is always available, because we are working on compact manifolds.

Theorem 3.4.1. If the maximal lifetime T of a solution gt to the Ricci flow on a compact manifold

M is finite, then

limt−→T

maxp∈M

‖R‖ = ∞.

Moreover, if ‖R‖2 ≤ C at time 0, then T ≥ 13√C

.

47

The proof follows the following guideline:

1. Assuming that the curvature operator is uniformly bounded on M × [0, T ) we will show that

the solution gt converges to a Riemannian metric gT on M in the C∞-topology while t is

tending to T.

2. We take gT as an initial metric for a solution to the Ricci flow. This solution lives at least for

some time ε > 0. From 1. follows that these solutions fit together to form a solution to the

Ricci flow on M × [0, T + ε), violating maximality of T.

We will use Gronwall’s lemma for several times.

Lemma 3.4.2 (Gronwall). Let I ⊂ R be an interval and f : I −→ [0,∞) a continuous map. Fix

constants A,B ≥ 0 and a t0 ∈ I. Suppose that f satisfies the inequality

f(t) ≤ A

∣∣∣∣∫ t

t0

f(s)ds∣∣∣∣+B

for all t ∈ I. Then

f(t) ≤ BeA|t−t0|

holds for all t ∈ I.

Proof. We proof the statement only for t > t0. Moreover, it is sufficient to prove it in a

neighborhood U of a point t1 ∈ I where f(t1) > 0. Letting F : U −→ R : F (t) := A∫ tt0f(s)ds+B

we get the inequality

F = Af ≤ AF.

This implies F (t) ≤ F (t0)eA(t−t0) = BeA(t−t0). But we have f ≤ F on U , so the claim follows.

Proof. According to Theorem 3.3.4 the uniform bound on the curvature operator on M × [0, T )

implies uniform bounds on all its spatial covariant derivatives. But generally, on different time slices

the bounds come from different metrics.

Our first task will be to show that the Riemannian metrics (gt)t∈[0,T ) are all bilipschitz equivalent

to each other. To see this, note that Ric is also bounded uniformly on M × [0, T ), so there is a

constant C0 such that for all X ∈ TM holds∣∣∣∣ ddtg(X,X)∣∣∣∣ = |−2Ric(X,X)| ≤ C0g(X,X).

48

Integrating both sides implies

e−C0tg0(X,X) ≤ gt(X,X) ≤ eC0tg0(X,X)

whenever t ∈ [0, T ), showing that any two of the Riemannian metrics we are considering are bilip-

schitz equivalent with bilipschitz constant eC0T .

Now we know that for all k ∈ N we can bound ∇kR uniformly on M × [0, T ) with respect to g0, say.

This leads to uniform g0-bounds on ∇kRic. Let D be an arbitrary connection on M .

We wish to show that for all k ∈ N and t0 ∈ [0, T ) the g0-norm of Dkgt − Dkgt0 grows at most

exponentially, or more precisely, that it satisfies a differential inequality of the form∣∣∣∣ ddt ∥∥Dkgt −Dkgt0∥∥2

0

∣∣∣∣ ≤ Ak∥∥Dkgt −Dkgt0

∥∥2

0+Bk

for some constants Ak, Bk. Moreover, we wish to show that we can choose Ak and Bk independent

of t0. Once we have shown this we finish the first step in the following way:

Gronwall’s Lemma implies ∣∣∣∣ ddt ∥∥Dkgt −Dkgt0∥∥2

0

∣∣∣∣ ≤ BkeAk|t−t0|

for all t ∈ [0, T ). Integrating both sides leads to

∥∥Dkgt −Dkgt0∥∥2

0≤ BkAk

(eAk(t−t0) − 1

)for all t ∈ [t0, T ). Hence, given some ε > 0, we simply choose t0 being so close to T , such that

BkAk

(eAk(T−t0) − 1

)< ε.

Then, using the same trick again, it follows that

∥∥Dkgt −Dkgs∥∥2

0< ε.

for all s, t ∈ (t0, T ). This shows that the family gt converges uniformly to a smooth tensor gT in the

C∞-topology. Clearly, the symmetry carries over to the limit. Using the uniform bilipschitz constant

from above we find that gT is positive definite. Therefore gT is a Riemannian metric.

To complete step 1, it remains to show that for all k exist constants Ak, Bk, such that for all

t0 ∈ [0, T ) holds ∣∣∣∣ ddt ∥∥Dkgt −Dkgt0∥∥2

0

∣∣∣∣ ≤ Ak∥∥Dkgt −Dkgt0

∥∥2

0+Bk

49

From Ric(X,Y ) = g(RicX, Y ) we get that for all k ∈ N and all vector fields Xk, ..., X1, Y, Z holds

(DkXk,...,X1

Ric)(Y, Z) =

∑l

∑k≥jl>...>j1≥1

(DlXjl

,...,Xj1g)((

Dk−lXk,...,dXji

,...,X1Ric

)Y, Z

)This implies∣∣∣∣ ddt (Dkgt

)(Xk, ..., X1, Y, Z)

∣∣∣∣ = ∣∣∣∣ ddt (DkXk,...,X1

gt)(Y, Z)

∣∣∣∣ = 2∣∣(Dk

Xk,...,X1Rict

)(Y,Z)

∣∣≤ 2

∑l

(k

l

)∥∥Dlgt∥∥

0‖gt‖0

∥∥Dk−lRict∥∥

0‖Xk‖0 · ... · ‖X1‖0 ‖Y ‖0 ‖Z‖0 .

Thus, using the uniform bilipschitz constant from above, we get∥∥∥∥ ddtDkgt

∥∥∥∥0

≤ C∑l

(k

l

)∥∥Dlgt∥∥

0

∥∥Dk−lRict∥∥

0,

for some constant C depending on k and the dimension. Hence,∣∣∣∣ ddt ∥∥Dkgt∥∥2

0

∣∣∣∣ ≤ 2C∥∥Dkgt

∥∥0

∑l

(k

l

)∥∥Dlgt∥∥

0

∥∥Dk−lRict∥∥

0.

Now we have to show that DkRic is uniformly bounded with respect to g0 on M × [0, T ). From the

evolution equation of the Levi-Civita-connection and the uniform bounds on the curvature operator

and its covariant derivatives we get∣∣∣∣ ddt ‖∇ −D‖20

∣∣∣∣ ≤ 2 ‖divR−∇Ric‖0 ‖∇ −D‖0 ≤ c ‖∇ −D‖0 ≤ c′ ‖∇ −D‖20 + c′′,

for some constants c, c′ and c′′, so all ∇t are uniformly equivalent by Gronwall’s lemma. As a

consequence, all induced connections are uniformly equivalent, too. But this is not sufficient for our

purpose. We must show that this is also true for the iterated covariant derivatives.

We need some notation:

Given an (r, 1) tensor µ (r > 0) and an (s, 1) tensor ν we define an (r+s−1, 1) tensor (µ, ν) = (µ, ν)r,s

by

(µ, ν)(X1, ..., Xr−1, Y1, ..., Ys) := µ(X1, ..., Xr−1, ν(Y1, ..., Ys))−∑i

ν(Y1, ..., µ(X1, ...Xr−1, Yi), ..., Ys).

(µ, ν) has the properties:

• it is tensorial in µ and ν

• ∇(µ, ν)r,s = (∇µ, ν)r+1,s + (µ,∇ν)r,s+1

50

We have the following lemma.

Lemma 3.4.3. 1. The induced connection on T r1M satisfies

∇ = (∇, · )2,r

2. The k-th iterated covariant derivative on End(TM) satisfies

d

dt∇kF =

k∑i=1

∇k−i(∇,∇i−1F )2,i

for all endomorphism fields F .

Proof. 1. Let µ ∈ T r1M . Pick vector fields X0, ...Xr. Then

(∇X0

µ)(X1, ..., Xr) = ∇X0

µ (X1, ..., Xr)−∑i

µ(X1, ...,∇X0

Xi, ..., Xr

)by definition. This gives(

∇X0µ)

(X1, ..., Xr) = ∇X0µ (X1, ..., Xr)−∑i

µ(X1, ..., ∇X0Xi, ..., Xr

)= (∇, µ)(2,r) (X0, ..., Xr)

2. This follows by induction over k.

The case k = 1 follows from 1.

Using 1. we get

d

dt

(∇k+1F

)= ∇∇kF +∇ d

dt∇kF =

(∇,∇kF

)(2,k+1)

+∇k∑i=1

∇k−i(∇,∇i−1F

)(2,i)

=k+1∑i=1

∇k+1−i(∇,∇i−1F )2,i

The lemma implies

∣∣∣∣ ddt ∥∥(∇k −Dk)F∥∥2

0

∣∣∣∣ ≤ 2∥∥(∇k −Dk

)F∥∥

0

k∑i=1

∥∥∥∇k−i(∇,∇i−1F )2,i∥∥∥

0

51

By induction, assuming uniform bounds on ∇lF for l < k, we get a uniform bound for the sum

k∑i=1

∥∥∥∇k−i(∇,∇i−1F )2,i∥∥∥

0

on M × [0, T ), using the product rule from above and the fact, that all derivatives

∇m∇ = ∇m (divR−∇Ric) are bounded uniformly. Thus we get∣∣∣∣ ddt ∥∥(∇k −Dk)F∥∥2

0

∣∣∣∣ ≤ c∥∥(∇k −Dk

)F∥∥2

0+ c′

for some constants c, c′. Using Gronwall’s lemma again, it follows that all the k-th iterated covariant

derivatives on End(TM), which we are considering here, are uniformly equivalent.

Hence, we get uniform bounds ∥∥DkRic∥∥

0≤ Ck

on M × (0, T ).

By induction, assuming uniform bounds on∥∥Dlgt

∥∥0

for l < k, we estimate∣∣∣∣ ddt ∥∥Dkgt∥∥2

0

∣∣∣∣ ≤ 2C∥∥Dkgt

∥∥0

∑l

(k

l

)∥∥Dlgt∥∥

0

∥∥Dk−lRict∥∥

0

≤ 2C∥∥Dkgt

∥∥2

0+ 2C

∥∥Dkgt∥∥

0C ′

≤ A0k

∥∥Dkgt∥∥2

0+B0k,

so by Gronwall’s lemma∥∥Dkgt

∥∥2

0grows at most exponentially. Now let t0 ∈M × [0, T ) be arbitrary.

Then ∣∣∣∣ ddt ∥∥Dkgt −Dkgt0∥∥2

0

∣∣∣∣ ≤ 2C∥∥Dkgt −Dkgt0

∥∥0

∑l

(k

l

)∥∥Dlgt∥∥

0

∥∥Dk−lRict∥∥

0

≤ 2C∥∥Dkgt −Dkgt0

∥∥0

∥∥Dkgt∥∥

0+ 2C

∥∥Dkgt −Dkgt0∥∥

0C ′

≤ 2C∥∥Dkgt −Dkgt0

∥∥2

0+ 2C

∥∥Dkgt −Dkgt0∥∥

0

(∥∥Dkgt0∥∥

0+ C ′

)≤ 2C

∥∥Dkgt −Dkgt0∥∥2

0+ C ′′

∥∥Dkgt −Dkgt0∥∥

0

≤ Ak∥∥Dkgt −Dkgt0

∥∥2

0+Bk,

where Ak and Bk do not depend on t0, because we can bound∥∥Dkgt0

∥∥0

uniformly on M × [0, T ).

Now the first step is done.

By Theorem 3.0.4 we get a unique solution g′ to the Ricci flow on M × [0, ε) with initial data gT ,

52

where ε > 0. Now we can extend g to M × [0, t+ ε), letting

gT+t := g′t

for t ∈ [0, ε). From the first step follows that this extension is smooth. Clearly, the extension g is a

solution to the Ricci flow on M × [0, T + ε). This contradicts the maximality of T .

Now we come to the second statement:

If we have ‖R‖2 ≤ C at time 0, then Theorem 3.3.1 tells us that

maxp∈M

‖R‖2

cannot become infinitely large before time 13√C

. This implies that the lifetime of g is at least 13√C

by the first statement.

So far, we have only shown

lim supt−→T

maxp∈M

‖R‖ = ∞.

But we claimed that

limt−→T

maxp∈M

‖R‖ = ∞.

We assume the contrary and establish a contradiction. Suppose that there is a sequence (ti) ∈ [0, T )

with ti −→ T as i −→∞, such that

maxp∈M

‖R‖ (p, ti) ≤ C ′ <∞

holds for all i. We fix an i with T − ti ≤ 16√C′

. Then the solution g′ to the Ricci flow with

initial metric gti has lifetime T ′ ≥ 13√C′

. By uniqueness of the solutions we have g′t = gt+ti for all

t ∈ [0, T − ti). Thus we have extended the solution g to M × [0, T + ε), where ε ≥ 16√C′

, violating

maximality of T.

We get the same result for solutions to the normalized Ricci flow.

Theorem 3.4.4. If the lifetime T of a solution g to the normalized Ricci flow on a compact manifold

is finite, then

limet−→eT

maxp∈M

∥∥∥R∥∥∥ = ∞.

53

Proof. Suppose that g has finite lifetime T. If the curvature operator would be uniformly bounded

on M × [0, T) then the scale factors ψ(t) would be bounded within positive constants which are

independent of time. Thus, we would get that the unnormalized Ricci flow has finite lifetime and

uniformly bounded curvature at the same time. This is impossible by Theorem 3.4.1.

At the end of this section we introduce some new terms, which will be used throughout the rest of

the text

Definition 3.4.1. A solution to the (normalized) Ricci flow is called nonsingular, if it has uni-

formly bounded curvature on the whole domain of definition. Otherwise it is called singular.

A nonsingular solution to the (normalized) Ricci flow which is defined on M × R is called eternal.

Note that Theorem 3.4.1 and Theorem 3.4.4 imply that nonsingular solutions the (normalized) Ricci

flow must have infinite lifetime.

