Notes from Math 538: Ricci ⁄ow and the Poincare...

Notes from Math 538: Ricci �ow and thePoincare conjecture

David Glickenstein

Spring 2009

Contents

1 Introduction 1

2 Three-manifolds and the Poincaré conjecture 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Examples of 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Spherical manifolds . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Sphere bundles over S1 . . . . . . . . . . . . . . . . . . . 42.2.3 Connected sum . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Idea of the Hamilton-Perelman proof . . . . . . . . . . . . . . . . 52.3.1 Hamilton�s �rst result . . . . . . . . . . . . . . . . . . . . 62.3.2 2D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.3 General case . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Background in di¤erential geometry 93.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Basics of tangent bundles and tensor bundles . . . . . . . . . . . 93.3 Connections and covariant derivatives . . . . . . . . . . . . . . . 12

3.3.1 What is a connection? . . . . . . . . . . . . . . . . . . . . 123.3.2 Torsion, compatibility with the metric, and Levi-Civita

connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.3 Higher derivatives of functions and tensors . . . . . . . . 16

3.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Basics of geometric evolutions 214.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Ricci �ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Existence/Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Basics of PDE techniques 295.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 The maximum principle . . . . . . . . . . . . . . . . . . . . . . . 295.3 Maximum principle on tensors . . . . . . . . . . . . . . . . . . . 33

iii

iv CONTENTS

6 Singularities of Ricci Flow 356.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Finite time singularities . . . . . . . . . . . . . . . . . . . . . . . 356.3 Blow ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.4 Convergence and collapsing . . . . . . . . . . . . . . . . . . . . . 386.5 �-noncollapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 Ricci �ow from energies 437.1 Gradient �ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.2 Ricci �ow as a gradient �ow . . . . . . . . . . . . . . . . . . . . . 447.3 Perelman entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.4 Log-Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . 527.5 Noncollapsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Ricci �ow for attacking geometrization 598.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Finding canonical neighborhoods . . . . . . . . . . . . . . . . . . 608.3 Canonical neighborhoods . . . . . . . . . . . . . . . . . . . . . . 608.4 How surgery works . . . . . . . . . . . . . . . . . . . . . . . . . . 638.5 Some things to prove . . . . . . . . . . . . . . . . . . . . . . . . . 63

9 Reduced distance 659.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659.2 Short discussion of W vs reduced distance . . . . . . . . . . . . . 659.3 L-length and reduced distance . . . . . . . . . . . . . . . . . . . . 669.4 Variations of length and the distance function . . . . . . . . . . . 709.5 Variations of the reduced distance . . . . . . . . . . . . . . . . . 74

10 Problems 79

Preface

These are notes from a topics course on Ricci �ow and the Poincaré Conjecturefrom Spring 2008.

v

vi PREFACE

Chapter 1

Introduction

These are notes from a topics course on Ricci �ow and the Poincaré Conjecturefrom Spring 2008.

1

2 CHAPTER 1. INTRODUCTION

Chapter 2

Three-manifolds and thePoincaré conjecture

2.1 Introduction

This lecture is mostly taken from Tao�s lecture 2.In this lecture we are going to introduce the Poincaré conjecture and the

idea of the proof. The rest of the class will be going into di¤erent details of thisproof in varying amounts of careful detail. All manifolds will be assumed to bewithout boundary unless otherwise speci�ed.The Poincaré conjecture is this:

Theorem 1 (Poincare conjecture) Let M be a compact 3-manifold which isconnected and simply connected. Then M is homeomorphic to the 3-sphere S3.

Remark 2 A simply connected manifold is necessarily orientable.

In fact, one can prove a stronger statement called Thurston�s geometrizationconjecture, which is quite a bit more complicated, but is roughly the following:

Theorem 3 (Thurston�s geometrization conjecture) Every 3-manifoldMcan be cut along spheres and �1-essential tori such that each piece can be givenone of 8 geometries (E3;S3;H3;S2 � R;H2 � R;fSL (2;R) ;Nil;Sol).We may go into this conjecture a little more if we have time, but certainly

elements of this will come up in the process of these lectures. We will take aquick look

2.2 Examples of 3-manifolds

Here we look in the topological category. It turns out that in three dimensions,the smooth category, the piecewise linear category, and the topological category

3

4CHAPTER 2. THREE-MANIFOLDS AND THE POINCARÉ CONJECTURE

are the same (e.g., any homeomorphism can be approximated by a di¤eomor-phism, any topological manifold can be given a smooth structure, etc.)

2.2.1 Spherical manifolds

The most basic 3-manifold is the 3-sphere, S3; which can be constructed inseveral ways, such as:

� The set of unit vectors in R4

� The one point compacti�cation of R3:

Using the second de�nition, it is clear that any loop can be contracted to apoint, and so S3 is simply connected. One can also look at the �rst de�nition andsee that the rotations SO(4) of R4 act transitively on S3; with stabilizer SO (3) ;and so S3 is a homogeneous space described by the quotient SO (4) =SO (3) :One can also notice that the unit vectors in R4 can be given the group structureof the unit quaternions, and it is not too hard to see that this group is isomorphicto SU (2) ; which is the double cover of SO (3) :As more examples of 3-manifolds, it is possible to �nd �nite groups � acting

freely on S3; and consider quotients of S3 by the action. Note that if the actionis nontrivial, then these new spaces are not simply connected. One can see thisin several ways. The direct method is that since there must be g 2 � and x 2 S3such that gx 6= x: A path from x to gx in S3 descends to a loop in S3=�: Ifthere is a homotopy of that loop to a point, then one can lift the homotopy to ahomotopy H : [0; 1]� [0; 1]! S3 such that H (t; 0) = (t) : But then, looking atH (1; s) ; we see that H (1; 0) = gx and H (1; 1) = x and H (1; s) = g0x for someg0 2 � for all s: But since � is discrete, this is impossible. A more high levelapproach shows that the map S3 ! S3=� is a covering map, and, in fact, theuniversal covering map and so �1

�S3=�

�= � if � acts e¤ectively. Hence each

of these spaces S3=� are di¤erent manifolds than S3: They are called sphericalmanifolds.The elliptization conjecture states that spherical manifolds are the only man-

ifolds with �nite fundamental group.

2.2.2 Sphere bundles over S1

The next example is to consider S2 bundles over S1; which is the same asS2 � [0; 1] with S2 � f0g identi�ed with S2 � f1g by a homeomorphism. Recallthat a homeomorphism

� : S2 ! S2

induces an isomorphism on homology (or cohomology),

�� : H2

�S2�! H2

�S2�

and sinceH2

�S2� �= Z

2.3. IDEA OF THE HAMILTON-PERELMAN PROOF 5

there are only two possibilities for ��: In fact, these classify the possible mapsup to continuous deformation, and so there is an orientation preserving and anorientation reversing homeomorphism and that is all (using �2 instead of H2).(See, for instance, Bredon, Cor. 16.4 in Chapter 2.)Notice that these manifolds have a map � : M ! S1: It is clear that

this induces a surjective homomorphism on fundamental group �� : �1 (M) !�1�S1� �= Z. The kernel of this map consists of loops which can be deformed

to maps only on S2 (constant on the other component), and since S2 is simplyconnected, this map is an isomorphism. Note that these manifolds are thus notsimply-connected.

2.2.3 Connected sum

One can also form new manifolds via the connected sum operation. Given two3-manifolds, M and M 0; one forms the connected sum by removing a disk fromeach manifold and then identifying the boundary of the removed disks. Wedenote this as M#M 0: Recall that in 2D, all manifolds can be formed from thesphere and the torus in this way.Now, we may consider the class of all compact, connected 3-manifolds (up

to homeomorphism) with the connected sum operation. These form a monoid(essentially a group without inverse), with an identity (S3). Any nontrivial (i.e.,non-identity) manifold can be decomposed into pieces by connected sums, i.e.,given any M; we can write

M �M1#M2# � � �#Mk

where Mj cannot be written as a connected sum any more (this is a theoremof Kneser). We call such a decomposition a prime decomposition and suchmanifolds Mj prime manifolds. The proof is very similar to the fundamentaltheorem of arithmetic which gives the prime decomposition of positive integers.

Proposition 4 Suppose M and M 0 are connected manifolds of the same di-mension. Then

1. M#M 0 is compact if and only if both M and M 0 are compact.

2. M#M 0 is orientable if and only if both M and M 0 are orientable.

3. M#M 0 is simply connected if and only if both M and M 0 are simplyconnected.

We leave the proof as an exercise, but it is not too di¢ cult.

2.3 Idea of the Hamilton-Perelman proof

In order to give the idea, we will introduce a few concepts which will be de�nedmore precisely in successive lectures.


Any smooth manifold can be given a Riemannian metric, denoted gij org (�; �) ; which is essentially an inner product (i.e., symmetric, positive-de�nitebilinear form) at each tangent space which varies smoothly as the basepoint ofthe tangent space changes. The Riemannian metric allows one to de�ne anglesbetween two curves and also to measure lengths of (piecewise C1-) curves byintegrating the tangent vectors of a curve. That is, if : [0; a]!M is a curve,we can calculate its length as

` ( ) =

Z a

0

g ( _ (t) ; _ (t))1=2

dt;

where _ (t) is the tangent to the curve at t: A Riemannian metric induces ametric space structure on M; as the distance between two points is given by thein�mum of lengths of all piecewise smooth curves from one point to the other. Itis a fact that the metric topology induces the original topology of the manifold.The main idea is to deform any Riemannian metric to a standard one. This is

the idea of �geometrizing.�How does one choose the deformation? R. Hamilton�rst proposed to deform by an equation called the Ricci �ow, which is a partialdi¤erential equation de�ned by

@

@tgij = �2Rij = �2Rc (gij)

where t is an extra parameter (not related to the original coordinates, so gij =gij (t; x) ; where x are the coordinates) and Rij = Rc (gij) is the Ricci curvature,a di¤erential operator (2nd order) on the Riemannian metric. That means thatthe Ricci �ow equation is a partial di¤erential equation on the Riemannianmetric. In coordinates, it roughly looks like

@

@tgij = �2gk`

@2

@xk@x`gij + F (gij ; @gij)

where gk` is the inverse matrix of gij and F is a function depending only on themetric and �rst derivatives of the metric.

2.3.1 Hamilton�s �rst result

The idea is that as the metric evolves, its curvature becomes more and moreuniform. It was shown in Hamilton�s landmark 1982 paper that

Theorem 5 (Hamilton) Given a Riemannian 3-manifold (M; g0) with pos-itive Ricci curvature, then the Ricci �ow with g (0) = g0 exists on a maximaltime interval [0; t�): Furthermore, the Ricci curvature of the metrics g (t) becomeincreasingly uniform as t! t�. More precisely,

Rij (t)� �R (t) gij (t) ;

where �R is the average scalar curvature, converges uniformly to zero as t! t0:

From this, one can easily show that a rescaling of the metric converges tothe round sphere.

2.3. IDEA OF THE HAMILTON-PERELMAN PROOF 7

2.3.2 2D case

In the 2D case it can be shown that any compact, orientable Riemannian man-ifold converges under a renormalized Ricci �ow (renormalized by rescaling themetric and rescaling time) to a constant curvature metric. This is primarilydue to Hamilton, with one case �nished by B. Chow. It is possible to use thismethod to prove the uniformization theorem, which states that any compact,orientable Riemannian manifold can be conformally deformed to a metric withconstant curvature. (The original proofs of Hamilton and Chow use the uni-formization theorem, but a recent article by Chen, Lu, and Tian shows how toavoid that).

2.3.3 General case

Hamilton introduced a program to study all 3-manifolds using the Ricci �ow.It was discovered quite early that the Ricci �ow may develop singularities evenin the case of a sphere if the Ricci curvature is not positive. An example is theso-called neck pinch singularity. Hamilton�s idea was to do surgery at these sin-gularities, then continue the �ow and continue to do this until no more surgeriesare necessary. Perelman�s work describes what happens to the Ricci �ow neara singularity and also how to perform the surgery. The new �ow is called RicciFlow with surgery. The main result of Perelman is the following.

Theorem 6 (Existence of Ricci �ow with surgery) Let (M; g) be a com-pact, orientable Riemannian 3-manifold. Then there exists a Ricci �ow withsurgery t ! (M (t) ; g (t)) for all t 2 [0;1) and a closed set T � [0;1) ofsurgery times such that:

1. (Initial data) M (0) =M; g (0) = g:

2. (Ricci �ow) If I is any connected component of [0;1) n T (and thus aninterval), then t! (M (t) ; g (t)) is the Ricci �ow on I (you can close thisinterval on the left endpoint if you wish).

3. (Topological compatibility) If t 2 T and " > 0 is su¢ ciently small, thenwe know the topological relationship M (t� ") and M (t) :

4. (Geometric compatibility) For each t 2 T; the metric g (t) on M (t) isrelated to a certain limit of the metric g (t� ") on M (t� ") by a certainsurgery procedure.

Note, we can express the topological compatibility more precisely. We havethatM (t� ") is homeomorphic to the connected sum of �nitely many connectedcomponents of M (t) together with a �nite number of spherical space forms(spherical manifolds), RP3#RP3, and S2 � S1: Furthermore, each connectedcomponent of M (t) is used in the connected sum decomposition of exactly onecomponent of M (t� ") :


Remark 7 The case of RP3#RP3 is interesting in that it is apparently the onlynonprime 3-manifold which admits a geometric structure (i.e., is covered by amodel geometry; it is a quotient of S2�R; this does not contradict our argumentabove because it is not a sphere bundle over S1). I have seen this mentionedseveral places, but I do not have a reference.

Remark 8 Morgan-Tian and Tao give a more general situation where nonori-entable manifolds are allowed. This adds some extra technicalities which we willavoid in this class.

The existence needs something more to show the Poincaré conjecture. Oneneeds that the surgeries are only discrete and that the �ow shrinks everythingin �nite time.