3.5 Limits of Ricci Flows

In the chapters 4 an 5 we have the following special situation:

there is a nonsingular solution g to the normalized Ricci flow which is noncollapsed1, i.e. g has

infinite lifetime, the curvature operators R have their norms uniformly bounded by a constant C

on the whole domain of definition and there is a sequence of points (pi, ti) ⊂ M × [0,∞) with the

following properties: ti −→ ∞ as i −→ ∞ and inj(pi,eti)

(M, geti

)≥ δ > 0 independent of i. For

i ∈ N and t ∈ [−ti,∞) we define gi(t) := g(ti + t) to be the normalized Ricci flow shifted by the

time ti. We wish to show that that the sequence(M, geti

, pi)

of the time shifted normalized Ricci

flows subconverges to a limit flow (M∞, g∞, p∞), which is defined on M × R and satisfies the same

curvature and injectivity radius bounds like the members of the sequence.

If g was a solution to the Ricci flow instead, then we were lucky, because then we could simply apply

Hamilton’s Compactness theorem for solutions to the flow and extract a limit flow.

Our task will be to prove a similar result for solutions to the normalized Ricci flow. Before we are

able to state the theorem some definitions are in order.1the precise definition is given in chapter 5

54

Definition 3.5.1. (M, g, p,Q) is called an evolving complete marked Riemannian manifold

if

1. M is a smooth n-dimensional manifold

2. there exist α < 0 < ω such that g = (g(t))t∈(α,ω) is a smooth family of complete Riemannian

metrics on M and

3. Q is a g0-orthonormal frame over p.

Definition 3.5.2. A sequence (Mi, gi, pi, Qi) of evolving complete marked Riemannian manifolds

converges to an evolving complete marked Riemannian manifold (M, g, p,Q) if

1. lim supi→∞ αi ≤ α and lim infi→∞ ωi ≥ ω for the corresponding αi’s, ωi’s, α and ω of the

sequence and the limit respectively

2. there is a collection Ui of open subsets of M with p ∈ Ui such that for any compact subset

K ∈M exists I = I(K) such that i ≥ I implies K ⊂ Ui

3. there are diffeomorphisms ϕi : Ui → Vi ⊂Mi such that pi ∈ Vi and

• each ϕi takes p to pi and pulls Qi back to Q

• the pullback flows ϕ∗i gi converge to g uniformly on all compact subsets of M × (α, ω)

together with all their derivatives

In the sequel, we will call “sequences of evolving complete marked Riemannian manifolds”

sequences of flows. Limits of sequences of flows will be called limit flows.

Remark 3.5.1. Note that fixing the frames Qi ensures that the limit flow is unique up to a unique

isometry. The precise formulation and proof of this fact are given in the following Lemma.

Lemma 3.5.1. If (Mi, gi, pi, Qi) → (M, g, p,Q) and (Mi, gi, pi, Qi) → (M ′, g′, p′, Q′), then there

exists exactly one diffeomorphism φ : M → M ′, such that φ(p) = p′, φ∗Q′p′ = Qp and φ∗g′ = g on

M × ((α, ω) ∩ (α′, ω′)).

Proof. We will proceed as follows:

55

1. we construct φ at time zero of the limit flows and show uniqueness

2. we exploit the way of constructing φ to see that it is an isometry of the limit flows

For R > 0 and t ∈ (α, ω) ∩ (α′, ω′) let BR(q, t) be the ball of radius R around q ∈ M with respect

to the metric g(t), BR(q′, t) be the ball of radius R around q′(t) ∈M ′ with respect to the metric g′

and, whenever defined, BR(pi, t) be the ball of radius R around pi ∈Mi with respect to the metric

gi(t). Moreover, let ϕi : Ui → Vi and ϕ′i : U ′i → V ′i be the corresponding diffeomorphisms from the

definition of convergence above.

Now fix R > 0 and δ > 0. Let ε > 0 be arbitrary. Using the uniform convergence of the pullback

metrics we find an I0, such that B2R(p, 0) ⊂ Ui, B2R(p′, 0) ⊂ U ′i and

11 + ε

g ≤ ϕ∗i gi ≤ (1 + ε)g

and1

1 + εg′ ≤ (ϕ′i)

∗g′i ≤ (1 + ε)g′

holds on the closure of the corresponding balls for all i ≥ I0. This implies(1

1 + ε

) 12

d ≤ ϕ∗i di ≤ (1 + ε)12 d

and (1

1 + ε

) 12

d′ ≤ (ϕ′i)∗d′i ≤ (1 + ε)

12 d′

on the closure balls around p and p′ of radius 32 for the corresponding path metrics on the balls of

radius 2R, if I0 has been chosen big enough. Thus we can achieve that the following inclusions are

valid for big i:

• BR(p, 0) ⊂ ϕ−1i (BiR+ε(pi, 0)) ⊂ BR+2ε(p, 0) ⊂ Ui

• B′R,(p, 0) ⊂ ϕ′i−1(BiR+ε(pi, 0)) ⊂ B′R+2ε(p, 0) ⊂ U ′i

Therefore i ≥ I0 implies ϕi(BR(p, 0)) ⊂ BiR+ε(p, 0) ⊂ Vi and hence the maps φi(q) := ((ϕ′i)−1◦ϕi)(q)

are defined on BR(p, 0) for i ≥ I0. We claim that the φi subconverge in the C2-topology to a uniquely

defined isometric embedding φ : BR(p, 0) −→M ′.

Using that 11+εg ≤ ϕ∗i gi ≤ (1 + ε)g and 1

1+εg′ ≤ (ϕ′i)

∗g′i ≤ (1 + ε)g′ holds on the balls of radius 2R

56

around p and p′ if i is big enough, we get that the pullback metrics φ∗i g′ and g are (1+ε)2 bilipschitz

equivalent on BR(p, 0), i.e. we get that

1(1 + ε)2

g(X,X) ≤ g′(dφiX, dφiX) ≤ (1 + ε)2g(X,X)

for big i and all vector fields X on BR(p, 0). This shows that the norms of differentials of the φi are

uniformly bounded in terms of the limit metrics g and g′ by (1 + ε)√n.

Let ∇, ∇i, ∇′ and ∇′i denote the Levi-Civita connections of the metrics g, ϕ∗i gi, g′ and ϕ′i

∗gi

respectively. From the Koszul formula and the convergence of the metrics we get

∇i −∇ −→ 0

and

∇′i −∇′ −→ 0

uniformly on the closure of the balls. Since φi is an isometry with respect to the pullback metrics

ϕ∗i gi and ϕ′i∗gi, its Hessian ∇idφi vanishes. Thus, taking the Hessian ∇dφi with respect to the

metrics g and g′ we get

(∇Xdφi) (Y ) = (∇Xdφi) (Y )−(∇iXdφi

)(Y ) =

(∇′ −∇′i)

dφiX(dφiY )− dφi

(∇−∇i

)X

(Y )

This implies

‖∇dφi‖ ≤(∥∥∇′ −∇′i∥∥+

∥∥∇−∇i∥∥) ‖dφi‖2

where we take the norm with respect to g and g′. But we already know that the norms of the

differentials of the φi are uniformly bounded by (1 + ε)√n and that the Levi-Civita-connections

converge uniformly. So the Hessian ∇dφi converges uniformly to 0. Now we use Arzela-Ascoli to

extract a C2 limit φ : BR(p, 0) −→M ′. Recall that we can achieve that

1(1 + ε)2

g(X,X) ≤ g′(dφiX, dφiX) ≤ (1 + ε)2g(X,X)

holds for any prescribed ε > 0 and all vector fields X on BR(p, 0) if we choose i big enough. This

implies that φ is an isometric embedding. Moreover, since each of the φi pulls the frame Q′ back to

the frame Q, so does φ. Thus it is uniquely determined. Running the usual diagonal argument we

find a uniquely determined isometry

φ : (M, g(0)) → (M ′, g′(0)).

57

This completes the first step.

Now let t0 6= 0 and K ⊂M a compact subset. Choosing i big enough we get that the maps φi from

above are defined on K. Let ε > 0. Arguing as above it follows that

1(1 + ε)2

gt0(X,X) ≤ g′t0(dφiX, dφiX) ≤ (1 + ε)2gt0(X,X)

holds on K for big i and all vector fields X on K. This implies that φ is also an isometry with

respect to the metrics gt0 and g′t0 . This proves the lemma.

Now we state Hamilton’s compactness theorem for solutions to the Ricci flow. A proof of this

theorem can be found in [H4].

Theorem 3.5.2 (R.S. Hamilton). Given any sequence (Mi, gi, pi, Qi) of evolving complete marked

Riemannian manifolds with the properties

1. ∂∂tgi = −2Rici on Mi × (α, ω), α < 0 < ω

2. ‖Ri‖i ≤ C <∞ uniformly on Mi × (α, ω) and independent of i and

3. injpi(Mi, gi(0)) ≥ δ > 0 independent of i

there is a subsequence converging to an evolving complete marked Riemannian manifold (M, g, p,Q)

satisfying 1.,2. and 3. 2

We will use this theorem to prove the following one.

Theorem 3.5.3. Given any sequence (Mi, gi, pi, Qi) of evolving complete marked Riemannian man-

ifolds with the properties

1. Mi is compact for all i

2. ∂∂tgi = 2

n rigi − 2Rici on Mi × (α, ω), α < 0 < ω

3.∥∥∥Ri

∥∥∥i≤ C <∞ uniformly on Mi × (α, ω) and independent of i and

4. injpi(Mi, gi(0)) ≥ δ > 0 independent of i

there is a subsequence converging to an evolving complete marked Riemannian manifold (M∞, g∞, p∞, Q∞)

satisfying 2,3 and 4, where r is the uniform limit of the ri in the C∞-topology 2

58

Remark 3.5.2. Note that in general we cannot expect r∞ to be the average scalar curvature of the

limit. Hence, the limit flow is not a normalized Ricci flow in general. But if M∞ is compact then

things are different and we will see that r∞ is indeed the average scalar curvature of the limit which

implies that the limit flow g is a normalized Ricci flow. Moreover, the restriction g∞|M∞×[0,∞) turns

out to be nonsingular in this case.

To prove the theorem we need the following lemmata:

Lemma 3.5.4. Let M be a compact n-dimensional manifold. Let g be a solution to the Ricci flow

on M × (α, ω), where α < 0 < ω and let g be the corresponding normalized Ricci flow, defined on

M × (α, ω).

1. If ‖R‖ ≤ C on M × (α, ω), then

e−2(n−1)C|t| ≤ ψ(t) ≤ e2(n−1)C|t|

and1

2(n− 1)

(1− e−2(n−1)C|t|

)≤ τ(t) ≤ 1

2(n− 1)

(e2(n−1)C|t| − 1

)holds for all t ∈ (α, ω).

2. If∥∥∥R∥∥∥ ≤ C on M × (α, ω), then

e−2(n−1) eC|et| ≤ ψ(t) ≤ e2(n−1) eC|et|

and1

2(n− 1)

(1− e−2(n−1) eC|et|

)≤ τ(t) ≤ 1

2(n− 1)

(e2(n−1) eC|et| − 1

)holds for all t ∈ (α, ω).

Proof. This follows immediately from the definition of the functions.

Lemma 3.5.5. If∥∥∥R∥∥∥2

≤ C on M × [0, T) then there are constants ck such that∣∣∣∣ dkdtk r∣∣∣∣ ≤ ck

holds on [0, T).

59

Proof. This follows from r = 1ψ r,

ddet = 1

ψddt , the previous lemma and Lemma 3.3.12.

Proof. First we assume that α and ω are finite. Let m ∈ N be arbitrary. We show that if m

is big enough, then it is possible to extract a limit flow (M∞, g∞, p∞, Q∞), which is defined on

M × (α+ 1m , ω −

1m ). Then the theorem follows applying the usual diagonal argument.

Pick m in a way such that the interval (α + 1m , ω −

1m ) isn’t empty. Then we get uniform bounds

on all time derivatives ddet ri ≤ ck on the interval [α + 1

m+1 , ω −1

m+1 ] by Lemma 3.5.5. Note that

these bounds are independent of i, because the bounds on the curvature operators are independent

of i. Therefore, restricting the ri to [α + 1m+1 , ω −

1m+1 ] and using Arzela -Ascoli, we get that the

ri converge uniformly to a smooth map r∞ in the C∞-topology of this interval after passing to a

subsequence. But then

• ψi(t) = e−2n

Ret0 eri(es)des → e−

2n

Ret0 er∞(es)des =: ψ∞(t)

• τi(t) =∫ et0ψi(s)ds→

∫ et0ψ∞(s)ds =: τ∞(t)

also, using dominated convergence. It is clear that ψ∞ and τ∞ are limits in the C∞-topology of

[α+ 1m+1 , ω −

1m+1 ].

Let Θ = max {−α, ω}. From Lemma 3.5.4 we know that

e−2(n−1)CΘ ≤ ψi ≤ e2(n−1)CΘ

holds on [α+ 1m+1 , ω −

1m+1 ] for all i, so this bound carries over to the limit. In particular, we get

∂∂tτ∞ = ψ∞ > 0 on this interval, which implies that τ∞ is a diffeomorpism onto its image.

Now we define A := limi−→∞ τi(α + 1m+1 ) and W := limi−→∞ τi(ω − 1

m+1 ). Moreover, we define

αm = τ∞(α + 1m ) and ωm := τ∞(ω − 1

m ). Then we get that the unnormalized Ricci flows gi with

initial metrics gi(0) = gi(0) are defined on Mi × (αm, ωm) if i is sufficiently large, since

A ≤ αm < 0 < ωm ≤ W . Using the uniform bounds on the scale factors again, we find that

the curvature operators Ri are uniformly bounded on Mi × (αm, ωM ) by a constant C, which

is independent of i. Furthermore, we have that inj(pi) (Mi, gi(0)) ≥ δ holds for all i, because

gi(0) = gi(0) by definition.