Theorem 9 (Discrete surgery times) Let t! (M (t) ; g (t)) be a Ricci �owwith surgery starting with an orientable manifold M (0) : Then the set T ofsurgery times is discrete. In particular, any compact time interval contains a�nite number of surgeries.

Theorem 10 (Finite time extinction) Let (M; g) be a compact 3-manifoldwhich is simply connected and let t! (M (t) ; g (t)) be an associated Ricci �owwith surgery. Then M (t) is empty for su¢ ciently large t:

With these theorems, one can conclude the Poincaré conjecture in the fol-lowing way. Given M a simply connected, connected, compact Riemannianmanifold, associate a Ricci �ow with surgery. It has �nite extinction time, andhence �nite surgery times. Now one can use the topological decomposition towork backwards and build the manifold backwards, which says that the man-ifold M is the connected sum of �nitely many spherical space forms, copiesof RP3#RP3, and S2 � S1: But since M is the simply connected, everythingin the connected sum must be simply connected, and hence every piece of theconnected sum must be simply connected, so M must be a sphere.

Chapter 3

Background in di¤erentialgeometry

3.1 Introduction

We will try to get as quickly as possible to a point where we can do somegeometric analysis on Riemannian spaces. One should look at Tao�s lecture 0,though I will not follow it too closely.

3.2 Basics of tangent bundles and tensor bun-dles

Recall that for a smooth manifold M; the tangent bundle can be de�ned inessentially 3 di¤erent ways ((Ui; �i) are coordinates)

� TM =Fi

(Ui � Rn) = � where for (x; v) 2 Ui � Rn; (y; w) 2 Uj � Rn we

have (x; v) � (y; w) if and only i¤ y = �j��1i (x) and w = d

��j�

�1i

�x(v) :

� TpM = fpaths : (�"; ")!M such that (0) = pg = � where � � � if(�i � �)0 (0) = (�i � �)0 (0) for every i such that p 2 Ui: TM =

Fp2M

TpM:

� TpM to be the set of derivations of germs at p; i.e., the set of linear func-tionals X on the germs at p such that X (fg) = X (f) g (p) + f (p)X (g)for germs f; g at p: TM =

Fp2M

TpM:

On can de�ne the cotangent bundle by essentially taking the dual of TpM ateach point, which we call T �pM; and taking the disjoint union of these to get thecotangent bundle T �M: One could also use an analogue of the �rst de�nition,where the only di¤erence is that instead of using the vector space Rn; one uses

9

10 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY

its dual and the equivalence takes into account that the dual space pulls backrather than pushes forward. Both of these bundles are vector bundles. One canalso take a tensor bundle of two vector bundles by replacing the �ber over apoint by the tensor product of the �bers over the same point, e.g.,

TM T �M =Gp2M

�TpM T �pM

�:

Note that there are canonical isomorphisms of tensor products of vector spaces,such as V V � is isomorphic to endomorphisms of V: Note the di¤erence betweenbilinear forms (V � V �), endomorphisms (V V �), and bivectors (V V ).It is important to understand that these bundles are global objects, but will

often be considered in coordinates. Given a coordinate x =�xi�and a point p in

the coordinate patch, there is a basis @@x1

��p; : : : ; @

@xn

��pfor TpM and dual basis

dx1��p; : : : ; dxnjp for T �pM: The generalization of the �rst de�nition above gives

the idea of how one considers the trivializations of the bundle in a coordinatepatch, and how the patches are linked together. Speci�cally, if x and y givedi¤erent coordinates, for a point on the tensor bundle, one has

T ij��kab��c (x)@

@xi @

@xj � � � @

@xk dxa dxb � � � dxc

= T ij��kab��c (x (y))

�@y�

@xi@y�

@xj� � � @y

�

@xk@xa

@y�@xb

@y�� @x

c

@y

�@

@y� @

@y� � � � @

@y� dy� dy� � � � dy ;

where technically everything should be at p (but as we shall see, one can considerthis for all points in the neighborhood and this is considered as an equation ofsections). Recall that a section of a bundle � : E ! B is a function f : B ! Esuch that � � f is the identity on the base manifold B: A local section may onlybe de�ned on an open set in B: On the tangent space, sections are called vector�elds and on the cotangent space, sections are called forms (or 1-forms). Ona tensor bundle, sections are called tensors. Note that the set of @

@xi form abasis for the vector �elds in the coordinate x, and dxi form a basis for the local1-forms in the coordinates. Sections in general are often written as � (E) or asC1 (E) (if we are considering smooth sections).Now the equation above makes sense as an equation of tensors (sections of

a tensor bundle). Often, a tensor will be denoted as simply

T ij��kab��c :

Note that if we change coordinates, we have a di¤erent representation T �� ofthe same tensor. The two are related by

T �� = T ij��kab��c (x (y))

�@y�

@xi@y�

@xj� � � @y

�

@xk@xa

@y�@xb

@y�� @x

c

@y

�:

One can also take subsets or quotients of a tensor bundle. In particular,we may consider the set of symmetric 2-tensors or anti-symmetric tensors (sec-tions of this bundle are called di¤erential forms). In particular, we have theRiemannian metric tensor.

3.2. BASICS OF TANGENT BUNDLES AND TENSOR BUNDLES 11

De�nition 11 A Riemannian metric g is a two-tensor (i.e., a section of T �MT �M) which is

� symmetric, i.e., g (X;Y ) = g (Y;X) for all X;Y 2 TpM; and

� positive de�nite, i.e., g (X;X) � 0 all X 2 TpM and g (X;X) = 0 if andonly if X = 0.

Often, we will denote the metric as gij ; which is shorthand for gijdxidxj ;where dxidxj = 1

2

�dxi dxj + dxj dxi

�: Note that if gij = �ij (the Kronecker

delta) then

�ijdxidxj =

�dx1�2+ � � � (dxn)2 :

One can invariantly de�ne a trace of an endomorphism (trace of a matrix)which is independent of the coordinate change, since

nXa=1

T aa =Xa

T��@xa

@y�@y�

@xa

=Xa

T��

=X�

T�� :

In fact for any complicated tensor, one can take the trace in one up index andone down index. This is called contraction. Usually, when there is a repeatedindex of one up and one down, we do not write the sum. This is called Einsteinsummation convention. The above sum would be written

T aa = T�� :

It is understood that this is an equation of functions.We cannot contract two indices up or two indices down, since this is not

independent of coordinate change (try it!) However, now that we have theRiemannian metric, we can use it to �lower an index�and then trace, so we get

T abgba = T aa :

In order to raise the index, we need the dual to the Riemannian metric, whichis gab; de�ned such that gabgbc = �ac (so g

ab is the inverse matrix of gab). Thenwe can use gab to raise indices and contract if necessary. Occasionally, extendedEinstein convention is used, where all repeated indices are summed with theunderstanding that the Riemannian metric is used to raise or lower indiceswhen necessary, e.g.,

Taa = Tabgab:

Since often we will be changing the Riemannian metric, it becomes importantto understand that the metric is there when extended Einstein is used.


3.3 Connections and covariant derivatives

3.3.1 What is a connection?

A covariant derivative is a particular way of di¤erentiating vector �elds. Whydo we need a new way to di¤erentiate vector �elds? Here is the idea. Supposewe want to give a notion of parallel vectors. In Rn; we know that if we takevector �elds with constant coe¢ cients, those vectors are parallel at di¤erentpoints. That is, the vectors @

@x1

��(0;0)

+2 @@x2

��(0;0)

and @@x1

��(1;�1)+2

@@x2

��(1;�1)are

parallel. In fact, we could say that the vector �eld @@x1 + 2

@@x2 is parallel since

vectors at any two points are parallel. One might say it is because the coe¢ cientsof the vector �eld are constant (not functions of x1 and x2). However, thisnotion is not invariant under a change of coordinates. Suppose we consider the

new coordinates�y1; y2

�=�x1;�x2�2�

away from x2 = 0 (where it is not a

di¤eomorphism). Then the vector �eld in the new coordinates is

@yi

@x1@

@yi+ 2

@yj

@x2@

@xj=

@

@y1+ 4x2

@

@y2=

@

@y1+ 4py2

@

@y2:

The coe¢ cients are not constant, but the vector �eld should still be parallel(we have only changed coordinates, so it is the same vector �eld)! So we needa notion of parallel vector �eld that is independent of coordinate changes (orcovariant).Remember that we want to generalize the notion that a vector �eld has

constant coe¢ cients. Let X = Xi @@xi be a vector �eld in a coordinate patch.

Roughly speaking, we want to generalize the notion that @Xi

@xj = 0 for all i andj: The problem occurred because @

@x1

�@@x1

�is di¤erent in di¤erent coordinates.

Thus we need to specify what this is. Certainly, since @@xi is a basis, we must

get a linear combination of these, so we take

ri@

@xj= �kij

@

@xk

for some functions �kij : These symbols are called Christo¤el symbols. To makesense on a vector �eld, we must have

ri (X) = ri�Xj @

@xj

�=@Xj

@xi@

@xj+Xj�kij

@

@xk

=

�@Xk

@xi+Xj�kij

�@

@xk:

Notice the Leibniz rule (product rule). One can now de�ne r for any vectorY = Y i @

@xi byrYX = rY i @

@xiX = Y i (riX) :

3.3. CONNECTIONS AND COVARIANT DERIVATIVES 13

This action is called the covariant derivative.One now de�nes �kij in such a way that the covariant derivative transforms

appropriately under change of coordinates. This gives a global object called aconnection. The connection can be de�ned axiomatically as follows.

De�nition 12 A connection on a vector bundle E ! B is a map

r : � (TB) � (E)! � (E)

(X;�)! rX�

satisfying:

� Tensoriality (i.e., C1 (B)-linear) in the �rst component, i.e., rfX+Y � =frX�+rY � for any function f and vector �elds X;Y

� Derivation in the second component, i.e., rX (f�) = X (f)�+ frX�:

� R-linear in the second component, i.e., rX (a�+ ) = arX (�)+rX ( )for a 2 R.

We will consider connections primarily on the tangent bundle and tensorbundles. Note that a connection r on TM induces connections on all tensorbundles (also denoted r) in the following way:

� For a function f and vector �eld X; we de�ne rXf = Xf

� For vector �elds X;Y and dual form !; we use the product rule to derive

rX (! (Y )) = X (! (Y )) = (rX!) (Y ) + ! (rXY )

and thus(rX!) (Y ) = X (! (Y ))� ! (rXY ) :

In particular, the Christo¤el symbols for the connection on T �M are thenegative of the Christo¤el symbols of TM; i.e.,

r @

@xidxj = ��jikdx

k

where �kij are the Christo¤el symbols for the connection r on TM:

� For a tensor product, one de�nes the connection using the product rule,e.g.,

rX (Y !) = (rXY ) ! + Y rX!

for vector �elds X;Y and 1-form !:


Remark 13 The Christo¤el symbols are not tensors. Note that if we changecoordinates from x to ~x; we have

r @

@xi

@

@xj= r� @~xk

@xi@

@~xk

��@~x`@xj

@

@~x`

�=

@~x`

@xi@xj@

@~x`+@~xk

@xi@~x`

@xjr @

@~xk

@

@~x`

which means that

�kij = ~�mp`

@~xp

@xi@~x`

@xj@xk

@~xm+

@~x`

@xi@xj@xk

@~x`:

One �nal comment. Recall that we motivated the connection by consideringparallel vector �elds. The connection gives us a way of taking a vector at apoint and translating it along a curve so that the induced vector �eld along thecurve is parallel (i.e., r _ X = 0 along ). This is called parallel translation.Parallel vector �elds allow one to rewrite derivatives in coordinates; that is,

if X = Xi @@xi is parallel, then

@Xi

@xj= �Xk�ijk:

3.3.2 Torsion, compatibility with the metric, and Levi-Civita connection

There is a unique metric associated with the Riemannian metric, called theRiemannian connection or Levi-Civita connection. It satis�es two properties:

� Torsion-free (also called symmetric)

� Compatible with the metric.

Compatibility with the metric is the easy one to understand. We want theconnection to behave well with respect to di¤erentiating orthogonal vector �elds.Being compatible with the metric is the same as

rX (g (Y;Z)) = g (rXY;Z) + g (Y;rXZ) :

Note that normally there would be an extra term, (rXg) (Y; Z) ; so compati-bility with the metric means that this term is zero, i.e., rg = 0; where g isconsidered as a 2-tensor.Torsion free means that the torsion tensor �; given by

� (X;Y ) = rXY �rYX � [X;Y ]

vanishes. (One can check that this is a tensor by verifying that � (fX; Y ) =� (X; fY ) = f� (X;Y ) for any function f). It is easy to see that in coordinates,the torsion tensor is given by

�kij = �kij � �kji;

3.3. CONNECTIONS AND COVARIANT DERIVATIVES 15

which indicates why torsion-free is also called symmetric.Tao gives a short motivation for the concept of torsion-free. Consider an

in�nitesimal parallelogram in the plane consisting of a point x; the �ow of xalong a vector �eld V to a point we will call x+ tV; the �ow of X along a vector�eld W to a point we will call x + tW; and then a fourth point which we willreach in two ways: (1) go to x+ tV and then �ow along the parallel translationof W for a distance t and (2) go to x + tW and then �ow along the paralleltranslation of V for a distance t: Note that using method (1), we get that thepoint is

(x+ tV + sW )js=0 + t@

@s

��s=0

(x+ tV + sW ) +O�t3�

= x+ tV + tW + t2@

@s

��s=0

V +O�t3�= x+ tV + tW � t2V iW j�kji

@

@xk+O

�t3�:

Note that using method (2), we get instead

x+ tV + tW � t2W iV j�kji@

@xk+O

�t3�;

Thus this vector is x+t (V +W ) up toO�t3�only if �kji = �

kij :Doing this around

every in�nitesimal parallelogram gives the equivalence of these two viewpoints.Here is another:

Proposition 14 A connection is torsion-free if and only if for any point p 2M;there are coordinates x around p such that �kij (p) = 0:

Proof. Suppose one can always �nd coordinates such that �kij (p) = 0: Thenclearly at that point, �kij = 0: However, since the torsion is a tensor, we cancalculate it in any coordinate, so at each point, we have that the torsion vanishes.Now suppose the torsion tensor vanishes and let x be a coordinate around p:Consider the new coordinates

~xi (q) = xi (q)� xi (p) + �ijk (p)�xj (q)� xj (p)

� �xk (q)� xk (p)

�:

Then notice that

@~xi

@xj= �ij + �

ik` (p) �

kj

�x` � x` (p)

�+ �ik` (p)

�xk � xk (p)

��`j

and so@~xi

@xj(p) = �ij :

Thus ~x is a coordinate patch in some neighborhood of p: Moreover, we havethat

@2~xi

@xj@xk= �ijk (p) :


One can now verify that at p;

�kij (p) = ~�mp`

@~xp

@xi@~x`

@xj@xk

@~xm+

@~x`

@xi@xj@xk

@~x`

= ~�kij (p) + �kij (p) :

The Riemannian connection is the unique connection which is both torsion-free and compatible with the metric. One can use these two properties to derivea formula for it. In coordinates, one �nds that the Riemannian connection hasthe following Christo¤el symbols

�kij =1

2gk`�

@

@xigj` +

@

@xjgi` �

@

@x`gij

�:

One can easily verify that this connection has the properties expressed. Notethat the gj` in the formula, etc. are not the tensors, but the functions. Thisis not a tensor equation since �kij is not a tensor. Also note that it is veryimportant that this is an expression in coordinates (i.e., that

�@@xi ;

@@xj

�= 0).