Thus, after passing to a subsequence the sequence (Mi, gi, pi, Qi) converges to a limit flow

(M∞, g∞, p∞, Q∞) by Theorem 3.5.2.

60

Using Lemma 3.3.12 , we find that for big i all time derivatives of ri are uniformly bounded on

[αm, ωm] and that these bounds are independent of i. By the same arguments as above we conclude

ri → r∞, ψi → ψ∞ and τi → τ∞ for a subsequence in the C∞-topology of [αm, ωm] and that the

latter limit is a diffeomorphism onto its image.

Now let ϕi : Ui −→ Vi be the diffeomorphisms from the definition of convergence, which have been

used implicitly when we took the limit. Then, putting all together, we see

ϕ∗i gi = (ψi ◦ τi)ϕ∗i (gi ◦ τi)

will converge to

g∞ := (ψ∞ ◦ τ∞)(g∞ ◦ τ∞)

uniformly on all compact subsets of M∞ × (α+ 1m , ω −

1m ) together with all their derivatives.

Since

r∞ = limi−→∞

ri

= limi−→∞

(ψi ◦ τi)(ri ◦ τi)

= (ψ∞ ◦ τ∞)(r∞ ◦ τ∞),

and

∂

∂tgi =

2nrigi − 2Rici

for all i, we have

∂

∂tg∞ =

2nr∞g∞ − 2Ric∞

in the limit.

Clearly, the bounds on curvature and the injectivity radii carry over to the limit, using the standard

theory of converging Riemannian manifolds. This completes the proof for finite α and ω.

In the general case we pick sequences (αm) and (ωm) of real numbers such that the αm converge

to α from above and such that the ωm converge to ω from below. Then we use the usual diagonal

argument to construct a subsequence converging on the prescribed domain.

61

Now we are able to extract limit flows from the sequence (M, gi, pi, Qi) of the time shifted

normalized Ricci flows from above on all finite time intervals. Applying the usual diagonal

argument we can construct a subsequence which converges to an eternal limit flow. Moreover, we

have the following corollary.

Corollary 3.5.6. If the sequence (M, gi, pi, Qi) from above subconverges to a limit flow (M∞, g∞, p∞, Q∞)

and M∞ is compact, then the restriction of g∞ to M∞ × [0,∞) is nonsingular.

Proof. Since the limit flow has uniformly bounded curvature and infinite lifetime, we only have to

show that it is a normalized Ricci flow. But this is clear, because if M∞ is compact, then for big

i the diffeomorpisms ϕi : Ui −→ Vi from the definition of convergence turn out to be isometries

(M∞, ϕ∗i gi) −→ (M, gi). Hence, the average scalar curvatures of the sequence converge to the

average scalar curvature of the limit.

Chapter 4

Compact Three-Manifolds with

Positive Ricci Curvature

In this chapter we prove the famous result of R.S. Hamilton.

Theorem 4.0.7. Let (M, g0) be a compact three dimensional Riemannian manifold with positive

Ricci curvature. Then M is diffeomorphic to a spherical space form.

This happens by proving the following stronger theorem, which is also due to Hamilton.

Theorem 4.0.8. Let (M, g0) be a compact three dimensional Riemannian manifold with positive

Ricci curvature. Then

1. the solution get to the normalized Ricci flow with initial metric g0 is nonsingular and

2. there is a constant κ > 0, such that for any choice (pi, ti) ∈ M × [0,∞) with ti −→ ∞ as

i −→∞ and any choice of geti-orthonormal frames Qi over pi the sequence

(M, gi, pi, Qi)

of time shifted flows subconverges to an eternal Normalized Ricci Flow

(M, g∞, p∞, Q∞)

with constant positive sectional curvature κ.

62

63

In the sequel gt will denote the solution to the Ricci Flow with initial metric g0. Its lifetime will be

denoted by T . The lifetime of get will be denoted by T.

4.1 Some Special Features of Dimension Three

Let (V, 〈·, ·〉) be an oriented three dimensional Euclidean vector space. Then we can define a linear

map ι : V −→∧2

V , letting ι(v) := (ιvω)∗, where ω denotes the volume form of V . If (e1, e2, e3) is a

positively oriented orthonormal basis of V , then ι(e1) = e2 ∧ e3,ι(e2) = −e1 ∧ e3 and ι(e3) = e1 ∧ e2.

Thus, ι is a linear isometry with respect to the given inner product on V and the induced inner

product on∧2

V . This has far reaching consequences, e.g. it implies that if (M, g) is a three

dimensional Riemannian manifold, then we can recover the whole curvature operator from the

knowledge of the Ricci tensor. We shall explain this in more detail in the following lemma.

Lemma 4.1.1. If (M, g) is a three dimensional Riemannian manifold, p ∈M and {ei}i=1,2,3 ⊂ TpM

is an orthonormal eigenbasis of Ricp, then {ei ∧ ej}1≤i<j≤3 is an orthonormal eigenbasis of Rp and

Rp =12scal(p) · id−

∑k

Ricp(ek, ek)g(ι(ek), · )⊗ ι(ek).

Proof. We have that

Ric(e1, e1) = g(R(e1, e2)e2, e1) + g(R(e1, e2)e2, e1) = sec(〈e1, e2〉) + sec(〈e1, e3〉).

Similar computations with the other members of the given basis lead toRicp(e1, e1)

Ricp(e2, e2)

Ricp(e3, e3)

=

1 1 0

1 0 1

0 1 1

secp(〈e1, e2〉)secp(〈e1, e3〉)secp(〈e2, e3〉)

The matrix relating the sectional curvatures with the Ricci curvatures has determinant −2, so it is

invertible. Its inverse is given by the matrix

12

1 1 −1

1 −1 1

−1 1 1

.

64

This gives

g (Rp(e1 ∧ e2), e1 ∧ e2) =12(Ricp(e1, e1) +Ricp(e2, e2)−Ricp(e3, e3))

=12scal(p)−Ricp(e3, e3)

Moreover, for 1 ≤ k < 3 holds

g (Rp(e1 ∧ e2), ek ∧ e3) = g(Rp(e1, e2)e3, ek) = 0

by the symmetries of the (4,0) curvature tensor. This implies

Rp(e1 ∧ e2) =(

12scal(p)−Ricp(e3, e3)

)e1 ∧ e2,

so e1 ∧ e2 is an eigenvector of Rp. Similar computations show that

Rp(e1 ∧ e3) =(

12scal(p)−Ricp(e2, e2)

)e1 ∧ e3

and

Rp(e2 ∧ e3) =(

12scal(p)−Ricp(e1, e1)

)e2 ∧ e3

and the lemma follows.

The following corollary is an immediate consequence of the lemma from above.

Corollary 4.1.2. A three dimensional Riemannian manifold has positive sectional curvarure if and

only if Ric < 12scal holds everywhere on M.

Another important observation is that in the three dimensional case every orthonormal basis of∧2V is of the form {ei ∧ ej}1≤i<j≤3, where {ei}i=1,2,3 is an orthonormal basis of V .

In consequence, if L ∈ End(∧2

V ) is self adjoint, then it has an orthonormal eigenbasis of the form

{ei ∧ ej}1≤i<j≤3, where {ei}i=1,2,3 is an orthonormal eigenbasis of V .

We apply this result to get information about the Ricci tensor from the knowledge of the curvature

operator on a three dimensional Riemannian manifold (M, g). Assume that R is diagonal in the

orthonormal basis {ei ∧ ej} ⊂ TpM , p ∈M . From Lemma 3.2.16 we know that

R(X,Y )Z = −ρ(R(X ∧ Y ))(Z)

65

holds for all tangent vectors X,Y, Z ∈ TpM . That implies

Ric(X) = −∑k

ρ(R(X ∧ ek)(ek).

Hence,

Ric(ei) = −3∑k=1

λik(g(ei, ek)ek − g(ek, ek)ei) =

(3∑k=1

λik

)ei,

where the λij stand for the eigenvalues of the curvature operator. This shows that Ric is diagonal

in the basis {ei} and looks like

Ric =

λ12 + λ13

λ12 + λ23

λ13 + λ23

.

Another application of this fact is that in dimension three we get a nice formula for the #-product

restricted to self adjoint endomorphisms L of∧2

V . Namely, we have

L#L =

λ13λ23

λ12λ23

λ12λ13

,

where the λij denote the eigenvalues of L, more precisely, L(ei ∧ ej) = λij · ei ∧ ej . This formula is

now immediate from the formula for the #-product

L#L(X ∧ Y ) =∑k

[L(X ∧ ek),L(Y ∧ ek)],

we gave in Lemma 3.2.16 and

[ei ∧ ek, ej ∧ ek] = ei ∧ ej .

The latter statement follows from the proof of the fifth statement in the lemma mentioned above.

Now we are ready to enter the long proof of Theorem 4.0.8.

4.2 Pinching the Sectional Curvatures

The curvature operator R evolves by the formula

∇ ∂∂t

R = ∆R + 2R#R + 2R2,

66

which gives rise to the vertical ODE

R = 2R#R + 2R2.

Using the results from the previous section, it follows that R and 2R#R + 2R2 are simultaneously

diagonal. Hence, the vertical ODE preserves the eigenspaces of R and therefore any of its solutions

remains diagonal, if it was diagonal in the beginning. This allows us write the vertical ODE as a

system of ODEs λ1 = 2λ2

1 + 2λ2λ3

λ2 = 2λ22 + 2λ1λ3

λ3 = 2λ23 + 2λ1λ2,

where λ1, λ2 and λ3 stand for eigenvalues of the curvature operator. Studying these equations in

more detail we will prove the following pinching results.

Proposition 4.2.1. 1. There are constants K > 0 and δ ∈ (0, 1), depending only on the initial

metric, such that

‖R◦‖ ≤ Kscal1−δ

on M × [0, T ), where R◦ = R− 13 traceR denotes the trace free part of R.

2. There is a constant k > 0 , depending only on the initial metric, such that

Ric− kscal > 0

on M × [0, T ).

Before proving the proposition we shall mention two important applications.

Corollary 4.2.2. The lifetime T of g is finite and

limt−→T

maxscal(t) = ∞.

Proof. Proposition 3.2.11 says that the scalar curvature evolves by the formula

˙scal = ∆scal + 2 ‖Ric‖2.

Using that

1m

(m∑i=1

xi

)2

≤m∑i=1

x2i

67

holds for all m ∈ N and x1, ..., xm ∈ R we find that the scalar curvature satisfies the PDI

˙scal ≥ ∆scal +23scal2.

This implies ddt+minscal > 0 on { t ∈ [0, T )|minscal(t) > 0}. Since we have positive scalar curvature

at the start, we must have positive scalar curvature in the future by Proposition 2.1.7 and Proposition

2.1.7. As the map x 7→ x2 is increasing on [0,∞) we can apply Corollary 2.1.8 to this situation and

compare scalar curvature with solutions to the initial value problem{ϕ = 2

3ϕ2

ϕ(0) = minscal(0).

This leads to minscal ≥ ϕ as long as the solution g exists. Since ϕ solves the explosion equation with

positive initial value, it must become infinitely large in finite time T0. But maxscal dominates ϕ, so

must reach infinity before. This tells us that the metric has only finite lifetime T ≤ T0. Otherwise,

we would have Riemannian metrics on M with unbounded scalar curvature, which is impossible,

because M is compact.

Now we prove the second statement.

Since T is finite, we know by Theorem 3.4.1 that curvature must blow up when time reaches T .

More precisely, we have that

limt−→T

maxp∈M

‖R‖ = ∞,

so at least one eigenvalue of the curvature operator becomes infinitely large in absolute value as time

reaches T , i.e. there must be a sequence of points (pi, ti) with ti −→ T when i −→∞, such that

limi−→∞

|λ(pi, ti)| = ∞,

where λ(pi, ti) denotes an eigenvalue of the curvature operator R(pi,ti). From the second statement

in Proposition 4.2.1 we know Ric − kscal > 0 which implies that the sum of any two eigenvalues

of the curvature operator is bigger than zero. This is the case, because M has dimension three.

Combining both arguments gives

limt−→T

maxscal(t) = ∞

as claimed.

68

Remark 4.2.1. The first statement is true in all dimensions, provided that the initial metric has

positive scalar curvature. It is even true, if we only assume that the initial metric has nonnegative

scalar curvature and scal(p, 0) > 0 in at least one point p ∈M . This is a consequence of the strong

maximum principle and the fact that nonnegative scalar curvature is preserved by the Ricci flow in

any dimension.

Corollary 4.2.3. We have ∥∥∥R◦ ◦ τ∥∥∥∥∥∥R ◦ τ∥∥∥ =

‖R◦‖‖R‖

≤ K ′scal−δ

on M × [0, T ) and

Ric − kscal > 0

on M × [0, T)

Proof. The inequality in the first statement comes from ‖R‖2 ≥ 13 (traceR)2 and scal = 2traceR.

The quotient ‖R◦‖‖R‖ is invariant under scaling with constant scale factors. This gives∥∥∥R◦ ◦ τ

∥∥∥∥∥∥R ◦ τ∥∥∥ =

‖R◦‖‖R‖

.

because the scale factors we use for normalizing depend only on time.

The second statement is true for the same reason.

The first part of Corollary 4.2.3 implies that the ratio ‖eR◦◦eτ‖‖eR◦eτ‖ becomes small at those points,

where scal becomes large. If in addition the norm of the curvature operator R ◦ τ remains bounded

at those points, then we know that the pinching will become optimal there. Until now we only

know that maxscal reaches infinity. But we will show later that scal becomes infinitely large

everywhere while∥∥∥R ◦ τ

∥∥∥ is bounded uniformly on M × [0, T).

Now we come to the proof of Proposition 4.2.1. We will need the following lemmata.

Lemma 4.2.4. Let X be a set and f = (fx)x∈X a family of convex maps fx : Rn −→ R. If the map

sup f : Rn −→ R : v 7→ supx∈X

fx(v)

is defined, then it is convex. Replacing “convex” by “concave” gives the corresponding result for

inf f .

69

Proof. We prove only the first statement. The proof for the second one is almost the same. Let

v, w ∈ Rn and λ ∈ [0, 1] be arbitrary. Then, using convexity and the most elementary properties of

taking suprema of real sets, we simply compute

sup f(λv + (1− λ)w) = supx∈X

fx (λv + (1− λ)w)

≤ supx∈X

(λfx(v) + (1− λ)fx(w))

≤ supx∈X

λfx(v) + supx∈X

(1− λ)fx(w)

≤ λ sup f(v) + (1− λ) sup f(w)

Lemma 4.2.5. Let (V, 〈·, ·〉) be a three dimensional Euclidean vector space. If F is a self adjoint

endomorphism of V then we denote its eigenvalues by λ1(F ) ≤ λ2(F ) ≤ λ3(F ).

Then the maps λ1, (λ1 + λ2) : Sym(V ) −→ R are concave

and the maps λ3, (λ2 + λ3) : Sym(V ) −→ R are convex.

Proof. Choose F ∈ Sym(V ). Then has an orthonormal eigenbasis {ei}, such that Fei = λi(F )ei.

We abbreviate λi(F ) by λi. For all v ∈ V with ‖v‖ = 1 holds

〈Fv, v〉 =∑i,j

〈Fviei, vjej〉 =∑i

λi(vi)2 ≥ λ1

∑i

(vi)2 = λ1,

showing that

λ1 = minv∈S2

〈Fv, v〉 .

This implies that the map F 7→ λ1(F ) is a minimum of linear functions, so it is concave by Lemma

4.2.4.

A similar computation shows that

λ3(F ) = maxv∈S2

〈Fv, v〉

which implies that λ3 is convex.

We know that (λ1 + λ2) = trace−λ3, so λ1 +λ2 is concave because trace is linear and −λ3 concave.

Finally, we have that (λ2 + λ3) = trace − λ1, which shows that λ3 + λ2 is convex because −λ1 is

convex.

70

Lemma 4.2.6. The following subsets of Sym(∧2

π∗TM)

are parallel and fiberwise convex. For

(p, t) ∈M × [0, T ) let

• C0(p,t) :={

L ∈ Sym(∧2

π∗TM)

(p,t)

∣∣∣∣λ1 ≥ 0}

and C0 =⋃

(p,t) C0(p,t)

• C1(p,t) :={

L ∈ Sym(∧2

π∗TM)

(p,t)

∣∣∣∣λ1 + λ2 ≥ 0}

and C1 =⋃

(p,t) C1(p,t)

• C2(p,t) :={

L ∈ Sym(∧2

π∗TM)

(p,t)

∣∣∣∣λ2 + λ3 − C(λ1 + λ2) ≤ 0}

and C2 =⋃

(p,t) C2(p,t)

• C3(p,t) :={

L ∈ Sym(∧2

π∗TM)

(p,t)

∣∣∣∣λ1 + λ2 + λ3 ≥ 0 and λ3 − λ1 ≤ K(λ1 + λ2 + λ3)1−δ}

and C3 =⋃

(p,t) C3(p,t)

where C,K > 0 and δ ∈ (0, 1) are constants and the λ1 ≤ λ2 ≤ λ3 stand for the eigenvalues of L.

Moreover, C1 and C1∩C2 are preserved by the flow φ+ ∂∂t , where φ(R) = 2R#R+2R2. Furthermore,

for all C > 0, we can find some δ > 0, such that for all K > 0 the set C1 ∩ C2 ∩ C3 is preserved by

the flow of φ+ ∂∂t .

Proof. Sym(∧2

π∗TM)

is a parallel subbundleof End(∧2

π∗TM). All the sets from above are

parallel, because if we transport an endomorphism along a curve via parallel translation, its eigen-

values remain unchanged.

Let (p, t) ∈M× [0, T ). We show that C0(p,t), C1(p,t), C2(p,t) and C3(p,t) are convex. The lemmata from

above imply that the maps λ1 and λ1 + λ2 are concave. So C0(p,t) and C1(p,t) are convex, because

superlevelsets of concave functions are convex.

Using the lemmata from above once more, we find that the map λ2 +λ3−C(λ1 +λ2) convex. Since

sublevelsets of convex functions are convex, C2(p,t) must be convex.

Again using the lemmata, it follows that the map λ3−λ1 is convex. Using that the function x 7→ x1−δ

is concave on [0,∞) for our choice of δ, we find that (λ1 + λ2 + λ3)1−δ is concave on the convex set

{L|λ1 + λ2 + λ3 ≥ 0}, because λ1 + λ2 + λ3 is linear. Therefore, C3(p,t) is convex, because it is the

sublevelset of a convex map.

Repapametrizing time by the factor 12 , the vertical ODE becomes

R = R#R + R2

71

and consequently the eigenvalues satisfyλ1 = λ2

1 + λ2λ3

λ2 = λ22 + λ1λ3

λ3 = λ23 + λ1λ2

.

It is clear that if the given sets are preserved by solutions to this system of ODEs, then also by

solutions of the original one, after adapting the constants.

First we will show that the ordering λ1 ≤ λ2 ≤ λ3 is preserved:

we haved

dt(λ3 − λ2) = λ2

3 − λ22 − λ1(λ3 − λ2) = (λ3 − λ2)(λ3 + λ2 − λ1)

so λ1 ≤ λ2 at the start implies λ1 ≤ λ2 in the future. To see this, let f, g : [0, ω) −→ R two functions,

such that f = fg. Then f is given by f(t) = f(0)eR t0 g(s)ds. Therefore f(0) ≥ 0 implies f ≥ 0. In

additiond

dt(λ2 − λ1) = λ2

2 − λ21 − λ3 (λ2 − λ1) = (λ2 − λ1)(λ2 + λ1 − λ3).

Thus, λ2 ≤ λ3 at the start implies λ2 ≤ λ3 in the future.

0. C0 is preserved:

on the set {λ1 ≥ 0} holds λ1 = λ21 + λ2λ3 ≥ 0. This implies that C0 is preserved. The same

calculation shows that the set {λ1 > 0} is also preserved.

1. C1 is preserved:

on the set {λ1 + λ2 ≥ 0} holds

d

dt(λ1 + λ2) = λ2

1 + λ22 + λ3(λ1 + λ2) ≥ 0,

because λ1 ≤ λ2 ≤ λ3 and λ1 + λ2 ≥ 0 implies λ3 >≥ 0. Therefore, C1 is preserved. The same

calculation shows that the set {λ1 + λ2 > 0} is also preserved.

2. C1 ∩ C2 is preserved for all C > 0:

We show that the ratio λ2+λ3λ1+λ2

decreases on the set {λ1 + λ2 > 0}. Note that from λ1 ≤ λ2 ≤ λ3 and

λ1 + λ2 > 0 follows that λ2, λ3 > 0.

On the one hand

d

dtlog (λ2 + λ3) =

λ22 + λ2

3 + λ1(λ2 + λ3)λ2 + λ3

= λ1 +λ2

2 + λ2λ3 −+λ2λ3 + λ23

λ2 + λ3

= λ1 + λ3λ2 + λ3

λ2 + λ3+ λ2

λ2 − λ3

λ2 + λ3≤ λ1 + λ3.

72

And on the other hand

d

dtlog (λ1 + λ2) =

λ21 + λ2

2 + λ3(λ1 + λ2)λ1 + λ2

= λ3 +λ2

1 + λ1λ2 − λ1λ2 + λ22

λ1 + λ2

= λ3 + λ1λ1 + λ2

λ1 + λ2+ λ2

λ2 − λ1

λ1 + λ2≥ λ3 + λ1.

This shows thatd

dtlog(λ2 + λ3

λ1 + λ2

)≤ 0

and we are done with C1 ∩ C2.

Now we come to the last statement:

Let C > 0 be arbitrary. Then for this C, the set C1 ∩ C2 is preserved. We compute

d

dt(λ3 − λ1) = λ2

3 − λ21 + λ2(λ1 − λ3) = (λ3 + λ1)(λ3 − λ1) + λ2(λ1 − λ3)

= (λ3 − λ1)(λ3 + λ1 − λ2)

and

d

dt(λ1 + λ2 + λ3) = λ2

1 + λ2λ3 + λ22 + λ1λ3 + λ2

3 + λ1λ2

= (λ1 + λ2 + λ3)(λ1 + λ3 − λ2) + λ22 + λ2(λ1 + λ2) + λ3(λ2 − λ1).

Therefore, with λ1 ≤ λ2 ≤ λ3 and λ1 + λ2 > 0 we get

d

dtlog(λ1 + λ2 + λ3) ≥ λ1 + λ3 − λ2 +

λ22

λ1 + λ2 + λ3.

Using that the inequality λ2 + λ3 − C(λ1 + λ2) ≤ 0 is preserved, we find that

λ3 ≤ λ3 + λ2 ≤ C(λ1 + λ2) ≤ 2Cλ2,

so

λ1 + λ3 − λ2 ≤ λ1 + λ2 + λ3 ≤ 3λ3 ≤ 6Cλ2.

Letting ε := 136C2 we estimate

d

dtlog(λ1 + λ2 + λ3) ≥ λ1 + λ3 − λ2 +

λ22

λ1 + λ2 + λ3

≥ λ1 + λ3 − λ2 +1

36C2

(λ1 + λ2 + λ3)2

λ1 + λ2 + λ3

≥ (1 + ε)(λ1 + λ3 − λ2)

73

Now we define δ ∈ (0, 1) by

δ := 1− 11 + ε

and compute

d

dtlog(

λ3 − λ1

(λ1 + λ2 + λ3)1−δ

)=

d

dtlog(λ3 − λ1)− (1− δ)

d

dtlog(λ1 + λ2 + λ3)

≤ λ1 + λ3 − λ2 + (1− δ)(1 + ε)(λ1 + λ1 − λ2)

≤ 0

because (1− δ)(1 + ε) = 1. This shows that the quotient

λ3 − λ1

(λ1 + λ2 + λ3)1−δ

is decreasing and we are done.

Proof of Proposition 4.2.1. In both cases, the claim follows applying the weak maximum principle

for tensors. To prove the first statement, we choose C > 0 such that at time 0 holds

maxp∈M

(λ2 + λ3 − C(λ1 + λ2)

)≤ 0.

This is possible for the following reasons:

• Ric > 0 at the start,

• λ1, λ2 and λ3 are continuous and

• M is compact.

Now we choose δ ∈ (0, 1) as in the lemma and a K > 0, such that at time 0 holds

maxp∈M

(λ3 − λ1 − 21−δK(λ1 + λ2 + λ3)1−δ

)≤ 0.

This is possible for the same reasons as above. By the lemma from above, these inequalities are

preserved by the flow of φ + ∂∂t , where φ(R) = 2R#R + 2R2, so by the weak maximum principle

for tensor we conclude that the eigenfunctions of the solution R to the PDE

∇ ∂∂t

R = ∆R + 2R#V + 2R2

74

satisfy these inequalities, too.

Applying

‖R◦‖2 =13

((λ1 − λ2)2 + (λ1 − λ3)2 + (λ2 − λ3)2

)and

(λ1 − λ2)2, (λ2 − λ3)2 ≤ (λ1 − λ3)2

we get

‖R◦‖ ≤ 21−δK(λ1 + λ2 + λ3)1−δ ≤ Kscal1−δ

on M × [0, T ) as claimed.

Now we define k := 12(1+C) , where C is the constant from above. Then on M × [0, T ) holds

(C + 1)(λ1 + λ2) ≥ (λ2 + λ3) + (λ1 + λ2) > λ1 + λ2 + λ3 =scal

2.

Since the smallest eigenvalue of Ric is given by λ1 + λ2, we have shown that

Ric > kscal

holds on M × [0, T ) and we are done.

Moreover, we have the following proposition.

Proposition 4.2.7. In dimension 3 the Ricci flow preserves

1. nonnegative and positive Ricci curvature

2. nonnegative and positive sectional curvature

Proof. It follows immediately from Lemma 4.2.6 0., 1. and the weak tensor maximum principle that

nonnegative sectional curvature and nonnegative Ricci curvature are preserved by the Ricci flow.

The rest follows applying the strong tensor maximum principle.

4.3 Estimating the Gradient of Scalar Curvature

Proposition 4.3.1. For all η > 0 there exists a constant C = C(g0, η) depending only on the initial

metric and η, such that

‖∇scal‖ ≤ ηscal32 + C

75

The proof follows applying the maximum principle to the right quantity. But how can we find it?

Let’s have a look at the evolution equation of ‖∇scal‖2. It is given by

∂

∂t‖∇scal‖2 = ∆ ‖∇scal‖2 − 2 ‖∇∇scal‖2 + 4g

(∇scal,∇‖Ric‖2

),

which is shown below. Because it’s hard to deal with the last term we try out another quantity:

∂

∂t

‖∇scal‖2

scal= ∆

‖∇scal‖2

scal− 2scal

∥∥∥∥∇∇scalscal

∥∥∥∥2

+4scal

g(∇scal,∇‖Ric‖2

)− 2 ‖Ric‖2

scal2‖∇scal‖2

.

This can be done because the scalar curvature is positve. The advantage of this evolution equation

is that one can get rid of the term

4scal

g(∇scal,∇‖Ric‖2

)by adding the term α ‖R◦‖2 − η2scal2 to ‖∇scal‖2

scal . We denote this sum by F . The constants α and

η will be determined later.

We have to compute the evolution equation of F

Lemma 4.3.2. We have the following evolution equations

1.∂

∂t‖∇scal‖2 = ∆ ‖∇scal‖2 − 2 ‖∇∇scal‖2 + 4g

(∇scal,∇‖Ric‖2

)2.