3.3.3 Higher derivatives of functions and tensors

One of the important reasons for having a connection is it allows us to takehigher derivatives. Note that one can take the derivative of a function withouta connection, and it is de�ned as

df = rfdf (X) = rXf = X (f)

df =@f

@xidxi:

One can also raise the index to get the gradient, which is

grad (f) = rif @

@xi=

@f

@xjgij

@

@xi:

However, to take the next derivative, one needs a connection. The secondderivative, or Hessian, of a function is

Hess (f) = r2f = rdfr2f = (ridf) dxi

=

�ri�@f

@xjdxj��

dxi

=

�@2f

@xi@xjdxj � @f

@xj�jikdx

k

� dxi

=

�@2f

@xi@xj� @f

@xk�kij

�dxj dxi:

3.4. CURVATURE 17

Often one will write the Hessian as

r2ijf = rirjf =@2f

@xi@xj� �kij

@f

@xk:

Note that if the connection is symmetric, then the Hessian of a function issymmetric in the usual sense. The trace of the Hessian, 4f = gijr2ijf; is calledthe Laplacian, and we will use it quite a bit.We also may use the connection to compute acceleration of a curve. The

velocity of a curve is _ ; which does not need a connection, but to computethe acceleration, r _ _ ; we need the connection (one also sometimes sees theequivalent notation D _ =dt). A curve with zero acceleration is called a geodesic.Finally, given any tensor T; one can use the connection to form a new tensor

rT; which has an extra down index.

3.4 Curvature

One can de�ne the curvature of any connection on a bundle E ! B in thefollowing way

R : � (TM) � (TM) � (E)! � (E)

R (X;Y )� = rXrY ��rYrX��r[X;Y ]�:

We will consider the curvature of the Riemannian connection on the tangentbundle. One can easily see that in coordinates, the curvature is a tensor denotedas

rirj@

@xk�rjri

@

@xk= Rìjk

@

@x`

which gives us that

ri��`jk

@

@x`

��rj

��ìk

@

@x`

�=

�@

@xi�`jk

�@

@x`+ �`jk�

mi`

@

@xm��

@

@xj�ìk

�@

@x`� �ìk�mj`

@

@xm

=

�@

@xi�`jk �

@

@xj�ìk + �

mjk�

ìm � �mik�`jm

�@

@x`

So the curvature tensor is

Rìjk =@

@xi�`jk �

@

@xj�ìk + �

mjk�

ìm � �mik�`jm:

Often we will lower the index, and consider instead the curvature tensor

Rijk` = Rmijkgm`:

The Riemannian curvature tensor has the following symmetries:

� Rijk` = �Rjik` = �Rij`k = Rkìj (These imply that R can be viewed as aself-adjoint (symmetric) operator mapping 2-forms to 2-forms if one raisesthe �rst two or last two indices).


� (Algebraic Bianchi) Rijk` +Rjki` +Rkij` = 0:

� (Di¤erential Bianchi) riRjk`m +rjRki`m +rkRij`m:

Remark 15 The tensor Rijk` can also be written as a tensor R (X;Y; Z;W ) ;which is a function when vector �elds X;Y; Z;W are plugged in. We willsometimes refer to this tensor as Rm : The tensor Rìjk is usually denoted byR (X;Y )Z; which is a vector �eld when vector �elds X;Y; Z are plugged in.

Remark 16 Sometimes, the up index is lowered into the 3rd spot instead ofthe 4th, This will change the de�nitions of Ricci and sectional curvature below,but the sectional curvature of the sphere should always be positive and the Riccicurvature of the sphere should be positive de�nite.

Remark 17 Note that �kij involved �rst derivatives of the metric, so Riemanniancurvature tensor involves �rst and second derivatives of the metric.

From these one can derive all the curvatures we will need:

De�nition 18 The Ricci curvature tensor Rij is de�ned as

Rij = R`ìj = Rìjmg`m:

Note that Rij = Rji by the symmetries of the curvature tensor. Ricci willsometimes be denoted Rc (g) ; or Rc (X;Y ) :

De�nition 19 The scalar curvature R is the function

R = gijRij

De�nition 20 The sectional curvature of a plane spanned by vectors X and Yis given by

K (X;Y ) =R (X;Y; Y;X)

g (X;X) g (Y; Y )� g (X;Y )2:

Here are some facts about the curvatures:

Proposition 21 1. The sectional curvatures determine the entire curvaturetensor, i.e., if one can calculate all sectional curvatures, then one cancalculate the entire tensor.

2. The sectional curvature K (X;Y ) is the Gaussian curvature of the surfacegenerated by geodesics in the plane spanned by X;Y:

3. The Ricci curvature can be written as an average of sectional curvature.

4. The scalar curvature can be written as an average of Ricci curvatures.

5. The scalar curvature essentially gives the di¤erence between the volumesof small metric balls and the volumes of Euclidean balls of the same radius.

3.4. CURVATURE 19

6. In 2 dimensions, each curvature determines the others.

7. In 3 dimensions, scalar curvature does not determine Ricci, but Ricci doesdetermine the curvature tensor.

8. In dimensions larger than 3, Ricci does not determine the curvature tensor;there is an additional piece called the Weyl tensor.

With this in mind, we can talk about several di¤erent kinds of nonnegativecurvature.

De�nition 22 Let x be a point on a Riemannian manifold (M; g) : Then x has

1. nonnegative scalar curvature if R (x) � 0;

2. nonnegative Ricci curvature at x if Rc (X;X) = RijXiXj � 0 for every

vector X 2 TxM ;

3. nonnegative sectional curvature if R (X;Y; Y;X) = g (R (X;Y )Y;X) � 0for all vectors X;Y 2 TxM ;

4. nonnegative Riemann curvature (or nonnegative curvature operator) ifRm � 0 as a quadratic form on 2 (M) ; i.e., if Rijk`!ij!k` � 0 forall 2-forms ! = !ijdx

i ^ dxj (where the raised indices are done using themetric g).

It is not too hard to see that 4 implies 3 implies 2 implies 1: Also, in 3dimensions, 3 and 4 are equivalent. In dimension 4 and higher, these are alldistinct.

Chapter 4

Basics of geometricevolutions

4.1 Introduction

This lecture roughly follows Tao�s Lecture 1. We will talk in general about �owsor Riemannian metrics and Ricci �ow.We will consider a �ow of Riemannian metrics to be a one-parameter family

of Riemannian metrics, usually denoted g (t) or gij (t) or gij (x; t) on a �xedRiemannian manifold M . There are more ingenious ways to de�ne such a �owusing spacetimes (called generalized Ricci �ows). However, at present I do notthink that they give a signi�cant savings over the more classical idea, since onestill needs to consider singular spacetimes. For more on generalized Ricci �ows,consult the book by Morgan-Tian.The family g (t) is a one-parameter family of sections of a vector bundle, and

one can take its derivative as

@

@tg (t) = lim

dt!0

g (t+ dt)� g (t)dt

since g (t) and g (t+ dt) are both sections of the same vector bundle, so thedi¤erence makes sense. In fact, we can di¤erentiate any tensor in this way.Similarly, we can try to solve di¤erential equations of the form

@

@tgij = _gij

for some prescribed _gij : The evolution of the metric induces an evolution of themetric on the cotangent bundle, using

@

@t

�gijgjk

�=

@

@t�ij

@

@tgij = �gik _gk`g`j :

21

22 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS

The Riemannian connection is also changing if the metric is changing. Thusfor a �xed vector �eld X; we have

@

@triX =

@

@t

�@Xj

@xi@

@xj+ �kijX

j @

@xk

�= Xj _�kij

@

@xk:

We can use the fact that the connection is torsion-free and compatible with themetric to derive the formula for _�kij :

0 =@

@t(rigjk) =

@

@t

�@

@xigjk � �ìjg`k � �ìkgj`

�= ri _gjk � _�ìjg`k � _�ìkgj`;

and

0 = _�kij � _�kji

so we can solve for _�kij as

ri _gjk = _�ìjg`k +_�ìkgj`

rj _gki = _�`jkg`k +_�`jigk`

rk _gij = _�`kig`j +_�`kjgi`

to get

_�kij =1

2gk` (ri _gj` +rj _gi` �r` _gij) : (4.1)

Remark 23 This mimics the proof of the formula for the Riemannian con-nection given that it is torsion-free and compatible with the metric. There areother ways to derive this formula, for instance by computing in normal coor-dinates and using the fact that although �kij is not a tensor,

@@t�

kij comes from

the di¤erence of two connections and is thus a tensor. We will use this methodbelow.

We may now look at the induced formula for evolution of the Riemanniancurvature tensor. Recall that, in coordinates,

Rìjk@

@x`= ri

��`jk

@

@x`

��rj

��ìk

@

@x`

�:

Since we are interested in the derivative of a tensor, @@tR

ìjk =

_Rìjk; we cancompute this in any coordinate system we want. Recall that there is a coordinatesystem around p such that all Christo¤el symbols vanish at p: Doing this reduces

4.1. INTRODUCTION 23

the equation to

_Rìjk@

@x`=

@

@t

�ri��`jk

@

@x`

��rj

��ìk

@

@x`

��=

@

@t

�@

@xi�`jk

@

@x`� @

@xj�ìk

@

@x`

�=

@

@xi_�`jk

@

@x`� @

@xj_�ìk

@

@x`

= ri _�`jk@

@x`�rj _�ìk

@

@x`:

This last piece is tensorial (recall that _�kij is a tensor), and thus only dependson the point, not the coordinate patch, so we must have that

_Rìjk = ri _�`jk �rj _�ìk:

We can now use the (4.1) to get

_Rìjk = ri�1

2g`m (rj _gkm +rk _gjm �rm _gjk)

��rj

�1

2g`m (ri _gkm +rk _gim �rm _gik)

�=1

2g`m(rirj _gkm +rirk _gjm �rirm _gjk �rjri _gkm �rjrk _gim +rjrm _gik)

=1

2g`m(rirj _gkm �rjri _gkm +rirk _gjm �rirm _gjk �rjrk _gim +rjrm _gik)

=1

2g`m(�Rpijk _gmp �R

pijm _gkp +rirk _gjm �rirm _gjk �rjrk _gim +rjrm _gik):

We can take the trace _Rjk = _Riijk to get

_Rjk =1

2gim(�Rpijk _gmp �R

pijm _gkp +rirk _gjm �rirm _gjk �rjrk _gim +rjrm _gik)

= �12gimRpijk _gmp +

1

2gpqRjp _gkp �

1

2gimrirm _gjk �

1

2gimrjrk _gim

+1

2gim(rkri _gjm �Rpikj _gpm �R

pikm _gjp +rjrm _gik)

= �gimRpijk _gmp +1

2gpqRjp _gkp +

1

2gpqRkq _gjp �

1

2gimrirm _gjk

� 12gimrjrk _gim +

1

2gim(rkri _gjm +rjrm _gik)

= �124L _gjk �

1

2gimrjrk _gim +

1


where

4L _gjk = gimrirm _gjk + 2gimRpijk _gmp � gpqRjp _gkp � gpqRkq _gjp

is the Lichnerowitz Laplacian (notice only the �rst term has two derivatives of_gjk). (Note: I think that T. Tao has an error in this formula with the sign ofthe last term of _Rjk:


Finally, we may take a trace to get

_R = gjk _Rjk � gpj _gpqgqkRjk= �gpqgjkRjp _gkp � gimrirmgjk _gjk+ gimgjkrkri _gjm= � hRc; _gi � 4 trg ( _g) + div div _g

where

(div h)j = gk`rkh`j

4.2 Ricci �ow

Note that in the evolution of Ricci curvature, if one considers

_g = �2Rc;

one gets

� 12gimrjrk _gim +

1


= gimrjrkRim � gimrkriRjm � gimrjrmRik= rjrkR� gimrjrmRik � gimrkriRjm

Note that the di¤erential Bianchi identity implies

0 = gjmgk` (riRjk`m +rjRki`m +rkRij`m)= riR� gjmrjRim � gk`rkRij`m

soriR = 2gjmrjRim

so

�12gimrjrk _gim+

1

2gim(rkri _gjm+rjrm _gik) = rjrkR�

1

2rjrkR�

1

2rkrjR = 0:

Thus under Ricci �ow,@

@tRc = 4LRc :

Furthermore, We see that

@R

@t= 2 hRc;Rci+ 24R� 2gi`gjkrirjRk`

= �2 hRc;Rci+ 24R� gi`rir`R= 4R+ 2 jRcj2

The important notion to get right now is that this looks very much like aheat equation with a reaction term. We will see how to make use of this in thenear future.

4.3. EXISTENCE/UNIQUENESS 25

4.3 Existence/Uniqueness

Note that the Ricci �ow equation,

@

@tg = �2Rc

is a second order partial di¤erential equation, since the Ricci curvature comesfrom second derivatives of the metric. To truly look at existence/uniqueness,one must write this as an equation in coordinates. We will look at the lineariza-tion of this operator in order to �nd the principle symbol (which is basicallythe coe¢ cients of the linearization of the highest derivatives). Analysis of theprinciple symbol will often allow us to determine that a solution exists for ashort time. Here is the meta-theorem for existence of parabolic PDE:Meta-Theorem (imprecise): A semi-linear PDE of the form

@u

@t� aij (x; t) @2u

@xi@xj+ F (x; t; u; @u) = 0:

on a compact manifold has a solution with initial condition u (x; 0) = f (x)

if there exists � > 0 such that aij�i�j � � j�j2 (this condition is called strictparabolicity) for t close to 0. Similarly, if we allow aij to depend on u (makingthe equation quasilinear, the same is true if we look at the linearization (whichis then semilinear).

Remark 24 aij is called the principal symbol of the parabolic di¤erential oper-ator. If one takes out the @

@t ; the di¤erential operator is said to be elliptic if itsatis�es the inequality.

Remark 25 We can replace @@xi with ri since the di¤erence has fewer deriva-

tives.

Remark 26 One should be able to prove a coordinate independent version, butthis is not usually done. All theory is based on theory of di¤erential equationson domains in the plane.

Remark 27 For an arbitrary, nonlinear second order PDE of the form

G�x; t; u; @u; @2u

�= 0;

one can consider the linearization of G with respect to u: This will look roughlylike �

@HijG

@2

@xi@xj+ @ViG

@

@xi+ @uG

�v

where G = G (x; t; u; V;H) and the operator is evaluated at some u (which iswhere it has been linearized. Notice this now gives a semilinear PDE. Theprinciple symbol is @Hij

G�i�j :


Example 28 Note that the equation

@u

@t=

@2

(@x1)2u+

@2

(@x2)2u = �ij

@2

@xi@xj

is already linear. For an equation like

@u

@t= u2

@2u

@x2;

the linearization is@v

@t= u2

@2v

@x2+ 2u

@2u

@x2v;

thus the principal symbol is u2 which is positive if u > 0:

Now, the Ricci operator is an operator on sections, not just functions, sohow do we make sense of the kind of result given above. We can make a similarde�nition in terms of the linearization, but now the principle symbol is a mapfrom sections of the symmetric 2-tensor bundle to itself. What we need is thatfor any � 6= 0; the principle symbol is a linear isomorphism.Recall that the linearization of Rjk is

_Rjk = �1

2gimrirm _gjk �

1

2gimrjrk _gim +

1


so the principle symbol of �2Rjk is

�̂ [DRc] (�) (h) = gim�i�mhjk + gim�j�khim � gim(�k�ihjm + �j�khik):

In order to see if this is an isomorphism, we can rotate � so that �1 > 0 and�2 = � � � = �n = 0 and by scaling we can assume �1 = 1: We can also assumethat at a point gij = �ij : Then we see that

�̂ [DRc] (�) (h)jk = hjk + �1j �1k (h11 + � � �+ hnn)� (�1khj1 + �1jh1k):

And so the matrix for the symbol gives0BBB@h22 + � � �+ hnn 0 � � � 0

0... h��0

1CCCAwhere 2 � �; � � n: We see immediately that there is an n-dimensional kernel(we can let h1k equal anything we want and if everything else is zero, we are inthe kernel).Now we will see how to overcome this issue. Rewrite the linearization as

_Rjk = �1

2gimrirm _gjk �

1

2gimrjrk _gim +

1


= �12gimrirm _gjk +

1

2rkVj +rjVk

4.3. EXISTENCE/UNIQUENESS 27

ifVj = gimri _gjm �

1

2rj�gim _gim

�:

The last term is equal to the Lie derivative LV gjk (where V = V i = gijVj ; oftendenoted V #) and so we get that the linearization of �2Rjk is

gimrirm _gjk � LV gjk:

Lie derivatives arise from changing by di¤eomorphisms, i.e., if �t are di¤eomor-phisms such that

d

dt�t (x) = X (x)

and �0 is the identity (i.e., �t is the �ow of X), then

d

dt

��t=0

��t g0 = LXg0:

One can pretty easily see that if we take the vector �eld V as above, we canlook at the �ow �t and ��t g (t) and we will see that ~g (t) = ��t g (t) evolves by

@

@t~g = �2Rc (~g) + LV ~g

and the linearization isgimrirm _gjk:

This is like looking at an equation roughly like

@

@th = gimrirmhjk;

which is a heat equation with a unique solution. This has principal symbol gij ;which is strictly positive de�nite. It can be shown that this implies that themodi�ed Ricci �ow (the equation above on ~g) has a unique solution. One canthen show that this implies the Ricci �ow has a unique solution too. I.e.,

Theorem 29 Given an initial closed Riemannian manifold (M; g0) ; there is atime T > 0 and Riemannian metrics g (t) on M for each t 2 [0; T ) such thatwhich satisfy the initial value problem

@

@tg = �2Rc (g)

g (0) = g0:

Moreover, given the initial condition, g (t) are uniquely determined, and thereis a maximal such T:

Remark 30 One can also show that this is true for complete manifolds withbounded curvature jRmj ; which was done by Shi. However, the proof is muchmore di¢ cult on noncompact manifolds.

Chapter 5

Basics of PDE techniques

5.1 Introduction

This section will roughly follow Tao�s lecture 3. We will look at some basic PDEtechniques and apply them to the Ricci �ow to obtain some important resultsabout preservation and pinching of curvature quantities. The important factis that the curvatures satisfy certain reaction-di¤usion equations which can bestudied with the maximum principle.

5.2 The maximum principle

Recall that if a smooth function u : U ! R where U � Rn has a local minimumat x0 in the interior of U; then

@u

@xi(x0) = 0

@2u

@xi@xj(x0) � 0

where the second statement is that the Hessian is nonnegative de�nite (has allnonnegative eigenvalues). The same is true on a Riemannian manifold, replacingregular derivatives with covariant derivatives.

Lemma 31 Let (M; g) be a Riemannian manifold and u :M ! R be a smooth(or at least C2) function that has a local minimum at x0 2M: Then

riu (x0) = 0rirju (x0) � 0

4u (x0) = gij (x0)rirju (x0) � 0:

Proof. In a coordinate patch, the �rst statement is clear since riu = @u@xi :

The second statement is that the Hessian is positive de�nite. Recall that in

29

30 CHAPTER 5. BASICS OF PDE TECHNIQUES

coordinates, the Hessian is

rirju =@2u

@xi@xj� �kij

@u

@xk;

but at a minimum, the second term is zero and the positive de�niteness followsfrom the case in Rn: The last statement is true since both g and the Hessianare positive de�nite.

Remark 32 There is a similar statement for maxima.

The following lemma is true in the generality of a smooth family of metrics,though is also of use for a �xed metric.

Lemma 33 Let (M; g (t)) be a smooth family of compact Riemannian manifoldsfor t 2 [0; T ]: Let u : [0; T ]�M ! R be a C2 function such that

u (0; x) � 0

for all x 2M: Also let A 2 R. Then exactly one of the following is true:

1. u (t; x) � 0 for all (t; x) 2 [0; T ]�M; or

2. There exists a (t0; x0) 2 (0; T ] such that all of the following are true:

u (t0; x0) < 0

riu (t0; x0) = 0;4g(t0)u (t0; x0) � 0;

@u

@t(x0; t0) < 0:

Proof. Consider the function

u (t; x) + "t:

If u (t; x) + "t > 0 for all " > 0 (small), then u (t; x) � 0: Otherwise there is an" > 0 and an initial t0 such that there is a x0 2M such that u (t0; x0)+"t0 = 0:At the �rst such time, we must have that x0 is a spatial minimum for thisfunction, and thus

u (t0; x0) = �"t0 < 0ru (t0; x0) = 04u (t0; x0) � 0@u

@t(t0; x0) � �" < 0:

5.2. THE MAXIMUM PRINCIPLE 31

Corollary 34 Let (M; g (t)) be a smooth family of compact Riemannian man-ifolds for t 2 [0; T ]: Let u; v : [0; T ]�M ! R be C2 functions such that

u (0; x) � v (0; x)

for all x 2M: Also let A 2 R. Then exactly one of the following is true:

1. u (t; x) � v (t; x) for all (t; x) 2 [0; T ]�M; or

2. There exists a (t0; x0) 2 (0; T ] such that all of the following are true:

u (t0; x0) < v (t0; x0)

riu (t0; x0) = riv (t0; x0) ;4g(t0)u (t0; x0) � 4g(t0)v (t0; x0) ;

@u

@t(x0; t0) <

@v

@t(t0; x0) +A [u (t0; x0)� v (t0; x0)] :

Proof. Replace u with e�At (u� v) :This will allow us to estimate subsolutions of a heat equation by supersolu-

tions of the same heat equation.

Corollary 35 Let the assumptions be the same as in Corollary 34, including

u (0; x) � v (0; x) :

Suppose u is a supersolution of a reaction-di¤usion equation, i.e.,

@u

@t� 4g(t)u+rX(t)u+ F (t; u)

and v is a subsolution of the same equation, i.e.,

@v

@t� 4g(t)u+rX(t)v + F (t; v)

for all (t; x) 2 [0; T ]�M; where X (t) is a vector �eld for each t and F (t; w) isLipschitz in w; i.e., there is A > 0 such that

jF (t; w)� F (t; w0)j � A jw � w0j :

Thenu (t; x) � v (t; x)

for all t 2 [0; T ] :

Proof. Consider

@

@t(u� v) � 4g(t) (u� v) +rX(t) (u� v) + F (t; u)� F (t; v)

� 4g(t) (u� v) +rX(t) (u� v)�A ju� vj :


The dichotomy in Corollary 34 says that either u�v � 0 for all t; x or else thereis a point (t0; x0) such that at that point,

u� v < 04 (u� v) = 0r (u� v) = 0@

@t(u� v) � A0 (u� v) = �A0 ju� vj

for any A0: But the inequality above says that at that same point

@

@t(u� v) � �A ju� vj ;

which is a contradiction if �A0 < �A:Usually, instead of making v a subsolution, we will just make v the subsolu-

tion to the ODEdv

dt� F (t; v) ;

where v = v (t) is independent of x and so this is also a subsolution to the PDE.Here is an easy application:

Proposition 36 Nonnegative scalar curvature is preserved by the Ricci �ow,i.e., if R (0; x) � 0 for all x 2 M and the metric g satis�es the Ricci �ow fort 2 [0; T ), then R (t; x) � 0 for all x 2M and t 2 [0; T ].

Proof. Recall that R satis�es the evolution equation

@R

@t= 4g(t)R+ 2 jRcj2 ;

thus it is a supersolution to the heat equation (with changing metric), i.e.,

@R

@t� 4R:

By Corollary 35, we must have that R � 0 for all t:We can actually do better. Notice that if Tij is a 2-tensor on an n-dimensional

Riemannian manifold (M; g), then

jT j2 � 1

n

�gijTij

�2since ��Tij � 1

n

�gk`Tk`

�gij

��2 � 0(expand that out and see it implies the previous inequality). Thus

jRcj2 � 1

nR2

5.3. MAXIMUM PRINCIPLE ON TENSORS 33

and so scalar curvature satis�es

@R

@t� 4R+ 2

nR2:

The maximum principle implies that R (t; x) � f (t) for all x 2 M; where f (t)is the solution to the ODE

df

dt=2

nf2

f (0) = minx2M

R (x; 0) :

This equation can be solved explicitly asZ1

f2df =

Z2

ndt

� 1f=2

nt� 1

f (0)

f (t) =f (0)

1� 2nf (0) t

as long as f (0) 6= 0: Notice that if f (0) > 0 then this says that R (t; x) goesto in�nity in �nite time T � n

2f(0) : If f (0) < 0; then this says that if the �owexists for all time, then the scalar curvature becomes nonnegative in the limit.

5.3 Maximum principle on tensors

Sometimes it may be useful to use a tensor variant, for a function u : [0; T ] !� (V ) where � (V ) are sections of a tensor bundle (such as if we wish to applythe maximum principle to the Ricci tensor, for instance). Here is the theorem(possibly due to Hamilton?)

Lemma 37 Let (M; g) be a d-dimensional Riemannian manifold and let V bea vector bundle over M with connection r: Let K be a closed, �berwise convexsubset of V which is parallel with respect to the connection. Let u 2 � (V ) be asection such that

1. u (x) 2 @Kx at some point x 2M; and

2. u (y) 2 Ky for all y in a neighborhood of x

(This is the notion that u attains a maximum at x:) Then rXu (x) is tangentto Kx at u (x) and the Laplacian 4u (x) = gij (x)rirju (x) is an inward ortangential pointing vector to Kx at u (x) :

Here are the relevant de�nitions.


De�nition 38 A subset K of a tensor bundle � : E ! M is �berwise convexif the �ber Kx = K \ Ex (where Ex = ��1 (x)) is a convex subset of the vectorspace Ex:

De�nition 39 A subset K is parallel to the connection r if it is preserved byparallel translation, i.e., if Px;y is parallel translation along a curve from x toy; then P �x;yKy � Kx (this is if the tensors are all contravariant).

Example 40 The set of positive de�nite two-tensors is �berwise convex andparallel with respect to the Levi-Civita connection.

The maximum principle on tensors can be used to show things like:

1. Nonnegative Ricci curvature is preserved by Ricci �ow in dimension 3.

2. Nonnegative curvature operator is preserved by Ricci �ow in all dimen-sions.

We will go into this in more detail in future lectures.