∂

∂t

‖∇scal‖2

scal= ∆

‖∇scal‖2

scal− 2scal

∥∥∥∥∇∇scalscal

∥∥∥∥2

+4scal

g(∇scal,∇‖Ric‖2

)− 2 ‖Ric‖2

scal2‖∇scal‖2

3.∂

∂tscal2 = ∆scal2 − 2 ‖∇scal‖2 + 4scal ‖Ric‖2

4.∂

∂t‖R◦‖2 = ∆ ‖R◦‖2 − 2 ‖∇R◦‖2 + 4trace

(R◦ ◦

(R#R + R2

))Proof. 1. We compute

∂

∂t‖∇scal‖2 = 2g

(∇ ∂

∂t∇scal,∇scal

)Using the evolution equations ∇ = 2Ric ◦ ∇ of the gradient operator and the evolution equation

˙scal = ∆scal + 2 ‖Ric‖2 we get

∇ ∂∂t∇scal =

d

dt(∇scal)−Ric(∇scal) = ∇scal +∇ ˙scal −Ric(∇scal)

= ∇∆scal + 2∇‖Ric‖2 +Ric(∇scal) = ∆∇scal + 2∇‖Ric‖2

76

In the last step we have used the formula

∆∇f = ∇∆f +Ric(∇f)

Now we can write

∂

∂t‖∇scal‖2 = 2g (∆∇scal,∇scal) + 4g

(∇scal,∇‖Ric‖2

)= ∆ ‖∇scal‖2 − 2 ‖∇∇scal‖2 + 4g

(∇scal,∇‖Ric‖2

),

where we have used Lemma 3.3.3 in the last step.

2. We do some auxiliary computations first:

Suppose that f and g are functions on a Riemannian manifold. Then

(I) ∆(fg) = (∆f)g + f∆g + 2g(∇f,∇g) and if f > 0 or f < 0 then

(II) ∆1f

=2f3

‖∇f‖2 − 1f2

∆f

(III) f∥∥∥∥∇∇ff

∥∥∥∥2

=‖∇f‖2

f3+‖∇∇f‖2

f+ g(∇ 1

f,∇‖∇f‖2),

(I) is a well known formula.

(II) We compute in a point p. Let {ei} be a Fermi frame with origin p

∆1f

= f∑i

g

(∇ei

∇ 1f, ei

)= −

∑i

g

(∇ei

1f2∇f, ei

)=

2f3

(∇f,∇f)− 1f2

∆f

(III) We compute in p

f

∥∥∥∥∇∇ff∥∥∥∥ =

∑i

g

(∇ei

∇ff,∇ei

∇ff

)= f

∑i

g

(−eiff2

∇f +1f∇ei

∇f,−eiff2

∇f +1f∇ei

∇f)

=‖∇f‖4

f3− 2f2g(∇∇f∇f,∇f

)+

1f‖∇∇f‖2

=‖∇f‖4

f3+(∇ 1f

)‖∇f‖2 +

1f‖∇∇f‖2

=‖∇f‖4

f3+ g

(∇ 1f,∇‖∇f‖2

)+

1f‖∇∇f‖2

77

Now we are ready to compute the evolution equation

∂

∂t

‖∇scal‖2

scal=

(∂∂t ‖∇scal‖

2)scal − ‖∇scal‖2 ∂

∂tscal

scal2

=∆ ‖∇scal‖2 − 2 ‖∇∇scal‖2 + 4g

(∇scal,∇‖Ric‖2

)scal

−‖∇scal‖2

(∆scal + 2 ‖Ric‖2

)scal2

.

Applying (I) gives

∆ ‖∇scal‖2

scal= ∆

‖∇scal‖2

scal−(

∆1scal

)‖∇scal‖2 − 2g

(1scal

,∇‖∇scal‖2

)and (II) implies

− 1scal2

∆scal = ∆1scal

− 2scal3

‖∇scal‖2.

Therefore we get

∂

∂t

‖∇scal‖2

scal= ∆

‖∇scal‖2

scal−2g

(∇ 1scal

,∇‖∇scal‖2

)− 2scal3

‖∇scal‖4 − 2scal

‖∇∇scal‖2

︸ ︷︷ ︸=:(∗)

+4scal

g(∇scal,∇‖Ric‖2

)− 2 ‖Ric‖2

scal‖∇scal‖2

From (III) follows that (∗) equals

−2scal∥∥∥∥∇∇scalscal

∥∥∥∥2

.

This shows 2.

3. Using (I) from above we compute

∂

∂tscal2 = 2scal

∂

∂tscal = 2scal

(∆scal + 2 ‖Ric‖2

)= ∆scal2 − 2 ‖∇scal‖2 + 4scal ‖Ric‖2

4. Note that ‖R◦‖2 equals ‖R‖2 − 13 (traceR)2 = ‖R‖2 − 1

12scal2. We compute

∂

∂t‖R‖2 = 2g

(∇ ∂

∂tR,R

)= 2g (∆R,R) + 4g

(R#R + R2,R

)= ∆ ‖R‖2 − 2 ‖∇R‖2 + 4trace

(R ◦

(R#R + R2

))where we have used Lemma 3.3.3 in the last step. By 3. we get that

∂

∂t‖R◦‖2 = ∆ ‖R◦‖2 − 2 ‖∇R‖2 − 1

6‖∇scal‖2 + 4trace

(R ◦

(R#R + R2

))− 1

3scal ‖Ric‖2

.

78

We have scal = 2traceR and ‖Ric‖2 = 2trace(R#R + R2

). The latter is not obvious. To see that

it’s true, we diagonalize the curvature operator. Then

R#R + R2 =

λ2

1 + λ2λ3

λ22 + λ1λ3

λ23 + λ1λ2

,

so

2trace(R#R + R2

)= (λ1 + λ2)2 + (λ1 + λ3)2 + (λ2 + λ3)2 = ‖Ric‖2

as claimed. Therefore we get

4trace(R ◦

(R#R + R2

))− 1

3scal ‖Ric‖2 = 4trace

(R◦ ◦

(R#R + R2

)).

It remains to show that ‖∇R‖2 − 112 ‖∇scal‖

2 = ‖∇R◦‖2. Let p ∈M and {ei} ⊂ TpM be an

orthonormal basis. Then

‖∇R◦‖2 =∑i

∥∥∇eiR◦∥∥2 =

∑i

∥∥∥(∇eiR)◦∥∥∥2

=∑i

(∥∥∇eiR∥∥2 − 1

3(trace∇ei

R)2)

= ‖∇R‖2 − 13‖∇traceR‖2

= ‖∇R‖2 − 112‖∇scal‖2

The following lemma provides the estimates we need to get F under control.

Lemma 4.3.3. We have the following estimates

E1. ‖∇scal‖2 ≤ 3 ‖∇Ric‖2

E2. ‖∇Ric‖2 ≤ 4 ‖∇R‖2

E3.4scal

∣∣∣g (∇scal,∇‖Ric‖2)∣∣∣ ≤ 8

√3 ‖∇Ric‖2

E4. ‖∇R◦‖2 ≥ 1109

‖∇R‖2

Proof. Pick a point p and a Fermi frame with origin p.

79

E1. We compute

‖∇scal‖2 =

∥∥∥∥∥∑i

(trace∇eiRic)(ei)

∥∥∥∥∥2

≤

(∑i

trace∇eiRic

)2

≤ 3∑i

(trace∇ei

Ric)2

= 3∑i

∥∥∇eiRic∥∥2 = 3 ‖∇Ric‖2

.

E2. We compute

‖∇Ric‖2 ≤∑i,j

‖(∇eiRic) (ej)‖2 ≤∑i,j,k,l

∥∥(∇eiR)(ej ∧ el), ek ∧ el

∥∥2

≤∑

i,j,k,l,m

∥∥(∇eiR)(ej ∧ el), ek ∧ em

∥∥2

= 4 ‖∇R‖2

E3. Note that Ric > 0 implies scal ≤ ‖Ric‖. Furthermore, Ric > 0 implies that ‖Ric‖ is smooth.

Thus, we are allowed to compute in the following way

4scal

∣∣∣g (∇scal,∇‖Ric‖2)∣∣∣ ≤ 8 ‖Ric‖

scal‖∇scal‖ ‖∇Ric‖ ≤ 8

√3 ‖∇Ric‖2

.

E4. We will use that 13

(∑3i=1 xi

)2

≤∑i x

2i holds for all x1, x2, x3 ∈ R.

‖∇R◦‖2 =14

∑i,j,k,r,s

(g((∇ei

R)(ej ∧ ek), er ∧ es

)− 1

3(dtraceR(ei)) g (ej ∧ ek, er ∧ es)

)2

≥ 14

∑i,k,r

(g((∇ei

R)(ei ∧ ek), er ∧ ek

)− 1

3(dtraceR(ei)) g (ei ∧ ek, er ∧ ek)

)2

≥ 136

∑r

∑i,k

g((∇ei

R)(ei ∧ ek), er ∧ ek

)− 1

3g ((dtraceR(ei))ei ∧ ek, er ∧ ek)

2

=136

∑r

(g

(∑i

(∇ei

Ric)ei, er

)− 1

6

∑k

(g (∇scal, er) g (ek, ek)− g (∇scal, ek) g (ek, er)

))2

=136

∑r

(12g (∇scal, er)−

12g (∇scal, er) +

16g (∇scal, er)

)2

=1

1296‖∇scal‖2

From ‖∇R◦‖2 = ‖∇R‖2 − 112 ‖∇scal‖

2 we get ‖∇R‖2 ≥ 112

(1 + 1

108

)‖∇scal‖2 = 1

12109108 ‖∇scal‖

2.

Hence,

‖R◦‖2 ≥(

1− 108109

)‖∇R‖2 =

1109

‖∇R‖2

and the result follows.

80

Now we are ready to estimate F and give the proof of the proposition.

Proof of Proposition 4.3.1. E3 implies

∂

∂t

‖∇scal‖2

scal≤ ∆

‖∇scal‖2

scal+ 8

√3 ‖∇Ric‖2

.

Using E2, E4, Proposition 4.2.1, the estimate for the #-product and ‖R‖2 ≤ scal2 we estimate

∂

∂t‖R◦‖2 ≤ ∆ ‖R◦‖2 − 1

218‖∇Ric‖2 + C0scal

3−δ

where C0 depends only on the initial metric. Furthermore, we can estimate

∂

∂tscal2 ≥ ∆scal2 − 6 ‖∇Ric‖2 +

43scal3.

Therefore we get

∂

∂tF ≤ ∆F +

(8√

3− α

218+ 6η2

)‖∇Ric‖2 + αC0scal

3−δ − 4η2

3scal3.

Thus, for any η > 0 we can find α = α(η) and a constant C1 = C1(η, α(η), go) such that the

corresponding F satisfies∂

∂tF ≤ ∆F + C

on M × [0, T ) which implies ddt+F ≤ C1. Using that T is finite we get that

F (t) ≤ F (0) + C1T

holds for all t ∈ [0, T ). Let C2 := F (0) + C1. From the definition of F follows that on M × [0, T )

holds‖∇scal‖2

scal≤ C2 + η2scal2.

This implies that we can find some constant C = C(η, g0), such that on M × [0, T ) holds

‖∇scal‖ ≤ ηscal32 + C.

81

4.4 Uniform Explosion of Curvature

We will use Proposition 4.3.1, Proposition 4.2.1 and Myers’ theorem to show that scalar curvature

blows up everywhere.

Theorem 4.4.1 (Myers). Let M be an n-dimensional complete Riemannian manifold and γ a

geodesic. If Ric|γ ≥ (n− 1)H for some constant H > 0 and the length of γ is at least π√H

, then γ

has conjugate points.

A proof of Myers’ theorem can be found in [C-E] on page 27.

Corollary 4.4.2 (Myers). Let M be an n-dimensional complete Riemannian manifold with

Ric ≥ (n− 1)H for some constant H > 0. Then M is compact and its fundamental group is finite.

Proposition 4.4.3.

limt−→T

maxscal

minscal= 1

Proof. We have to show that for all η > 0 we can find θ = θ(η) ∈ [0, T ), such that

minscal(t) ≥ (1− η)maxscal(t)

holds for all t ∈ [θ, T ).

Proposition 4.2.1 provides a constant k > 0, such that on M × [0, T ) holds

Ric− kscal > 0.

On [0,∞) we have that η2

1−η decreases if η decreases and that η2

1−η −→ 0 as η −→ 0, so it is

sufficient to find the θ’s for those η’s which satisfy

η2

1− η≤ k

2π2.

Let η > 0 be one of those. Then Proposition 4.3.1 provides a constant C = C(η, g0) such that

‖∇scal‖ ≤ 12η2scal

32 + C

on M × [0, T ). By Corollary 4.2.2 we have that

limt−→T

maxscal(t) = ∞,

82

so there exists some time θ ∈ [0, T ) such that on M × [θ, T ) holds

12maxscal

32 ≥ C.

This gives

‖∇scal‖ ≤ η2maxscal32

on M × [θ, T ).

Now we fix t ∈ [θ, T ) and choose a point p ∈M , where scal(p, t) = maxscal(t). In addition, let

r = r(η, t) = 1

η√maxscal(t)

. We claim that at time t on the closure of the ball Br(p) of radius r

around p holds

scal ≥ (1− η)maxscal.

To see this, let γ be a unit speed geodesic starting in p at time t and s0 ≤ r. We estimate

maxscal − scal(γ(s0)) = scal(γ(0))− scal(γ(s0)) =∫ s0

0

gt(∇scal, γ)ds

≤∫ s0

0

‖∇scal‖ ds ≤∫ r

0

η2maxscal32 ds

=η2maxscal

32

η√maxscal

= η ·maxscal

and the claim follows.