Remark 41 Why do we need convex? Consider the scalar case where we replaceu (t; x) � 0 with u (t; x) � 1� x2 nearby x = 0: Consider the �rst time u (t; x)

Chapter 6

Singularities of Ricci Flow

6.1 Introduction

This roughly covers Lecture 7 of Tao. We will look at analysis of singularitiesof Ricci �ow.

6.2 Finite time singularities

Recall that we proved short time existence for Ricci �ow, which means the �owexists until it reaches a singularity. The �rst, most basic result on singularitiesis the following result of Hamilton:

Theorem 42 Let (M; g (t)) be a solution to the Ricci �ow on a compact man-ifold on a maximal time interval [0; T ): If T <1; then

limt!T�

maxx2M

jRm(t)j2g(t) =1:

Remark 43 Uniqueness of the Ricci �ow is necessary for there to exist a max-imal time interval. The idea is that if there is a solution on [0; T1) and anothersolution on [T1�"; T2); then they must agree on the overlap, so one can considerthe �ow on [0; T2) which extends both the �ows. Now simply take the sup of allT2 for which the �ow exists and this forms the maximal time interval.

Remark 44 N. Sesum was able to replace jRmj with jRcj :

Proof (idea). Suppose T < 1 and jRmj remains bounded. Then one canshow that all derivatives

��rk Rm�� are uniformly bounded for t 2 ["; T ): One canuse this to derive uniform bounds on the metric and its derivatives and thento extract a smooth limit metric at T (of a subsequence using an Arzela-Ascolitype compactness theorem). The existence/uniqueness result tells us that wecan extend the �ow for a short time, contradicting the fact that [0; T ) wasmaximal.

35

36 CHAPTER 6. SINGULARITIES OF RICCI FLOW

This is quite a useful theorem, but it is still possible to develop singularities ina �nite time (for instance, the sphere or a neck pinch). It will be very importantto analyze what is happening at the singular times so that we can do surgery toremove the problems and then continue the �ow. We will do this using a PDEtechnique called blowing up around the singularity, which uses scaling to allowus to see the precise behavior of the �ow near the singular time.

6.3 Blow ups

Here is the idea. At the singular time, we know that jRmj2 is going to in�nity.If we scale the metric g to cg; then we get

jRm(cg)j2 = 1

c2jRm(g)j2

or

jRm(cg)j = 1

cjRm(g)j ;

so if we want to prevent jRmj from going to in�nity, we choose a scaling

L�2n = maxx2M

jRm(g (tn))j

and rescale the metric g (tn) by L�2n : We can actually rescale to a sequence ofsolutions of the di¤erential equation (Ricci �ow) by looking at

gn (t) =1

L2ng�tn + tL

2n

�;

where tn ! T; the singular time. Notice that

@

@tgn =

@

@t

�L�2n g

�tn + tL

2n

��= �2Rc

�g�tn + tL

2n

��= �2Rc [gn (t)]

so gn is a sequence of solutions to the Ricci �ow whose initial value is gettingcloser to the singular time. Furthermore, the initial curvatures jRm(gn (0))j areall bounded by 1:On the down side, Ln ! 0 and so the metric is being multipliedby larger and larger scaling factors and thus it is quite likely that a limit willbecome noncompact. We will need to have a good notion of convergence whichallows convergence to noncompact manifolds.

Remark 45 We do not necessarily have to choose Ln as above. The fact thatL�2n ! 1 is why this is called a blow-up. If we take L�2n ! 0 then we havewhat is called a blow-down.

6.3. BLOW UPS 37

Notice that if the original Ricci �ow is de�ned on an interval [0; T ); then therescaled solutions gn (t) exist on the interval�

� tnL2n

;T � tnL2n

�:

Thus if we can extract a limit and tn ! T and Ln ! 0; then the limit metricswill be ancient, i.e., will start at t = �1 (the �nal endpoint depends a littlemore on how we choose the tn with respect to how we pick the Ln; allowing itto be 0; positive, or +1; we may have reason to choose any of these).The main goal is to �nd quantities which

1. become better as we go to the limit (Tao calls these critical or subcritical)and

2. severely restrict the geometry of the limit (Tao calls these coercive).

Examples of quantities to study or not:

� The volume of the manifold. If we look at

V (M; g) =

ZM

dVg;

we see that

V

�M;

1

L2ng

�=

1

L2=dn

V (M; g)

and so if Ln ! 0; this quantity does not persist to the limit. Tao callsthis behavior supercritical.

� The total scalar curvature. If we look at

F (M; g) =

ZM

RdV;

we see that

F

�M;

1

L2ng

�= L2�dn F (M; g) :

Thus it is preserved in dimension 2 (this is the Gauss-Bonnet theorem,which implies critical behavior) and does not persist in higher dimensions(supercritical).

� The minimum scalar curvature. If we look at

Rmin (M; g) = minx2M

R (x) ;

then we see that

Rmin

�M;

1

L2ng

�= L2nRmin (M; g)

and so this goes to zero as Ln ! 0: This is subcritical behavior. Unfortu-nately, Rmin = 0 does not give su¢ cient information of the limit to classify(not coercive enough).


� Lowest eigenvalue of the operator �44u+R: Notice that this is subcriticalsince if we call this functional H (M; g)

H

�M;

1

L2ng

�= L2H (M; g) :

It turns out that this quantity obeys a monotonicity and has some nicecoercivity, though not quite enough. We will soon look at a particularcritical (i.e., scale invariant) quantity which is related.

6.4 Convergence and collapsing

Manifolds may converge in a number of ways. Here are some examples:

� Sphere converging to a point

� Sphere converging to a cylinder

� Cylinder collapsing to a line

� Torus collapsing to a circle

� Torus collapsing to a line

� 3-sphere collapsing to a 2-sphere by shrinking Hopf �bers

There are several issues here, notably:

� Is there collapse?

� Does convergence involve noncompact manifolds?

The most obvious notion of collapse involves the injectivity radius goingto zero. Recall that the exponential map is the map from the tangent spaceat one point to the manifold where a vector v is taken to the point one unitfrom the origin along a geodesic starting with velocity v: This map is a localdi¤eomorphism, and there is an r > 0 such that the ball B(0; r) is mappeddi¤eomorphically to a ball on the manifold. The largest such r is called theinjectivity radius. As this goes to zero, there is collapse.We will see another way to measure this collapse soon.For noncompact manifolds, one needs to consider pointed convergence. This

generally involves looking at convergence of balls of larger and larger size. Ifall manifolds and their limit have a uniform diameter bound, then one does notneed to consider pointed convergence.

6.5. �-NONCOLLAPSE 39

6.5 �-noncollapse

One de�nition of a collapsing sequence is the following:

De�nition 46 A pointed sequence (Mn; gn; pn) of Riemannian manifolds is col-lapsing if injpn ! 0 as n!1.

We may wish to rescale the manifolds (Mn; gn) to be of some uniform size,say by making jRm(pn)j = 1: Then one can consider a rescaled collapsing ifjRm(pn)j1=2 injpn ! 0:When one assumes that the sectional curvature is bounded, then the collapse

is restricted. Let V (U) = Vg (U) denote the Riemannian volume of the Borelsubset U �M .

Theorem 47 (Cheeger) Suppose that jRmjg � Cr�20 on B (p; r0) � Md andthat

V (B (p; r0)) � �rd0

for some � > 0: Then the injectivity radius of p; injp; is at least

injp � cr0

for some constant c = c (C; �; d) > 0:

Let�s think about this theorem for a minute, to see if it is ever applicablesince it seems the assumptions are quite strong. Note that as r0 ! 0; the balllooks more and more Euclidean. That means that for very small r0 � injp;

V (B (p; r0)) � !rd;

where ! = ! (d) is the correct constant for a Euclidean ball, and

jRmjg � jRm(p)jg :

So as r0 ! 0; we see that

limr0!0

r20 sup

x2B(p;r0)jRm(x)j

!= 0

limr0!0

V (B (p; r0))

!rd= 1:

In particular, for any C > 0 and 0 < � < 1; there is a r� > 0 such that theassumptions are satis�ed if r0 � r�:Note that the converse is also true from more classical results.

Theorem 48 If jRmjg � C and injp � �; then there is a � = � (C; �; d) suchthat

V (B (p; r)) � �rd

for all r � �:


Collapse generally refers to the injectivity radius going to zero. Cheeger�stheorem tells us that when curvature is bounded, volume of balls getting smalland injectivity radius getting small are essentially the same. This roughly mo-tivates the following noncollapsing de�nition, which is not quite the de�nitionwe will use.

De�nition 49 A Riemannian manifold�Md; g

�is �-collapsed at p 2 M at

scale r0 if

1. (Bounded normalized curvature) jRmjg � r�20 for all x 2 B (p; r0) and

2. (Volume collapsed) V (B (p; r0)) � �rd0 :

If these are not satis�ed, then we say the manifold is �-noncollapsed at p atthe scale r0: Is this a reasonable de�nition? Here are some observations:

� By Cheeger�s theorem, this would imply a lower bound on injectivity ra-dius.

� Note that if the one is on a ball where the curvature is large, then themanifold is automatically �-noncollapsed at that scale.

� By the discussion above, every manifold is �-noncollapsed at a smallenough scale and � smaller than the constant for a Euclidean ball.

� This de�nition is scale independent in the following sense. If we consider�g = r�20 g; then the conditions are

jRm(�g)j�g � 1V (B�g (p; 1)) � �:

� The sphere Sn is �-noncollapsed at scales r0 less than the diameter (fora suitable choice of �). The bounded normalized curvature assumption ofthe de�nition are satis�ed for r0 � 1; although one might argue that thereis really no local collapsing at scales less than �: Apparently this will notbe important for our argument. At large scales, the curvature assumptionfails.

� The �at torus has jRmj = 0; and so it is noncollapsed at scales less thanthe injectivity radius. The curvature assumption is valid for large scales,but for a given �; if r0 is taken large enough, the torus must be collapsed,since the volume is never larger than the volume of the torus.

� We want to consider whether there exists a � such that a manifold is �-noncollapsed at large and small scales. Certainly, one can make � smallenough (say, less than the constant for the area of a Euclidean ball) sothat a manifold is �-collapsed at many scales, but this is not of use to us.

6.5. �-NONCOLLAPSE 41

Perelman adapted these ideas to Ricci �ow (time dependent metrics) asfollows.

De�nition 50 Let�Md; g (t)

�be a solution to Ricci �ow and let � > 0: Then

Ricci �ow is �-collapsed at a point (t0; x0) in spacetime at a scale r0 if:

1. (Bounded normalized curvature)

jRm(t; x)jg(t) � r�20

for all (t; x) 2�t0 � r20; t0

��Bg(t0) (x0; r0) ; and

2. (Collapsed volume)V�Bg(t0) (x0; r0)

�� rd0 :

Otherwise, we say that the solution is �-noncollapsed at p at scale r0:

Remark 51 Notice that the assumptions require Ricci �ow to exist on the timeinterval [t0 � r20; t0]:

Remark 52 As remarked above, for each � smaller than the volume of a unitball in Euclidean space, there is a r� so that M is �-noncollapsed at scales lessthan r�: For Ricci �ow, one can still �nd r�; but it will, in general depend ont0: As t0 goes to a singular time, it may be possible that r� ! 0: This is whatwe would like to rule out, as we shall see.

Remark 53 Notice that � is dimensionless.

Remark 54 If g (t) is �-noncollapsed at (t0; x0) at the scale r0; we see thatKg

�t� +

tK

�is �-noncollapsed at (K (t0 � t�) ; x0) at the scale of K�1=2r0:

Here is a typical noncollapsing theorem along the lines of Perelman.

Theorem 55 (Perelman�s noncollapsing theorem, �rst version) Let (M; g (t))be a solution to the Ricci �ow on compact 3-manifolds for t 2 [0; T0] such thatat t = 0 we have

jRm(p)jg(0) � 1V�Bg(0) (p; 1)

�� !

for all p 2 M and ! > 0 �xed. Then there exists � = � (!; T0) > 0 suchthat the Ricci �ow is �-noncollapsed for all (t0; x0) 2 [0; T0] �M and scales0 < r0 <

pt0.

A big point of this theorem is that it rules out a limit of � � R where � isthe cigar soliton solution of Hamilton. The manifold ��R is essentially a �xedpoint of Ricci �ow (when we consider it a �ow on metric spaces, not Riemannianmetrics). � is a positive curvature metric on R2 which has maximum curvatureat the origin and is asymptotic to a cylinder as one moves away from the origin.


This implies that � � R has volume V (B (0; r)) asymptotic to Cr2 for large r(For a cylinder, notice that large balls of radius r have volumes asymptotic toCr; not Cr2:) Thus, �� R is not �-noncollapsed at large scales (r0 large).Consider the blowups (M; gn (t)) de�ned above. Note that we have a � such

that the Ricci �ow g (t) is �-noncollapsed for time interval [0; T0] for any T0 < T:Thus, by Remark 54 we must have that gn (t) is �-noncollapsed at the scale of

L�1n r0 for the time intervalh� tnL2n; T�T0L2n

�: As n becomes large, we see that gn (t)

becomes �-noncollapsed at all scales, which is not true for �� R.We will describe this in more detail later in the course. We will now move

to proving this theorem, the subject of the next few lectures.

Chapter 7

Ricci �ow from energies

7.1 Gradient �ow

Formulating an equation as a gradient �ow has many advantages. Consider theheat equation

@f

@t= 4f

on a compact Riemannian manifold. It is easy to see that if one considers theenergy

E (f) =1

2

ZM

jrf jg dV;

that if we take the time derivative of the energy when f satis�es the heatequation, we get

dE

dt(f) =

ZM

rf � r�@

@tf

�dV

= �ZM

4f�@

@tf

�dV

= �ZM

(4f)2 dV � 0:

Thus we immediately get that the energy is decreasing and that stationary pointsare harmonic functions, i.e., functions which satisfy 4f = 0: This monotonicityalso tells us that f cannot have periodic solutions which are not �xed points,for if f (t1; x) = f (t2; x) for all x; then E (f (t1; �)) = E (f (t2; �)), and themonotonicity implies that 4f = 0 for t 2 [t1; t2]:The monotonicity is true in general for a gradient �ow. If one has an energy

E (f) ; one de�ned the gradient �ow as

@f

@t= �grad (E) ;

43

44 CHAPTER 7. RICCI FLOW FROM ENERGIES

where the gradient vector grad (E) is given so that

dE (X) = g (X; grad (E)) ;

for some metric g on the space of functions. In our case that g is the L2 metric(which uses the Riemannian metric g on M).It would be nice to represent Ricci �ow in this way. It is not at all trivial to

do this.