Recall that we have Ric− kscal > 0 on M × [0, T ). Thus, on Br(p) at time t holds

Ric > k(1− η)maxscal.

From our choice of η we get

r(η, · ) ≥ π√k(1−η)

2 maxscal

for all times. By Myers’ theorem we conclude that every geodesic staring in p at time t must

develop a conjugate point before leaving the closure of the ball Br(p). This implies that the whole

manifold is contained in the closure of Br(p), because geodesics can only minimize up to the first

conjugate point. This shows

minscal ≥ (1− η)maxscal

on [θ, T ) because t ∈ [θ, T ) was arbitrary.

83

Corollary 4.4.4.

limet−→eT

maxscal

minscal= 1

Proof. This is clear because we scale with factors depending only on time.

4.5 Conclusion

Proposition 4.5.1. There is a constant l > 0 such that on M × [0, T)

scal ≤ l.

The proof of Proposition 4.5.1 uses the following theorem of Bishop, Cheeger and Gromov. A

proof of this theorem can be found in [P].

Theorem 4.5.2 (Relative Volume Comparison, Bishop-Cheeger-Gromov). Suppose (M, g) is a

complete n-dimensional Riemannian manifold with Ric ≥ (n− 1)k for some real constant k. Then

r 7→ vol(Br(p))v(n, k, r)

is a nonincreasing function whose limit as r −→ 0 is 1.

(The quantity v(n, k, r) denotes the volume of the ball of radius r in the simply connected space

form with constant curvature k.)

Proof of Proposition 4.5.1. We have that the volume vol of the normalized Ricci flow is

constant. Therefore we get a uniform lower bound on the diameter diam(M, get) using Ric ≥ 0 and

Theorem 4.5.2 , namely

diam(M, get) ≥3

√vol

ω3,

where ω3 denotes the volume of the Euclidean ball B1(0) ⊂ R3.

On the other hand Corollary 4.2.3 tells us that on on M × [0, T) holds

Ric − kscal > 0

for some constant k > 0. But then

Ric − kminscal > 0

84

also. Thus,

diam(M, get) ≤π√

k2minscal(t)

.

by Myers’ theorem.

Both inequalities together imply that minscal is bounded from above on [0, T) by a constant

0 ≤ l0 <∞. Using Corollary 4.4.4 we find a constant 0 ≤ l <∞ such that on [0, T) holds

maxscal ≤ l. This completes the proof.

Proposition 4.5.3. g is nonsingular, i.e. T = ∞ and∥∥∥R∥∥∥et ≤ L uniformly on M × [0,∞) for

some constant L > 0

Proof. We show T = ∞ first.

From maxscal −→∞ as t −→ T and maxscal ≤ l on [0, T) follows that the scale factor ψ becomes

infinitely large when time reaches T . Recall that

ψ(t) = e23

R t0 r(s)ds

which gives

limt−→T

∫ t

0

r(s)ds = ∞.

Now we use ddt τ = ψ and ψ(r ◦ τ) = r to get∫ eτ(t)

0

r(s)ds =∫ t

0

r(s)ds

for all t ∈ [0, T ). But r is bounded by some constant l′ on [0, T) by Proposition 4.4.4, so

limt−→T

∫ eτ(t)0

r(s)ds = ∞

implies

T = limt−→T

τ(t) = ∞.

Now we show that∥∥∥R∥∥∥ is uniformly bounded.

Applying Proposition 4.2.1, Proposition 4.5.1 and ψ −→∞ as t −→ T we get∥∥∥R◦ ◦ τ∥∥∥ =

1ψ‖R◦‖ ≤ Kscal1−δ

ψ=K(scal ◦ τ)1−δ

ψδ−→ 0

85

as t −→ T uniformly. Therefore we can bound this expression uniformly by constant L0 ≥ 0. Using

Proposition 4.5.1 again we find∥∥∥R∥∥∥2

=∥∥∥R◦

∥∥∥2

+13(traceR)2 ≤ L0 +

112scal

2≤ L

for some constant L ≥ 0.

Now we have shown the first part of the theorem.

Proposition 4.5.4.

limet−→∞

maxp∈M

∥∥∥R◦∥∥∥et = 0

Proof. There is nothing left to be done. We refer to the proof of the previous proposition.

Corollary 4.5.5. There exist θ > 0 such that on M × [θ,∞)

0 <14Let < secet ≤ Let ≤ L

for suitable constants Let .

Proof. We must show that the sectional curvatures become positive in finite time. Then the result

follows from the previous proposition.

Let λ1 ≤ λ2 ≤ λ3 denote the eigenvalues of the curvature operator of the normalized Ricci flow.

We show that the ratio eλ2+eλ3eλ1+eλ2of the largest to the smallest eigenvalue of Ric converges to one as

time goes to infinity. In consequence we will get that for big times the sum of any two eigenvalues

of the Ricci tensor will be bigger than the third one, which is equivalent to Ric < 12 scal. This is

equivalent to positive sectional curvature by Corollary 4.1.2.

We note that the ratio eλ2+eλ3eλ1+eλ2is scale invariant. So it equals λ2+λ3

λ1+λ2, where λ1 ≤ λ2 ≤ λ3 stand for

the eigenvalues of the curvature operator of the unnormalized Ricci flow. We compute

λ2 + λ3

λ1 + λ2= 1 +

λ3 − λ1

λ1 + λ2≤ 1 +

Kscal1−δ

kscal= 1 +

K

kscal−δ

where we used Lemma 4.2.1. This gives

1 ≤ λ2 + λ3

λ1 + λ2

≤ 1 +K

kscal−δ,

and the claim follows because Kk scal

−δ converges to 0 uniformly on M as time goes to infinity.

86

Lemma 4.5.6. Let n ≥ 3. If a complete n-dimensional Riemannian manifold M has its sectional

curvatures bounded within

0 <14L < sec ≤ L,

then its injectivity radius is bounded from below by

inj(M) ≥ π

N√L,

where N := |π1M |.

Proof. From Myers’ theorem follows that M is compact and that π1M is finite, so the proposition

makes sense. To prove the lemma we will use the following results of Klingenberg.

Lemma 4.5.7 (Klingenberg). If M is a complete Riemannian manifold with L ≥ sec ≥ H > 0,

then its injectivity radius is bounded from below by

inj(M) ≥ min{

π√L, half of length of the shortest smooth closed geodesic in M

}.

Theorem 4.5.8 (Klingenberg). If M is a complete and simply connected Riemannian manifold of

dimension n ≥ 3 such that its sectional curvatures are bounded within 14L < sec ≤ L, then its

injectivity radius is bounded from below by inj(M) ≥ π√L.

The proofs of these statements can be found in [C-E], chapter 5.

Let γ be a smooth closed geodesic in M. Then some power γm with m ≤ N of γ is nullhomotopic,

because the fundamental group of M is finite. γm lifts to a smooth closed geodesic γm in the

universal covering M of M . Otherwise γm is not trivial in π1M . Using Klingenberg’s results we

get that the length L(γm) is at least 2π√L

and hence the length of γ is bounded from below by

L(γ) = L(γ) =1mL(γm) ≥ 2π

N√L.

Now we apply Klingenberg’s lemma to M and get that the injectivity radius is bounded from

below by

inj(M) ≥ π

N√L

as claimed.

87

Corollary 4.5.9. With I := min{inj(M, get)|t ∈ [0, θ]

}and N := |π1M | . we get

I0 := inj(M, get) ≥ min{I,

π

N√L

}.

Proof. Clear.

Now we have shown that every point (p, t) ∈M × [0,∞) is at least I0-thick. Therefore, if we take

an arbitrary sequence of points (pi, ti) with ti −→∞ and geti-orthonormal frames over i, then by

Theorem 3.5.3 we can extract an eternal limit flow (M∞, g∞, p∞, Q∞). The uniform diameter

bound carries over to the limit. By completeness, M∞ is compact and therefore diffeomorphic to

M . Moreover, the limit flow is a normalized Ricci flow again by Corollary 3.5.6. From

limet−→∞

maxp∈M

∥∥∥R◦∥∥∥et = 0

and Schur’s Lemma it is immediate that each time slice M ×{t}

of the limit flow has constant

positive sectional κ(t). Clearly, κ is constant, because otherwise the volume of the limit flow would

change in time, which is impossible!

It is clear that κ is universal in the sense that every limitflow must have curvature κ, no matter

which sequence is used. This completes the proof.

Chapter 5

Nonsingular Solutions

In this chapter we will study the behavior of nonsingular solutions to the normalized Ricci flow on

compact three-manifolds.

5.1 Evolution of Scalar Curvature Under the Normalized

Ricci Flow

Let M be a compact n-dimensional manifold and g a solution to the normalized Ricci flow on M .

Proposition 5.1.1. The scalar curvature scal evolves by the formula

∂

∂tscal = ∆scal + 2

∥∥∥Ric◦∥∥∥2

+2nscal(scal − r)

Proof. Recall that scal evolves by

∂

∂tscal = ∆scal + 2 ‖Ric‖2

.

Now scal = 1ψ scal,

∂∂t

= 1ψ∂∂t by construction and ∆ = 1

ψ∆ by Lemma 3.1.1. We use that to

compute

∂

∂tscal =

1ψ

∂

∂t

(1ψscal

)= − ψ

ψ3scal +

1ψ2

∂

∂tscal

=1ψ2

(∆scal + 2 ‖Ric‖2 − 2

nrscal

)= ∆scal + 2

∥∥∥Ric∥∥∥2

− 2nrscal

88

89

Note that∥∥∥Ric◦∥∥∥2

=∥∥∥Ric∥∥∥2

− 1n scal.

Hence,

2∥∥∥Ric∥∥∥2

− 2nrscal = 2

∥∥∥Ric◦∥∥∥2

+2nscal

(scal − r

)and we are done.

Corollary 5.1.2. 1. minscal is strictly increasing on{t ∈ [0, T)

∣∣∣minscal(t) < 0},

2. minscal(t0) > 0 for some t0 ∈ [0, T) implies minscal > 0 on [t0, T) and

3. minscal(t0) = 0 for some t0 ∈ [0, T) implies minscal ≥ 0 on [t0, T). In particular, if in addition

scal(p, t0) > 0 for some point p ∈M , than minscal is positive on (t0, T).

Proof. minscal is differentiable to the right and the right hand derivative is given by

d

dt+

∣∣∣∣et=et0 scal(t) = min{∂

∂tscal(p, t0)

∣∣∣∣ scal(p, t0) = minscal(t0)}.

By the previous proposition follows

d

dt+

∣∣∣∣et=et0 scal(t) ≥ 2n

min{scal(p, t0)

(scal(p, t0)− r(t0)

)∣∣∣ scal(p, t0) = minscal(t0)}

=2nminscal(t0)

(minscal(t0)− r(t0)

).

1. Since the minimum lies below the average we get minscal − r <≤ 0 for all times. So

minscal(t0) < 0 implies that

d

dt+

∣∣∣∣et=et0 scal(t) ≥ 2nminscal(t0)

(minscal(t0)− r(t0)

)> 0

and the first claim follows. To prove 2. and 3. let ϕ be the solution of the ODE

ϕ = 2nϕ(minscal − r

)with initial value ϕ(t0) = minscal(t0). Then

ϕ(t) = minscal(t0)e2n

Retet0

(mingscal−er)des.

This shows 2. and the first part of 3. because minscal dominates ϕ.

Note that we have minscal(t0) = 0 for the unnormalized Ricci flow at the corresponding time.

And if scal(p, t0) > 0 for some point p ∈M , then scal(p, t0) > 0 also. Applying the strong

maximum principle to the evolution of scal, which is given by

∂

∂tscal = ∆scal + 2 ‖Ric‖2

90

leads to scal > 0 on M × (t0, T ), where T denotes the lifetime of unnormalized flow. This implies

minscal > 0 on this interval by compactness of M . This gives minscal > 0 on (t0, T).

The corollary implies that one of the following three situations must occur:

1. The minimum of scalar curvature becomes positive before T. Then it remains positive forever.

2. The minimum of scalar curvature converges from below to zero as time goes to T

3. The minimum of scalar curvature converges from below to a negative number as time goes to T.

5.2 Long Time Pinching

The following theorem is due to Hamilton ([H5]). It is an improvement of the Hamilton-Ivey

Pinching Theorem which can be found in [H3] and in [I].

Theorem 5.2.1. Suppose that (g)t∈[0,T ) is a solution to the Ricci flow on a compact three

dimensional manifold M . Let λ1 ≤ λ2 ≤ λ3 denote the eigenvalues of the curvature operator and

assume λ1 ≥ −1 to start. Moreover, let X := −λ1. Then

scal

2≥ X (log (X) + log(1 + t)− 3)

holds on { (p, t) ∈M × [0, T )|X(p, t) > 0}

Proof. This is an application of the weak tensor maximum principle.

The curvauture operator evolves by the formula

∇ ∂∂t

R = ∆R + φ(R),

where φ(R) = 2R#R + 2R2. We know from the last chapter that after reparametrizing time by

the factor 12 the associated vertical ODE is equivalent to the following system of ODEs for the

eigenvalues λ1 ≤ λ2 ≤ λ3 of the curvature operatorλ1 = λ2

1 + λ2λ3

λ2 = λ22 + λ1λ3

λ3 = λ23 + λ1λ2

91

Now we define a subset C ⊂ Sym(∧2

π∗TM)

which has convex fibers, is invariant under parallel

translation in spatial direction and preserved by the flow of φ+ ∂∂t .

For (p, t) ∈M × [0, T ) let

C(p,t) :=

{L ∈ Sym

(∧2π∗TM

)(p,t)

∣∣∣∣∣trace(L)

2 ≥ − 31+t and X (L) ≥ 1

1+t impliestrace(L)

2 ≥ X (L) (log (X (L)) + log(1 + t)− 3)

}and

C :=⋃

(p,t)∈M×[0,T )

C(p,t).