7.2 Ricci �ow as a gradient �ow

An obvious choice of functional is the Einstein-Hilbert functional:

EH (g) =

ZM

RdV:

To calculate its variation, recall that if we have a variation of the metric (�g)ij =hij ; then we get

�R (h) = � hRc; hi � 4 trg (h) + div div h:

It is not hard to see that since

� log det g = gijhij

so

�pdet g = � exp

�1

2log det g

�=1

2(trg h)

pdet g

so� (dV ) =

1

2(trg h) dV:

Using the above formula we get

� (EH) (h) =

ZM

�� hRc; hi � 4 trg (h) + div div h+

1

2R (trg h)

�dV

=

ZM

�� hRc; hi+ 1

2R (trg h)

�dV

=

ZM

�1

2Rg � Rc; h

�dV:

Remark 56 Here we used the divergence theorem for a compact Riemannianmanifold, which says thatZ

hdiv T; Si dV = �ZhT;rSi dV;

7.2. RICCI FLOW AS A GRADIENT FLOW 45

for any n-tensor T and (n� 1)-tensor S: More explicitly,Zgi1j1 � � � ginjngji0rjTi0i1��inSj1j2��jndV =

Zgi1j1 � � � ginjngji0Ti0i1��inrjSj1j2��jndV:

So critical points of the Einstein-Hilbert functional satisfy the Einstein equa-tion. The gradient �ow would be

@

@tg = �2

�Rc�1

2Rg

�:

The problem is that this �ow is not parabolic and there is no existence theoryfor such equations.Let�s try a new tactic. Replace dV by a �xed measure dm: Then the func-

tional

H (g) =

ZM

Rdm

satis�es the variation

�H (h) =

ZM

(�hRc; hi � 4 trg (h) + div div h) dm:

You don�t lose the last two terms with the divergence theorem, since that onlyworks with the volume measure. However, we can consider the Radon-Nikodymderivative and write

dm =dm

dVdV

for a positive function dmdV : We can write

dm

dV= e�f :

Then,

�H (h) =

ZM

(�hRc; hi � 4 trg (h) + div div h) e�fdV

can be integrated by parts to get

�H (h) =

ZM

(�hRc; hi � hr trg (h) ;rfi+ div h � rf) e�fdV

=

ZM

��hRc; hi+ trg (h)4f � trg (h) jrf j2 �

h;r2f

�+ h (rf;rf)

�e�fdV

=

ZM

D�Rc�r2f +

�4f � jrf j2

�g +rfrf; h

Ee�fdV:

Remark 57 Tao often uses h�; �i to denote the Euclidean metric locally, andso explicitly puts in gij�s in this case. We will understand that h�; �i requiresthe metric, and so when quantities like this are di¤erentiated, we also need todi¤erentiate the metric, as we will see below.


We need to add another term, and so we get

F (g) =

ZM

�R+ jrf j2

�e�fdV =

ZM

R+

��r log dmdV��2!dm:

We now need that

0 = � (dm) = ��e�fdV

�= � (�f) e�fdV + 1

2trg he

�fdV

=

��f + 1

2trg h

�e�fdV

Thus we have that�f =

1

2trg h

Let

E (g) =

Z �gijrifrjf

�e�fdV

where dm = e�fdV is �xed (so that f = � log dm=dV and �f is expressed asabove). We can then compute

�E (h) = �

�Z �gijrifrjf

�e�fdV

�=

Z ��hh;rfrfi+ 2 hrf;r (�f)i � jrf j2 �f + 1

2jrf j2 trg h

�e�fdV

=

Z ��hh;rfrfi � 24f (�f) + jrf j2 �f + 1

2jrf j2 trg h

�e�fdV

=

Z �Dh;�rfrf � (4f) g + jrf j2 g

E�e�fdV:

Now, since F = H + E; we have

�F (h) =

ZM

�Rc�r2f; h

�e�fdV:

Thus the gradient �ow of �2F is

@g

@t= �2Rc (g)� 2r2f:

This is almost Ricci �ow, but not quite. The f is changing, too, by the equation

@f

@t= �4f �R:

This is a backward heat equation, which it turns out will make it useful to probebackwards.Notice that

2r2f = Lrfg;


the Lie derivative of g in the direction rf: This means that the �ow abovedi¤ers from Ricci �ow by a di¤eomorphism. Instead, we can consider �g = ��g;where � is the �ow of di¤eomorphisms generated by rf; and we will see that �gevolves by Ricci �ow. Furthermore, f will di¤er by a Lie derivative, and

Lrff = df (rf) = jrf j2

(where is the metric in here? It is in rf; which is a vector �eld gotten by raisingthe index on df) and so under the new �ow, �f = f � � evolves by

@ �f

@t= �4�g

�f � �R+��r �f ��2

�g:

Example 58 (Fundamental, important example) Let (M; g) be Euclideanspace M = Rd and let

f (t; x) =jxj2

4�+d

2log 4�� = � log

h(4��)

�d=2e�jxj

2=(4�)i

where � = t0 � t; Notice that e�fdx is the Gaussian measure, which solvesthe backward heat equation (the fundamental solution to the heat equation). Ift < t0; this choice of g and f satisfy the equations

@g

@t= �2Rc (g) (7.1)

@f

@t= �4f �R+ jrf j2 : (7.2)

We can check:

@f

@t=

@

@t

"jxj2

4�+d

2log 4��

#

=jxj2

4�2� d

2�:

rf = x

2�

jrf j2 = jxj2

4�2

4f = d

2�

so it works.

Notice that when we pull back by �; the measure ��dm is not static. Thusit makes sense to rewrite the functional as

F (M; g; f) =

ZM

�R+ jrf j2

�e�fdV:


Notice that this functional is invariant under di¤eomorphism, i.e.,

F (M;��g; f � �) = F (M; g; �)

for any di¤eomorphisms �:We also have that under the coupled �ows (7.1) and(7.2),

@F

@t(M; g; f) = 2

Z ��Rc+r2f ��2 e�fdV:Thus we have that F is monotone increasing under Ricci �ow. Unfortunately,now we have to explicitly deal with this quantity f: To eliminate this, we takethe in�mum:

� (M; g) = inf

�F (M; g; f) :

ZM

e�fdV = 1

�(the in�mum is over f). Using the following exercise, we can show that � is�nite.

Exercise 59 Show that � (M; g) is the smallest number for which one has theinequality Z

M

�4 jruj2g +Ru

2�dV � �

ZM

u2dV;

where u is in H1 (M) =W 1;2 (M), the Sobolev space of functions with 1 deriva-

tive in L2 (so it has norm kfkH1 =R �jrf j2g + f2

�dV for C1 functions). Hint:

show that we can assume u is positive and then write u = e�f=2: Thus � is thesmallest eigenvalue of the operator �44gu+R:

Using the exercise, one sees that the fact that every compact manifold sat-is�es a Poincaré inequality,Z

M

jruj2 dV � c (d)

ZM

u2dV;

implies that � is bounded below, basically, by the best constant in the Poincaréinequality plus minR: Note that the Poincaré inequality constant depends onthe dimension. We will see later a similar inequality for which the constant doesnot depend on dimension.Furthermore, one can prove that � is realized by a positive function u =

e�f=2 with kukL2(M) = 1: Note that H1 (M) embeds compactly into L2 (M)since M is compact. Thus if we take a minimizing sequence fung in H1; thereis a subsequence (which we also denote by fung which converges in L2 to afunction u: Now consider:Z �

jrunj2 +Ru2n�dV +

Z �jrumj2 +Ru2m

�dV

=1

2

Z �jr (un � um)j2 +R (un � um)2

�dV +

1

2

Z �jr (un + um)j2 +R (un + um)2

�dV


so

1

2

Z �jr (un � um)j2 +R (un � um)2

�dV =

Z �jrunj2 +Ru2n

�dV +

Z �jrumj2 +Ru2m

�dV

� 12

Z �jr (un + um)j2 +R (un + um)2

�dV:

The right side goes to zero in the limit since the terms go to �; � and �2�: Also,we know that

�min jRj kun � umkL2 �ZR (un � um)2 dV � min jRj kun � umkL2

and each term goes to zero. Thus we know that the sequence fung is Cauchyin H1 and since H1 is complete, it must converge to a function in H1:Since � is attained at a function, we can sometimes prove inequalities like

the following (taken from the notes of Kleiner and Lott):Let h (s; t; x) be a two-parameter family of functions such that

h (s; t0) = f� (t0 + s)

@h

@t= �4h+�R+ jrhj2 :

There is a solution to this for t � t0 since the equation is backwards elliptic.(Note: we have only shown that f� is in H1; so we mean a weak solution tothe parabolic equation. We could also show that f�; as a minimizer, satis�es aparticular elliptic equation, which implies that f� is smooth by elliptic regularitytheory.)

� (t0) � F (M; g (t0) ; h (s; t0 � s))

= F (M; g (t0 + s) ; h (s; t0))� 2Z s

0

�ZM

��Rc (g (t0 + �)) +r2h (s; t0 + �)��2 e�h(s;t0+�)dV � d�:We then have

� (t0) � � (t0 + s)�2Z s

0

�ZM

��Rc (g (t0 + �)) +r2h (s; t0 + �)��2 e�h(s;t0+�)dV � d�and so

@�

@t(t0) = lim

s!0+

� (t0 + s)� � (t0)s

� lims!0+

2

s

Z s

0

�ZM

��Rc (g (t0 + �)) +r2h (s; t0 + �)��2 e�h(s;t0+�)dV � d�= 2

ZM

��Rc (g (t0)) +r2h (0; t0)��2 e�h(0;t0)dV= 2

ZM

��Rc (g (t0)) +r2f� (t0)��2 e�f�dV


We can now derive

@�

@t� 2

Z ��Rc+r2f��2 e�f�dV� 2

3

Z(R+4f�)2 e�f�dV

� 2

3

�Z(R+4f�) e�f�dV

�2=2

3

�Z �R+ jrf�j2

�e�f�dV

�2=2

3�2:

7.3 Perelman entropy

We now wish to make our functional scale invariant (so that we get a criticalquantity, not just subcritical). In particular, we know that

dFmdt

(M; g) = 2

Z ��Rc+r2f ��2 e�fdVis �xed (under the gradient �ow) if

Rc = �r2f;

i.e., if (M; g) is a gradient Ricci soliton. We wish to have a new functional whichis �xed if (M; g) is a gradient shrinking soliton, i.e.,

Rc = �r2f + 1

2�g

for some � > 0: A round sphere is a gradient shrinking soliton, so it makes sensethat we would want something like this. Under the Ricci �ow, this structure ispreserved except that � decreases at a constant rate.First note that if we consider the Nash entropy

Nm (M; g) =

ZM

dm

dVlog

dm

dVdV =

ZM

�log

dm

dV

�dm = �

Zfdm

then

dNmdt

= �ZM

(�4f �R) dm

=

ZM

�R+ jrf j2

�dm = Fm (M; g) :

7.3. PERELMAN ENTROPY 51

This will come in handy. Now, suppose we want a quantity W (M; g) such that

dW

dt=

ZM

��Rc+r2f � 1

2�g

��2 dm:But in this case, would not have scale invariance forW; since t scales like distancesquared, so the integrand should scale like distance squared. We will �x this byassuming

d�

dt= �1

and trying fordW

dt= 2�

ZM

��Rc+r2f � 1

2�g

��2 dm: (7.3)

Now, to �nd such a quantity, consider��Rc+r2f � 1

2�g

��2 = ��Rc+r2f ��2 � 1

�(R+4f) + d

4�2:

Thus we have that

2�

ZM

��Rc+r2f � 1

2�g

��2 dm = �dFmdt

� 2Fm +d

2�

=d

dt

��Fm �Nm �

d

2log �

�:

This is what our Wm would be. However, as we did last time, we wanted toreparametrize so that

dm = e�fdV

where dV is evolving according to Ricci �ow evolution. This time, we will change

~f = f � d

2log (4��)

so thatdm = e�fdV = (4��)

�d=2e�

~fdV:

Remark 60 This looks like the heat kernel for Euclidean space, which is whythis particular normalization is given.

Note that the preservation of dm implies that

d

2�� @ ~f

@t+1

2trh = 0:

Under the gradient �ow@g

@t= �2Rc�2r2f;


we have@ ~f

@t=

d

2��R�4f:

Thus we get for

Wm (M; g; �) = �Fm �Nm �d

2log �

=

Z ��

�R+

��r ~f ��2�+ ~f � d

2log (4�)

�(4��)

�d=2e�

~fdV

Actually, we usually renormalize this to vanish in the Euclidean case, and so wecan change the d term appropriately to

Wm (M; g; �) =

Z ��

�R+

��r ~f ��2�+ ~f � d�(4��)

�d=2e�

~fdV:

We can also de�ne the Perelman entropy as the functional

W (M; g; f; �) =

Z h��R+ jrf j2

�+ f � d

i(4��)

�d=2e�fdV

where g is a Riemannian metric, f is a function on M , and � is a positiveconstant.If g satis�es the Ricci �ow, then we need to pull back f to get the three

evolutions

@g

@t= �2Rc (7.4)

@f

@t=

d

2��R�4f + jrf j2 (7.5)

d�

dt= �1:

Under these three, the Perelman Entropy satis�es

dW

dt(M; g; f; �) = 2�

ZM

��Rc+r2f � 1

2�g

��2 (4��)�d=2 e�fdV:We would like to �nd a minimum over all functions f and constants � so

that we have an invariant of the Riemannian manifold. However, it is not yetclear that such an in�mum exists. Recall that last time, the existence followedfrom a Poincaré inequality. In this scale invariant setting, the existence of aminimizer will follow from a log-Sobolev inequality.

7.4 Log-Sobolev inequalities

Let�s �rst consider what happens if g is the Euclidean metric. We would like toswitch to a function which looks like the heat kernel, namely,

u = (4��)�d=2

e�f :

7.4. LOG-SOBOLEV INEQUALITIES 53

Recall that our model case is when f = jxj2 = (4�) ; in which case this is preciselythe backwards heat kernel. The backwards heat kernel satis�es:

@u

@t= �4u

for � > 0 andlim�!0�

u (�; x) = �0 (x) ;

weakly, where �0 is the delta function. Furthermore, one can check thatZRdudx =

ZRd(4��)

�d=2e�jxj

2=(4�)dx = 1 (7.6)

for any �: The backwards heat kernel can be used to solve the heat equationwith some given �nal conditions, e.g., to solve

du

dt= �4u

u (T; x) = f (x) ;

we see that the convolution

u (t; x) =

ZRdf (y) (4��)

�d=2e�jx�yj

2=(4�)dy

is a solution.

Exercise 61 Show that all of this is true. Hint: to show (7.6), turn the integralinto polar coordinates and assume the dimension is at least 2: For the dimension1 case, there is a trick involving turning it into a dimension 2 integral andseparating.

We can check that for g Euclidean and f as above, we have

W (M; g; f; �) =

Z " jxj22�

� d#(4��)

�d=2e�jxj

2=(4�)dx:

One can show that this is zero since

W (M; g; f; �) =

Z h2��jrf j2 �4f

�i(4��)

�d=2e�fdx;

and integrating by parts (needs to be justi�ed) shows this is equal to zero.Now we re-write W by replacing f with u: We see that (remembering still

we are in Euclidean space),

W =

Z "�jruj2

u2� u log u

#dx� d

2log (4��)� d


using identities such as

u = (4��)�d=2

e�f :

log u = �d2log (4��)� f

jruj2 = (4��)�d jrf j2 e�2f

jrf j2 = jruj2

u2:

Tao shows that one can show that W � 0; which implies a log-Sobolevinequality

�

Z jruj2

u2dx �

Zu log udx+

d

2log (4��) + d;

or as it is usually stated, with �2 = u;

4�

Zjr�j2 dx � 1

�

Z�2 log �2dx+

d

2log (4��) + d:

For the general case, we have

W (M; g; f; �) =

Z "�

Ru+

jruj2

u2

!� u log u

#dV � d

2log (4��)� d

One can show that the

W (M; g; f; �) � �C (M; g; �) :

This implies essentially a log-Sobolev inequality, i.e.,

�

ZR�2dV + �

Z4 jr�j2 dV � �C +

Z�2 log �2dV +

d

2log (4��) + d:

In fact, we can take the

� (M; g; �) = inf

�W (M; g; f; �) :

Z(4��)

�d=2e�fdV = 1

�;

which is the best possible constant �C: It can be shown that � is �nite, whichis what we could call a log-Sobolev inequality.We can now show that if g (t) is a solution to Ricci �ow on t 2 [0; T0] and

� = T0 � t, then � (M; g; �) is increasing. The �rst exercise is important:

Exercise 62 Show that � (M; g; �) =W (M; g; f�; �) for a function f� 2 H1 (M) :(Not quite true... What is the true statement? Hint: you need to change to anew function �:)

7.5. NONCOLLAPSING 55

Once we know that � is realized by a function, we can show that � (M; g (t) ; � (t))is increasing as follows. Calculate � (M; g (t0) ; � (t0)) =W (M; g (t0) ; f� (t0) ; � (t0))for some minimizer f� (t0) : For any time t � t0; we can solve the equation forf in 7.4 backwards to t with initial condition f (t0; x) = f� (t0; x) (since f� is inH1 (M) ; there exists a weak solution to this parabolic �ow). We know that

� (M; g (t) ; � (t)) �W (M; g (t) ; f (t) ; � (t))

�W (M; g (t0) ; f� (t0) ; � (t0)) = � (M; g (t0) ; � (t0)) :

7.5 Noncollapsing

We will now show that log-Sobolev inequalities imply noncollapsing. Supposewe have a ball B (p;

p�) with bounded normalized curvature, i.e.,

jRm(x)j � 1

�

for x 2 B (p;p�) : Then jRj � � c (d) for some constant depending only on

dimension. Then the log-Sobolev inequality can be rewritten as

c (d)

Z�2dV + �

Z4 jr�j2 dV � � (M; g; �) +

Z�2 log �2dV +

d

2log (4��) + d:

Suppose � is a function supported on B (p;p�) such that

RM�2dV = 1: Then

Jensen�s inequality implies that

1

V (B)

ZB

�2 log �2dV ��

1

V (B)

ZB

�2dV

�log

�1

V (B)

ZB

�2dV

�=

1

V (B)log

1

V (B);

where B = B (p;p�) : (Recall that Jensen�s inequality requires a probability

measure.) So ZM

�2 log �2dV � log 1

V (B):

We now get, for this particular choice of �;

4�

Zjr�j2 dV � � (M; g; �) + log

�d=2

V (B)� c0 (d) :

Now we will specialize � even more. Suppose

� (x) = c

�d (x; p)p

�

�for some bump function on the real line which is 1 on [0; 1=2] and supported on[0; 1] (technically, we only need half the bump function, which is how I describedit). Thus � (x) = c on B (p;

p�=2) and c is such thatZ

B

�2dV = 1;


so c � V (B (p;p�=2))

�1=2: We can choose � so that jr�j � c00c=

p� on the

ball (for some constant c00), and so

4c00V (B)

V�B1=2

� � � (M; g; �) + log�d=2

V (B)� c0 (d) :

Finally, we can use a Bishop-Gromov volume comparison theorem:

Theorem 63 (Bishop-Gromov comparison) If�Md; g

�is a complete Rie-

mannian manifold withRc � (n� 1)Kg

for some K 2 R; then for any p 2M ; the volume ratio

V (B (p; r))

VK (B (pK ; r))

is non-increasing as a function of r; where pK is a point in the d-dimensionalsimply connected space of constant sectional curvature K; and VK (B (pK ; r)) isthe volume of a ball of radius r in that space.

In particular, we have that

V (B)

V�1=��B�p�1=� ;

p�� V

�B1=2

�V�1=�

�B�p�1=� ;

p�=2�� ;

and thus there is a � = � (�; d) such that

V (B)

V�B1=2

� � �:

In fact, � is independent of � since

V�1=��B�p�1=� ;

p��= V�1 (B (p�1; 1))

V�1=��B�p�1=� ;

p�=2��= V�1 (B (p�1; 1=2)) :

Thus there is a constant c000 which depends on d such that

c000 � � (M; g; �) � log �d=2

V (B);

i.e.,

V (B) ��e��c

000��d=2;

which implies �-noncollapsing at a scalep� for � = exp (�� c000) : Let�s formu-

late this into a proposition:

Proposition 64 There is a constant c = c (d) depending only on dimensionsuch that if �

�Md; g; �

�is �nite, then for � = exp (� (M; g; �)� c) ; the Rie-

mannian manifold (M; g) is �-noncollapsed at the scale ofp� :

7.5. NONCOLLAPSING 57

Let�s collect the facts about �:

Proposition 65 The following are true about �:

1. � (M; g; �) > �1 for any �xed manifold (M; g) and � > 0:

2. If (M; g (t)) satis�es the Ricci �ow for t 2 [0; T0] and � (t) = T0 � t; then� (M; g (t) ; � (t)) is increasing.

3. There is a constant c = c (d) depending only on dimension such that theRiemannian manifold (M; g) is �-noncollapsed at the scale of

p� at every

point for � = exp (� (M; g; �)� c) :

We can now prove:

Theorem 66 (Perelman�s noncollapsing theorem, �rst version) Let (M; g (t))be a solution to the Ricci �ow on compact 3-manifolds for t 2 [0; T ) such thatat t = 0 we have

jRm(p)jg(0) � 1V�Bg(0) (p; 1)

�� !

for all p 2 M and ! > 0 �xed. For any � > 0; there exists � = � (!; T; �) > 0such that the Ricci �ow is �-noncollapsed for all (t0; x0) 2 [0; T )�M and scales0 < r0 < �. We could also take � = � (t) and get a similar result, as long as� (t) is uniformly bounded on [0; T ):

Proof. We already showed that for a given � and metric, � (M; g; �) has a lowerbound. For any r20; we see by monotonicity that

��M; g (t) ; r20

��

�M; g (0) ; r20 + t

�:

Thus we have that if

�0 = inf��M; g (0) ; r2

�: r2 2 (0; �+ T )

then

��M; g (t) ; r20

�� 0:

Thus (M; g (t)) is �-noncollapsed at the scale of r0 for all

� = exp (�0 � c) � exp��M; g (t) ; r20

�� c�:

We need to see that �0 is not �1: Since T is �nite, there is no problem at thetop of the interval for r2: It can be shown that as r2 ! 0+; �

�M; g (0) ; r2

�! 0

(in the interest of time, we will not show this) and so there is no problem at theother side.

Remark 67 This is a bit stronger than what I proposed in an earlier lecture. Ithink Tao was thinking about future incarnations of this theorem, which is whyhe formulated as he did.

Chapter 8

Ricci �ow for attackinggeometrization

8.1 Introduction

I would like to go back to the general program and see what we need to learnabout. Much of this if from Morgan-Tian, with help from other sources.

Recall that we wish to perform surgery when the Ricci �ow comes to asingularity. We will consider a Ricci �ow with surgery (M; g (t)) de�ned for0 � t < T <1 which satis�es the following properties:

1. (Normalized initial conditions) We have

jRm(g (0))j � 1

andV (B (x; 1)) � 1

2V (BE3 (0; 1))

for any x 2M:

2. (Curvature pinching) The curvature is pinched towards positive. Thismeans that as the scalar curvature R ! +1; the ratio of the absolutevalue of the smallest eigenvalue of the Riemannian curvature tensor to thelargest positive eigenvalue goes to zero.

3. (Noncollapsed) There is a � > 0 so that the Ricci �ow is �-noncollapsed.

4. (Canonical neighborhood) Any point with large curvature has a canonicalneighborhood.

The key is to show that these conditions both allow surgery and then persistafter the surgery. In order to do this, we will need to be more precise with 3and especially 4.

59

60 CHAPTER 8. RICCI FLOW FOR ATTACKING GEOMETRIZATION

8.2 Finding canonical neighborhoods

The main way to �nd these is to take blow-ups as one goes to a singularity. Ifone takes a sequence ti ! T; where T is a singular time, then the blow ups

gi (t) =Mig

�ti +

t

Mi

�:

If Mi is comparable to sup jRm(g (ti))j ; then Mi ! 1: Furthermore, sinceT <1; we have that gi (t) is de�ned on the interval [�Miti; (T � ti)Mi): Thusthe limit will de�nitely be de�ned on (�1; 0]; and is thus ancient. Moreover,if we show that g is �-noncollapsed at some scale r0, then gi (t) will be �-noncollapsed at a scale r0

pMi; and so the limit is �-noncollapsed on all scales.

Finally, one can show that any 3-dimensional ancient solution has nonnegativecurvature.

De�nition 68 A �-solution of Ricci �ow is a solution de�ned for t 2 (�1; 0]which is �-noncollapsed on all scales and has nonnegative curvature.

These are the limits as you go to a �nite time singularity. The key is thatPerelman was able to classify these solutions in the following way:

1. �-solutions look like gradient shrinking solitons as t! �1:

2. Gradient shrinking solitons in dimension 3 must have �nite covers isomet-ric to (i) 3-spheres or (ii) 2-spheres cross R.

3. �-solutions have canonical neighborhoods.

Now the idea is that as one goes to a singularity, the manifold is like a�-solution, and thus has canonical neighborhoods.

8.3 Canonical neighborhoods

What is a canonical neighborhood? This is where the surgery should be done,so it needs to be classi�ed su¢ ciently to allow us to do a careful surgery. Itturns out that there are 4 types of canonical neighborhoods:

De�nition 69 A point x 2 M is in a (C; ")-canonical neighborhood if one ofthe following holds:

1. x is contained in a C-component.

2. x is contained in an open set which is within " of round in the Cb1="c-topology.

3. x is contained in the core of a (C; ")-cap.

4. x is in the center of a strong neck.

8.3. CANONICAL NEIGHBORHOODS 61

De�nition 70 De�ne the Ck (X; g0) �norm�on (X; g) to be

k(X; g)k2Ck(X;g0)= sup

x2X

(jg (x)� g0 (x)j2g0 +

kX`=1

��r`g0g (x)��2g0):

Remark 71 Technically, the norm should be

jjj (X; g) jjj2Ck(X;g0)= sup

x2X

(jg (x)j2g0 +

kX`=1

��r`g0g (x)��2g0):

The function k�k de�ned above is essentially k(X; g)k = jjj (X; g � g0) jjj: Withour current de�nition, k(X; g)kCk(X;g0)

= 0 if g = g0:

Problem 72 Here is something to think about. Fix X � Rn; say bounded.Given a sequence of Riemannian metrics fgig on X; under what conditionsdoes there exist a subsequence such that (M; gi) converge to some limit (X; g1)for some Riemannian metric g1 (i.e., k(X; gi)kCk(X;g1)

! 0 as i!1:

Problem 73 How could one use this norm idea to compare (X; g) and (X 0; g0)where X 6= X 0?