It is clear that C is invariant under parallel translation in spatial direction. Why are the fibers

convex?

For each t ∈ [0, T ) we define a convex set Σt ⊂ R2 by

Σt :={

(x, y) ∈ R2

∣∣∣∣ y

2≥ − 3

1 + tand x ≥ 1

1 + timplies

y

2≥ x

(log(x) + log(1 + t)− 3

)}.

Consider the map

Γ = Γ(p,t) : Sym(∧2

π∗TM)

(p,t)−→ R2 : L 7→

(trace(L)

2 , X (L)).

Then we get L ∈ C(p,t) if and only if Γ (L) ∈ Σt. Moreover, if we pick L and L′ in C(p,t), then we

get that Γ (λL + (1− λ)L′) ∈ Σt for all λ ∈ [o, 1] because trace is linear, X is convex and

(x, y0) ∈ Σt implies (x, y) ∈ Σt for all y ≤ y0. This shows that C(p,t) is convex.

Now we show that C is preserved by the flow of φ+ ∂∂t :

Let Y := −λ2. Then we can write

• X = −X2 + Y λ3

• Y = −Y 2 +Xλ3 and

• ˙scal2 = X2 + Y 2 + λ2

3 +XY − λ3 (X + Y ).

First we show that X ≥ 11+t implies scal

2 ≥ X (logX + log(1 + t)− 3). This is equivalent to

W := scal2X − log(X) ≥ log (1 + t)− 3. We want to show that W ≥ X whenever X > 0.

Using the formulas from above we compute

˙scal2X −

(X +

scal

2

)X = X

(X2 + Y 2 + λ2

3 +XY − λ3 (X + Y ))− (λ3 − Y )

(−X2 + Y λ3

)= X3 + I,

92

where I = XY 2 + λ23 (X − Y ) + Y λ3 (Y −X). We claim that X > 0 implies I ≥ 0.

If Y ≤ 0 then λ3 ≥ 0 follows because −Y ≤ λ3. In addition we have X > 0 and X − Y ≥ 0. All

together this implies I ≥ 0.

Now let Y > 0. We can rewrite I as

I = Y 3 − Y 3 +XY 2 + λ23 (X − Y ) + Y λ3 (Y −X)

= Y 3 + (X − Y )(λ2

3 − Y λ3 + Y 2)

Now we get I > 0 from Y > 0, X − Y ≥ 0 and λ23 + Y 2 ≥ 2 |Y λ3|.

Hence, X > 0 gives˙scal2X −

(X +

scal

2

)X > X3

which implies

W =˙scal2 − scalX

X2− X

X=

˙scal2 X −

(X + scal

2

)X

X2≥ X

In particular, if X ≥ 11+t then W ≥ 1

1+t = ddt (log(1 + t)− 3). Thus, if 0 < X ≤ 1 to start then

W (t) ≥ log (1 + t)− 3 in the future. Or equivalently, if 0 < X ≤ 1 to start then

scal2 ≥ X (logX + log(1 + t)− 3) in the future.

We claim that ddtscal2 ≥ 1

3

(scal2

)2.

On the one hand we have

d

dt

scal

2= λ2

1 + λ22 + λ2

3 + λ1λ2 + λ1λ3 + λ2λ3

and13

(scal

2

)2

=13(λ2

1 + λ22 + λ2

3 + 2λ1λ2 + λ1λ3 + λ2λ3

)on the other. Subtracting the lower from the upper equation gives

d

dt

scal

2− 1

3

(scal

2

)2

=16

((λ1 − λ2)

2 + (λ1 − λ3)2 + (λ2 − λ3)

2)≥ 0.

But ddt

(− 3

1+t

)≤ 1

3

(− 3

1+t

)2

. So if we have scal2 ≥ −3 at the start, then we get scal

2 ≥ − 31+t in the

future.

Now we have shown that C is preserved by the flow of φ+ ∂∂t . The theorem follows applying the

weak tensor maximum principle.

93

5.3 Three-Manifolds with Nonnegative Ricci Curvature

The main theorem of this section is a rigidity theorem for solutions to The Ricci flow on compact

three-manifolds. This theorem is due to Hamilton and has been published in [H2]

Theorem 5.3.1. Suppose that M is a compact three dimensional manifold and that (gt)t∈[0,T ) is a

solution to the Ricci flow on M where the initial metric has nonnegative Ricci curvature. Then for

all positive times t holds one of the following cases

• (M, gt) is flat, i.e. a quotient of T3

• (M, gt) is a quotient of S2 × S1, where the first factor has strictly positive sectional curvature

• (M, gt) has strictly positive Ricci curvature.

Proof. The Ricci tensor evolves by the formula

∇ ∂∂t

= ∆Ric+ φ(Ric),

where φ(Ric) = 2∑iR( · , ei)(Ricei). We have already shown that the Ricci flow preserves

nonnegative Ricci curvature. Theorem 2.3.3 provides a δ > 0, such that the kernel of Ric is parallel

on M × (0, δ). If Ric = 0 then sec = 0 follows, because M is three dimensional. By compactness

(M, gt) is a quotient of the three torus for t ∈ (0, δ). The result follows for all times, because the

solution is stationary. If two of the eigenvalues of Ric are zero, then so is the third. To see this let

λ1 ≤ λ2 ≤ λ3 denote the eigenvalues of the curvature operator. Then the eigenvalues of the Ricci

tensor are given by λ1 + λ2 ≤ λ1 + λ3 ≤ λ2 + λ3. At the beginning of section 4.2 we saw that the

eigenvalues of the curvature operator satisfy the vertical ODEλ1 = 2λ2

1 + 2λ2λ3

λ2 = 2λ22 + 2λ1λ3

λ3 = 2λ23 + 2λ1λ2

So the eigenvalues of the Ricci tensor satisfyddt (λ1 + λ2) = 2λ2

1 + 2λ22 + 2λ3 (λ1 + λ2)

ddt (λ1 + λ3) = 2λ2

1 + 2λ23 + 2λ2 (λ1 + λ3)

ddt (λ2 + λ3) = 2λ2

2 + 2λ23 + 2λ1 (λ2 + λ3)

94

Theorem 2.3.3 says that on M × (0, δ) the kernel of Ric is contained in the kernel of φ(Ric). Thus,

if λ1 + λ2 = 0 then 0 = φ(λ1 + λ2) = 2λ21 + 2λ2

2 + 2λ3 (λ1 + λ2), which implies λ1 = λ2 = 0.

Similarly we conclude λ1 = λ3 = 0 from λ1 + λ3 = 0. Thus if λ1 + λ2 = λ1 + λ3 = 0, then

λ2 + λ3 = 0 follows. Hence, it cannot happen that the kernel of Ric has dimension two.

Now suppose that the kernel is one dimensional. Then the same is true for the universal covering

(M, gt) of (M, gt) with the induced metric. Clearly, the kernel of Rict is also parallel. This gives a

parallel splitting of the tangent bundle TM = kerRic⊕(kerRic

)⊥. Thus, (M, gt) splits off a line

by the de Rham Decomposition Theorem. Therefore we have that (M, gt) is isometric to a product(S2 × R, g′t ⊕ can

), where can denotes the canonical Riemannian metric on R and the first factor

has strictly positive sectional curvature. By compactness (M, gt) is a quotient of S2 × S1 as

claimed. Since the Ricci flow preserves products and positive sectional curvature in dimension 2,

we get that the result for all times.

Now the only remaining case is that of strictly positive Ricci curvature.

Theorem 5.3.2. If M is a noncompact complete Riemannian n-manifold with nonnegative Ricci

curvature, then M has infinite volume.

We need the following

Lemma 5.3.3. If M is a complete Riemannian n-manifold with Ric ≥ (n− 1)κ for some κ ∈ R

then for all p, q ∈M and r, s > 0 satisfying r + s ≤ dist(p, q)

V (s)V (d+ r)− V (d− r)

vol(Br(p)) ≤ vol(Bs(q)),

where d := dist(p, q) and V (r) denotes the volume of the ball of radius r in the n-dimensional space

form of constant curvature κ

Proof. Let p, q ∈M and r, s > 0 with r + s ≤ d be arbitrary. The Bishop-Gromov Relative Volume

Comparison Theorem tells us that the function

R 7→ vol(BR(q))V (R)

is monotonically non increasing, i.e. 0 < R ≤ S implies

vol(BR(q))V (R)

≤ vol(BS(q))V (S)

,

95

or equivalentlyvol(BS(q))− vol(BR(q))

V (S)− V (R)≤ vol(BR(q))

V (R).

From the triangle inequality we get that

Br(p) ⊂ Bd+r(q) \Bd−r(q),

so

vol(Br(p)) ≤ vol(Bd+r(q))− vol(Bd−r(q)).

Therefore, putting all together,

V (s)V (d+ r)− V (d− r)

vol(Br(p)) ≤V (s)

V (d+ r)− V (d− r)(vol(Bd+r(q))− vol(Bd−r(q)))

≤ V (s)vol(Bd−r(q)V (d− r)

≤ V (s)vol(Bs(q)V (s)

= vol(Bs(q))

Now we are ready to prove the theorem.

Proof. Let p ∈M be arbitrary. We show that limR−→∞ vol(BR(p)) = ∞.

Let ρ : [0,∞) −→M be a unit speed ray emanating at p and pick some r0 > 0. For R > r0 we let

• r = r(R) := R−r02

• q = q(R) := ρ(r0 + r(R))

• d = d(R) := dist(q(R), p)

Since d = r0 + r, we can apply the above lemma with κ = 0 to this situation and get

rn

(d+ r0)n − (d− r0)nvol(Br0(p)) ≤ vol(Br(q)).

Nowrn

(d+ r0)n − (d− r0)n=

(R− r0)n

(R+ 3r0)n − (r − r0)n,

96

where the right hand side goes to infinity as R goes to infinity.

By the triangle inequality Br(q) ⊂ BR(p) for all R ≥ R0, so

vol(Br(q)) ≤ vol(BR(p))

for all R ≥ R0. Hence

limR−→∞

vol(BR(p)) = ∞,

and the theorem follows.

5.4 A Certain Singular Solution

Lemma 5.4.1. If g0 is a product metric on S2 × S1, where the S2-factor has positive curvature,

then the solution to the normalized Ricci flow with initial metric g0 is singular.

Proof. Let g denote the normalized Ricci flow with initial metric g0. If the lifetime of g is finite,

then g is singular by Theorem 3.4.4. Now we treat the nontrivial case of infinite lifetime.

The initial metric has positive scalar curvature. This has two consequences, namely:

1. the solution g = ψg to the unnormalized Ricci flow has only finite lifetime T

2. the scale factor ψ(t) tends to zero as time tends to infinity.

The first point is already known. The second follows from the first in the following way: On the one

hand the unnormalized Ricci flow preserves positive scalar curvature. Then so does the normalized

Ricci flow, which forces the scale factor ψ to be decreasing in time. Recall that ψ(t) = e−23

Ret0 erdes.

On the other hand, we know that the lifetime of the unnormalized Ricci flow is given by

T = limet−→∞

∫ et0

ψ(s)ds,

which is finite. The second point follows.

Since the unnormalized Ricci flow preserves products, so does the normalized Ricci flow, i.e. we

have g = g2 + g1 for all times, where gi denotes the Si-part of g. By the way, the gis are no

normalized Ricci flows!

Now we show that the volume vol2 of the S2-factor tends to zero while time tends to infinity:

97

From the evolution equation of g and Ric = Ric2 + Ric1 we get that

∂

∂tg2 =

23rg2 − Ric2.

But

r(t) =1

vol(t)

∫S2×S1

scal(·, t)dvol(t)

=vol1(t)

vol(t)

∫S2scal2(·, t)dvol2(t)

= r2(t),

so we get that the volume of the S2 factor evolves by the equation

∂

∂tvol2 =

23rvol2 − r2vol2

= −13rvol2

This gives

vol2(t) = e−13

Ret0 er(es)des vol2(0)

= (ψ(t))12 vol2(0)

by separation of variables. Hence

limet−→∞

vol2 = 0.

Using the Gauss-Bonnet formula we get that

r2(t) =1

vol2(t)

∫S2scal2(·, t)dvol2(t)

=2

vol2(t)

∫S2sec2(·, t)dvol2(t)

=8π

vol2(t).

This implies that the average scalar curvature r = r2 is not bounded in time. In addition, we know

that the volume vol doesn’t change under the normalized Ricci flow. Therefore, the average scalar

curvature r becomes infinitely large only if scalar curvature scal becomes infinitely large in at least

one point. This shows that g is singular.

98

5.5 Spherical Limits

Before we state the theorem of this section, we introduce some new terms which we will use

throughout the rest of the text.

Definition 5.5.1. A solution (g)et∈eT to the normalized Ricci flow on a compact manifold M is

called collapsed, if

lim infet−→eT

maxp∈M

injp(M, get

)= 0.

Otherwise it is called noncollapsed. If g is noncollapsed then we can find a sequence

(pi, ti) ⊂M × [0, T) with ti −→ T as ı −→∞ and a δ > 0, such that injpi

(M, geti

)≥ δ. Such a

sequence will be referred to as a sequence of thick points. If g is nonsingular and noncollapsed,

then it is always possible to extract an eternal limit flow from a sequence of time shifted

normalized Ricci flows which comes from a sequence of thick points. Such a limit flow will also be

called a noncollapsed limit flow.

Theorem 5.5.1. Suppose that g is a nonsingular, noncollapsed solution to the normalized Ricci

flow on a three dimensional compact manifold M . If the scalar curvature becomes positive in finite

time, then M is diffeomorphic to a spherical space form.

Proof. Let (pi, ti), where ti −→∞ as i −→∞, be a sequence of thick points,

i.e. injpi(M, gi(0)) ≥ δ > 0 for all i and some δ > 0. Then (M, gi, pi, Qi) −→ (M∞, g∞, p∞, Q∞)

after passing to a subsequence. First we show that (M∞, g∞, p∞, Q∞) has nonnegative sectional

curvature:

From minscal(t0) > 0 we get min scal(τ(t0)) > 0 for the unnormalized flow g. Therefore g has

finite lifetime T > 0. This implies limt−→T max scal(t) = ∞. Moreover, we get limet−→∞ ψ(t) = 0

because nonnegativity of scalar curvature makes the scale factor ψ decrease and

T = limet−→∞∫∞0ψ(s)ds.

Rescaling the initial metric, we can achieve that to start the smallest eigenvalue λ1 of the

curvature operator R is bounded from below by −1. Then at time 0 we get the same estimate for

λ1. We will apply Theorem 5.2.1 to show that the negative sectional curvatures of g must vanish

as time tends to infinity.

99

Adopting the notation of 5.2.1 we define X(t) := maxpM X(p, t) and assume that there is an

unbounded sequence (ti) ⊂ [0,∞) such that X(ti) ≥ ε holds for all i and some ε > 0. This implies

that X(τ(ti)) =eX(eti)eψ(eti

) −→∞ as i −→∞, where X(t) := maxp∈M X(p, t) for t ∈ [0, T ). Dropping

the time dependence we get

maxp∈M

scal(p, ti) = ψ(ti) maxp∈M

scal(p, τ(ti))

≥ X(ti)(log(X(τ(ti)))− 3),

which is a contradiction, because log(X(τ(ti))) −→∞ as t −→∞, but scal is uniformly bounded

in absolute value.

This shows that the limit flow (M∞, g∞, p∞, Q∞) has nonnegative sectional curvature.

In particular the limit flow has nonnegative Ricci curvature. In addition the limit flow has finite

volume, so M∞ is compact by Theorem 5.3.2 and therefore diffeomorphic to M . Moreover, the the

restriction of g to M∞ × [0,∞) is nonsingular by Corollary 3.5.6.

Applying Theorem 5.3.1 to the unnormalized limit flow we see that there are only three possible

cases

• (M∞, g∞) is flat, i.e. a quotient of T3

• (M∞, g∞) is a quotient of S2 × S1, where the first factor has strictly positive sectional

curvature

• (M∞, g∞) has strictly positive Ricci curvature.

Note that in general Theorem 5.3.1 only makes a statement about what is happening in the future.

But here we have Ric∞ ≥ 0 for all times. Therefore, we get the same result for the whole flow in

this case.

The first case is impossible. This follows from a work of Gromov and Lawson ([G-L]), which

implies that the n-torus doesn’t admit metrics with positive scalar curvature. But M∞ is

diffeomorphic to M and we have minscal(τ(t0)) > 0.

The second case is impossible either, since there is no nonsingular solution to the normalized Ricci

flow on S2 × S1, where the initial metric is a product and the first factor has strictly positive

sectional curvature. See Lemma 5.4.1 for the proof of this statement.

100

Now we treat the third case:

Since the limit flow has strictly positive Ricci curvature the original solution g must become

strictly Ricci positive after finite time. Hence, the limit flow has constant positive sectional

curvature by Theorem 4.0.8.

5.6 Flat Limits

Theorem 5.6.1. Let g be a nonsingular, noncollapsed solution to the normalized Ricci flow on a

three dimensional compact manifold M . If minscal converges to 0 from below as time goes to

infinity then M is diffeomorphic to a flat space form.

Proof. We show that there exists a sequence (pi, ti) of thick points such that the corresponding

sequence (M, gi, pi, Qi) of time shifted normalized Ricci flows converges to a flat limit flow

(M∞, g∞, p∞, Q∞). This happens by showing that there exist a sequence of thick points, such

that the limit flow has nonnegative sectional curvature. Once we have shown this, we are done in

the following way:

Using the same arguments as in the last section we conclude that Minfty is compact, diffeomorphic

to M , g∞ is nonsingular and that there are only the three possible cases

• (M∞, g∞) is flat, i.e. a quotient of T3

• (M∞, g∞) is a quotient of S2 × S1, where the first factor has strictly positive sectional

curvature

• (M∞, g∞) has strictly positive Ricci curvature.

The second case isn’t possible for the same reason as before. The third case isn’t possible, because

it implies that minscal must become positive in finite time. Therefore, the limit flow is flat.

Now we concentrate ourselves on finding the desired sequence of thick points.

Let g be the unnormalized flow, defined on M × [0, T ), 0 < T ≤ ∞. Its volume changes in time.

We consider the following cases:

1. 0 < c ≤ vol(t) ≤ C <∞ for all t ∈ [0, T ), where c and C are constants

2. not the first case

101

Case 1. The scale factors ψ are uniformly bounded within positive constants. Hence g has infinite

lifetime T = ∞.

If there exists A ≥ 0, such that

maxp∈M

X(p, t) ≤ A

1 + t

for all t, then

limt−→∞

maxp∈M

X(p, t) = 0,

so

limet−→∞

maxp∈M

X(p, t) = 0

also, using the bounds on ψ.

If not, then there exists a sequence (ti), ti −→∞, such that

limi−→∞

maxp∈M

X(p, ti)(1 + ti) = ∞.

Now we choose pi ∈M with X(pi, ti) = maxp∈M X(p, ti). The uniform bound on the scale factors

and the uniform bound on scal imply a uniform bound on scal. If necessary, we scale the initial

metric in so that we can apply the Long Time Pinching Theorem to this situation. Then we get:

scal(pi, ti) ≥ X(pi, ti)(

log (X(pi, ti)(1 + ti))− 3).

This implies that

X(pi, ti) −→ 0

as i −→∞, using the uniform bound on scal. Again using the bounds on the scale factors, we see

X(pi, ti) −→ 0

for the normalized flow at the corresponding times ti also.

This shows that

limi−→∞

maxp∈M

X(p, ti) = 0.

Since g is noncollapsed, we find points pi ∈M , such that the points (pi, ti) form a sequence of

thick points. Hence, if we extract a limit flow (M∞, g∞, p∞, Q∞) from the sequence (M, gi, pi, Qi)

of time shifted normalized Ricci flows which corresponds to this particular sequence of thick

points, then the limit flow has nonnegative sectional curvature.

Case 2. There are two possibilities, namely

102

1. vol(ti) −→ 0 for a sequence ti −→∞

2. vol(ti) −→∞ for a sequence ti −→∞

1. From

vol(ti) = (ψ(ti))32 vol

we get

ψ(ti) −→ 0

as i −→∞ because the volume vol is constant. Hence,

maxp∈M

X(p, ti) −→ 0,

using Theorem 5.2.1 in the same way as above.

From the assumption of noncollapse we know that we can choose a sequence of points (pi), such

that (pi, ti) is a sequence of thick points. Extracting a limit from the corresponding sequence

(M, gi, pi, Qi) of time shifted normalized Ricci flows yields a limit flow with nonnegative sectional

curvature.

2. The volume of the unnormalized flow evolves by the equation

d

dtvol = −rvol,

so

vol(ti) = e−R ti0 r(s)dsvol(0).

This implies the existence of another sequence si −→ T with r(si) < 0, where T is the lifetime of g.

Otherwise we have that the volume vol is bounded. Therefore r(si) < 0 either. Hence,

limi−→∞

r(si) = 0,

using minscal ≤ r and that minscal converges to 0 from below. Again we can find a sequence

(pi, si) of thick points and take a limit (M∞, g∞, p∞, Q∞).

Now we show that this limit is flat.

We have that

limi−→∞

ri(0) = limi−→∞

∫M

scali(·, 0)dvoli = 0,

103

because ri(0) = r(si) by definition and limi−→∞ r(si) = 0. We claim that∫M∞

scal∞(·, 0)dvol∞ = 0

in the limit.

If not, then there exist q ∈M∞, such that scal∞(q, 0) > 0, where we use that scal∞ ≥ 0. This

implies scal∞(·, 0) > 0 on a neighborhood of q. Now we choose r > 0, such that the closure of the

g∞(0)-ball B of radius r around q is contained in this neighborhood. Then

scal∞(, 0)|B ≥ δ

for some δ > 0. Now let ϕi : Ui −→ Vi be the diffeomorphisms from the definition of convergence.

If we choose 0 < ε < δ, and i0 so big such that B ⊂ Ui for all i ≥ i0, then we get∫M

scali(·, 0)dvoli =∫M\ϕi(B)

scali(·, 0)dvoli +∫ϕi(B)

scali(·, 0)dvoli

=∫M\ϕi(B)

scali(·, 0)dvoli +∫B

ϕ∗i (scali(·, 0)dvoli)

≥ minscali(0)vol + (δ − ε)∫B

ϕ∗i dvoli(0).

Letting i to infinity gives

0 ≥ (δ − ε)∫B

dvol∞(0).

This is impossible, since B has positive volume w.r.t. g∞(0).

It is clear that the limit flow has nonnegative scalar curvature, so scal∞ ≡ 0 because otherwise we

get ∫M∞

scal∞(·, 0)dvol∞ > 0.

Denormalizing g∞ to g∞ doesn’t change this property. g∞ is defined on M∞ × (α, ω), where

α < 0 < ω. Applying the strong maximum principle to the evolution equation of scal∞, namely

∂

∂tscal∞ = ∆scal∞ + 2 ‖Ric∞‖2

∞ ,

we see that

scal∞|M∞×(α,0] ≡ 0,

which implies

‖Ric∞‖2∞ ≡ 0

104

on M∞ × (α, 0]. Therefore g∞ is flat on M∞ × (α, 0], because M∞ is three dimensional. Hence, g∞

is flat for all times and therefore g∞ is flat as well. This completes the theorem.

5.7 Hyperbolic Limits

Theorem 5.7.1. Let g be a nonsingular, noncollapsed solution to the normalized Ricci flow on a

compact three dimensional manifold M . If minscal converges from below to a negative number κ

as time goes to infinity, then every noncollapsed limit flow is hyperbolic with constant sectional

curvature κ6 .

Proof. After rescaling we can assume that minscal(t) converges from below to −6. From the

evolution equation of scal

∂

∂tscal = ∆scal + 2

∥∥∥Ric◦∥∥∥2

+23scal(scal − r)

we getd

dt+minscal ≥ 2

3minscal(minscal − r),

and henced

dt+minscal ≥ 4(r −minscal),

using that minscal ≤ −6. This implies

0 ≤∫ ∞

0

(r −minscal)dt

≤ 14

∫ ∞

0

(d

dt+minscal)dt

<∞

since minscal is locally lipschitz, using Proposition 2.1.3.

Now for all ε > 0 we can choose Θ, such that t ≥ Θ implies

0 ≤∫ et+1

et(r −minscal)ds ≤ ε.

Since minscal converges to −6 we can choose Θ so big, such that minscal ≥ −6− ε holds on

[Θ,∞), too. But then

−ε ≤∫ et+1

et(r + 6)ds ≤ ε

105

for t ≥ Θ. This implies

−ε ≤∫ et+1

et(ri + 6)ds ≤ ε

for all t ∈ (Θ − ti,∞), so ∫ et+1

et(r∞ + 6)ds = 0

for all t ∈ R, which gives

r∞ ≡ −6

in the limit.

Now we show that scal∞ ≡ −6 in the limit. Then we conclude the theorem by regarding the

evolution equation of scal∞.

First consider g again. From the triangle inequality and the fact that the minimum of scalar

curvature lies below its average it follows that∣∣∣scal − r∣∣∣ ≤ (scal −minscal) + (r −minscal).

Therefore ∫M

∣∣∣scal − r∣∣∣ dvol ≤ ∫

M

(scal −minscal)dvol +∫M

(r −minscal)dvol

= 2∫M

(r −minscal)dvol

= 2(r −minscal)vol

Using one of our previous calculations we find∫ ∞

0

∫M

∣∣∣scal − r∣∣∣ dvolds ≤ 2vol

∫ ∞

0

(r −minscal)ds <∞.

Hence, ∫ et+1

et

∫M

∣∣∣scal − r∣∣∣ dvolds −→ 0

as t −→∞, which implies

scal( · , t) −→ −6

almost everywhere as t −→∞ for all. But then we get

scali( · , t0) −→ −6

106

almost everywhere as −→∞ for all fixed t0 ∈ R as well. On the other hand we have that scali ◦ ϕi

converges uniformly to scal∞ on all compact subsets of M∞, where the ϕi : Ui −→ Vi denote the

diffeomorphisms from the definition of convergence, which have been used implicitly when we took

the limit. By continuity we conclude that scal∞ ≡ −6 in the limit, as claimed.

Recall that each scali satisfies the evolution equation equation

∂

∂tscali = ∆scali + 2

∥∥∥Ric◦i ∥∥∥2

i+

23scali(scali − ri).

Using the convergence of the flows, the evolution equation carries over to the limit flow, i.e. scal∞

satisfies∂

∂tscal∞ = ∆scal∞ + 2

∥∥∥Ric◦∞∥∥∥2

∞+

23scal∞(scal∞ − r∞).

But scal∞ is constantly −6. Hence, putting all together we get

0 =∥∥∥Ric◦∞∥∥∥2

∞=∥∥∥Ric∞ + 2id

∥∥∥2

∞

in the limit. Thus, for all t the Riemannian metric g∞(t) is Einstein and all eigenvalues of Ric∞(t)

are given by −2. This forces the limit flow to have constant sectional curvature -1 because M∞

three-dimensional.

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109

Erklarung

Hiermit erklare ich, Florian Schmidt, dass ich die vorliegende Diplomarbeit mit dem Titel ”On the

Longtime Behaviour of Solutions to the Ricci Flow on Compact Thee-Manifolds” selbstandig

verfasst habe und nur die angegebenen Quellen und Hilfsmittel benutzt habe.

Eggmuhl, den 30.03.2006