De�nition 74 Let (N; g) be a Riemannian manifold and x 2 N a point. Thenan "-neck structure on (N; g) centered at x consists of a di¤eomorphism

� : S2 ��1";1

"

�! N;

with x 2 ��S2 � f0g

�; such that

k(N;R (x)��g)k < "

where the norm is with respect to Cb1="c�S2 �

�� 1" ;

1"

�; gstd

�; where gstd is the

product of the metric with curvature 1=2 on S2 with the Euclidean metric onthe interval. We say N is a "-neck centered at x:

De�nition 75 A compact, connected, Riemannian manifold (M; g) is called aC-component if

1. M is di¤eomorphic to S3 or RP3;

2. (M; g) has positive sectional curvature,

3. For every 2-plane P in TX;

1

C<

infP K (P )

supy2M R (y);


4.

C�1 supy2M

1pR (y)

< diam (M) < C infy2M

1pR (y)

:

De�nition 76 A compact, connected 3-manifold (M; g) is within " of round inthe Cb1="c-topology if there exists a constant � > 0; a compact manifold (Z; g0)of constant curvature +1; and a di¤eomorphism

� : Z !M

such thatk(Z; �� (�g))kCb1="c(Z;g0)

� ":

Finally, we have the complicated de�nition of a cap. The last conditions es-sentially say that the diameter, volume, and curvature di¤erences are controlledand are technical conditions needed in some arguments.

De�nition 77 Let (M; g) be a Riemannian 3-manifold. A (C; ")-cap in (M; g)is a noncompact submanifold (C; gjC) together with an open submanifold M � Cwith the following properties:

1. C is di¤eomorphic to an open 3-ball or to a punctured RP3:

2. N is a "-neck.

3. �Y = CnN is a compact submanifold with boundary. Its interior Y is calledthe core of C.

4. The scalar curvature R (y) > 0 for every y 2 C and

diam (C; gjC) <Cq

supy2C R (y):

5.

supx;y2C

R (x)

R (y)< C:

6.

V (C) < C�supy2C R (y)

�3=2 :7. For any y 2 Y; let ry de�ned by the the condition that

supy02B(y;ry)

R (y0) =1

r2y:

Then for each y 2 Y the ball B (y; ry) has compact closure in C and

1

C< inf

y2Y

V (B (y; ry))

r3y:

8.4. HOW SURGERY WORKS 63

8.

supy2C

jrR (y)jR (y)

3=2< C

and

supy2C

��@R@t (y)

��R (y)

2 < C

8.4 How surgery works

The key observations are this:

1. For every � and every small " > 0; there is a C1 = C1 ("; �) such that a�-solution is the union of (C1; ") canonical neighborhoods.

2. For every small " > 0; there is a C2 = C2 (") and a standard solution ofRicci �ow with is the union of (C2; ") canonical neighborhoods.

3. We can do surgery on canonical neighborhoods if they are su¢ ciently smalland positively curved.

Consider a Ricci �ow which becomes singular at a time T: Fix T� < T sothat there are no surgeries in the interval [T�; T ): By the assumptions, there isan open set � M such that the curvature is bounded for all t 2 [T�; T ); sothere is a limiting metric on as t ! T: Every end is the end of a canonicalneighborhood, which looks like a tube. We call these ends "-horns. We can then�x a constant � and consider the subset � � in which the scalar curvatureis bounded above by ��2: One can then show that " horns with boundaries in� are �-necks. We then do surgery on these �-necks by gluing in a standardsolution.

8.5 Some things to prove

Here are some things we will need to do in order for this procedure to work:1) Derive a description of �-solutions. This will be with regard to what the

asymptotic shrinking soliton is, and so we will need to show that there is one.2) Show that solutions have canonical neighborhoods.3) Describe the canonical solution4) Show �nite time extinction.

Chapter 9

Reduced distance

9.1 Introduction

We �rst wish to show that a �-solution has a limit which is a gradient-shrinkingsoliton. To do this, we will need to introduce a new notion, the reduced distance.

9.2 Short discussion of W vs reduced distance

We were able to show quite a bit using the W functional, so why introduce thereduced distance (whatever that is)? Let�s �rst think a bit about the case ofEuclidean space. Recall that in Euclidean space, W becomes

W (M; g; u; �) =

Z "�jruj2

u2� u log u

#dx� d

2log (4��)� d

and that W is minimized for

u� (x; �) = (4��)�d=2

exp�� jxj2 = (4�)

�:

We see that this formula essentially gives the distance function d (x; 0)2 = jxj2by

jxj2

4�= � log u� (x; �)�

d

2log (4��) :

However, we have very little control over this function. One can also derive thedistance function as follows:

d (x; 0) = inf (0)=0 (a)=x

Z a

0

j _ j d�:

This strong relationship is only true in Euclidean space. In general, there aretwo di¤erent concepts: the heat kernel u� (x; �) and the Riemannian distance

65

66 CHAPTER 9. REDUCED DISTANCE

function d (x; x0) : The function u� is essentially de�ned as the solution to aPDE and the distance function is de�ned by minimizing over paths. Thus,often d (x; x0) is easier to work with and easier to get more precise informationabout. The two things are closely related, but not the same concept. We willtry to do something similar for the functional W:

9.3 L-length and reduced distanceConsider a curve : [0; � ]!M . One can de�ne the length of a curve as

` ( ) =

Z �

0

j _ (s)j ds

and the energy as

E ( ) =1

2

Z �

0

j _ (s)j2 ds:

Note that the length is independent of reparametrization, but the energy is not.These notions give rise to the function rp which is the function representing thedistance to the point p; i.e.,

rp (x) = d (x; p) :

It turns out that the distance function satis�es some di¤erential equations andinequalities, in particular

4r � d� 1r

for Ricci nonnegative. Note that if r (x) =qP

(xi)2; then 4r = d�1

r : We willshow this inequality later in this lecture, but for now, let�s assume this and seethe implications on the volume. Notice that

d

dRV (B (p;R)) = A (S (p;R))

where S (p;R) is the geodesic sphere of radius R centered at p; and A is thearea (d � 1 dimensional volume measure) function. By Gauss lemma, we havethat

jrrj = 1and also that we can decompose the metric to be

dr2 + gS(p;r)

where gS(p;r) is the metric on the geodesic sphere. Denote the measure on thesphere of radius r as dAS(p;r): We see thatZ

B(p;R)

4rdV =ZS(p;R)

(rr � n) dAS(p;r)

=

ZS(p;R)

jrrj dAS(p;r)

= A (S (p;R)) :

9.3. L-LENGTH AND REDUCED DISTANCE 67

The decomposition above implies that the volume measure can be decomposedas

dV = dAS(p;r)dr

and so

d

dR

ZB(p;R)

4rdV = d

dR

Z R

0

ZS(p;r)

4rdAS(p;r)dr

= Rd�1ZS(p;R)

4rdAS(p;R):

Thus,

d

dR

A (S (p;R))

Rd�1=

1

R2(d�1)

"Rd�1

d

dR

ZB(p;R)

4rdV � (d� 1)Rd�2A (S (p;R))#

=1

R2(d�1)

"Rd�1

ZS(p;R)

4rdAS(p;R) � (d� 1)Rd�2A (S (p;R))#

� 1

R2(d�1)

"(d� 1)Rd�2

ZS(p;R)

dAS(p;R) � (d� 1)Rd�2A (S (p; r))#

= 0:

This implies an inequality on volume ratios.

Theorem 78 (Bishop-Gromov theorem, simpli�ed version) If Rc � 0then V (B (p; r)) =rd is a nonincreasing function of r:

Proof. We have already showed that

d

dR

A (S (p;R))

Rd�1� 0:

We now consider 0 � r1 � r2: We have

V (B (p; r2))� V (B (p; r1))rd2 � rd1

=

R r2r1A (S (p; r)) drR r2r1d rd�1dr

=

R r2r1

A(S(p;r))rn�1 rd�1drR r2

r1d rd�1dr

� A (S (p; r1))

d rd�11

:


Furthermore,

V (B (p; r1))

rd1=

R r10A (S (p; r)) drR r10d rd�1dr

=

R r10

A(S(p;r))rn�1 rd�1drR r1

0d rd�1dr

� A (S (p; r1))

d rd�11

:

Thus we have

V (B (p; r1))

rd1� V (B (p; r2))� V (B (p; r1))

rd2 � rd1;

which implies

0 � V (B (p; r1))

rd1� V (B (p; r2))� V (B (p; r1))

rd2 � rd1

=rd2V (B (p; r1))� rd1V (B (p; r2))

rd1�rd2 � rd1

�=

rd2�rd2 � rd1

� �V (B (p; r1))rd1

� V (B (p; r2))

rd2

�:

We now consider Ricci �ows. Let (M; g (�)) be a solution to backwards Ricci�ow, i.e.,

@g

@�= Rc (g (�))

and let : [0; � ]!M be a curve in M: We de�ne the L-distance to be

L ( ) =Z �

0

p��j 0 (�)j2g(�) +Rg(�)

�d�:

This may need a bit of explanation. First, 0 = d d� is the tangent vector to the

curve : Note that it is measured at � by the metric g (�) (so at di¤erent s; itis measured using di¤erent metrics!) The �rst term looks almost like the termin the energy

Rj 0j2 d�, which can be used to derive geodesics on a Riemannian

manifold. However, the addition of the termp� makes the integral scale like

length, not energy (think about this).

Remark 79 Tao uses �� in some places because of the understanding that� = �t and he wants g (t) to be a solution to Ricci �ow and V (t) (de�ned later)to be de�ned on t: We will allow g to be parametrized by � and so there is noneed for this.

9.3. L-LENGTH AND REDUCED DISTANCE 69

In the smooth, �xed manifold case, one considers the length functional

L ( ) =

Z �1

0

j 0 (�)jg d�

and then one can �nd the distance d (x0; x) between points by taking the in�-mum of length over all curves connecting those two points. One can de�ne thedistance function rx0 (x) which is the function which returns the distance to a�xed point x0: The analogue of this in the Ricci �ow case is the `-distance (alsocalled reduced length):

`(0;x0) (�; x) =1

2p�inf fL ( )g

where the inf is over all paths from x0 to x:

Remark 80 Note there is also the L-distance, which is 2p�`:

One can �nally de�ne the reduced volume

~V(0;x0) (�) =

ZM

��d=2 exp��`(0;x0) (�; x)

�dVg(�):

Our main goal will be to show the following:

Theorem 81@

@�`(0;x0) �4g(�)`(0;x0) +

��r`(0;x0)��2g(�) �R+ d

2�� 0:

As a corollary, we get monotonicity of the reduced volume if @@� g = 2Rc (g) :

Corollary 82 (Reduced Volume is monotone)

@

@�~V(0;x0) (�) � 0:

Proof.@

@�~V(0;x0) (�) =

@

@�

ZM

��d=2 exp��`(0;x0) (�; x)

�dVg(�)

= � d

2�~V(0;x0) (�)�

ZM

@`(0;x0)

@��d=2 exp

��`(0;x0) (�; x)

�dVg(�)

+

ZM

��d=2 exp��`(0;x0) (�; x)

�RdVg(�)

� � d

2�~V(0;x0) (�) +

ZM

��d=2 exp��`(0;x0) (�; x)

�RdVg(�)

+

ZM

��4g(�)`(0;x0) + jr`j

2g(�) �R+

d

2�

��d=2 exp

��`(0;x0) (�; x)

�dVg(�)

= 0:

Of course, all of this assumes su¢ cient regularity on `; which is not, ingeneral true. However, this argument can be made rigorous in some generality,including past the �conjugate radius.�


9.4 Variations of length and the distance func-tion

We start with the energy functional given above. We can do calculus of varia-tions as follows. Consider a variation � (t; s) such that

� (t; s = 0) = (t)

@

@s� (t; s = 0) = X (t) ;

so we consider the variation to be X: Furthermore, we consider variationswhich �x the endpoints, i.e., X (0) = X (�) = 0: Compute

�E (X) = �1

2

Z �

0

g ( _ ; _ ) dt

=1

2

@

@s

��s=0

Z �

0

g

�@�

@t;@�

@t

�dt

=

Z �

0

g

�Ds

@�

@t;@�

@t

�dt

=

Z �

0

g (DtX; _ ) dt

= g (X; _ )j�0 �Z �

0

g (X;Dt _ ) dt:

Remark 83 The notion DtX means DtX = r @@tX: It only depends on the

values along the curve : (Why?)

At a critical point for the energy, we must have �E (X) = 0 for all X; sowe get that Dt _ = 0 if the endpoints are �xed, i.e., if X (0) = X (�) = 0: Thisis the geodesic equation. Notice that if we restrict to a geodesic curve, (i.e.,Dt _ = 0) and �x the initial point (X (0) = 0), then

�E (X) = g (X (�) ; _ (�)) :

On a geodesic,d

dtj _ (t)j2 = 2g (Dt _ ; _ ) = 0

so has constant velocity. So along a geodesic, we have

E ( ) =1

2

Z �

0

j _ j2 ds = 1

2j _ j2 �

` ( ) =

Z �

0

j _ j ds = j _ j �

and so if is a geodesic, then

E ( ) =1

2�[` ( )]

2

9.4. VARIATIONS OF LENGTH AND THE DISTANCE FUNCTION 71

and so we have that if is the gives the distance d (p; x), then

E ( ) =1

2�d (p; x)

2:

If we want to compute a variation

d

dsd (p; x (s))

2

away from the cut locus (so variations in geodesics stay minimizing), we have

d

dsd (p; x (s))

2= 2��E (X) = 2�g

�dx

ds; _ (�)

�;

where X is a variation of geodesics (one must show that these exist, but theydo!) Note that we can always reparametrize so that � = 1, and then we get that��rd (p; x)2��2 = 4 j _ (1)j2 ;or

4d (p; x)2 jrd (p; x)j2 = 4 j _ (1)j2 = 4d (p; x)2

and sojrd (p; x)j = 1:

Now compute the second variation of energy when is a geodesic. We get

�2E (X;X) =@

@s

Z �

0

g (DtX; _ ) dt

= g (X; _ )j�0 �Z �

0

g (X;Dt _ ) dt:

�2E (X;X) =1

2

@2

@s2

��s=0

Z �

0

g

�@�

@t;@�

@t

�dt

=

Z �

0

g

�DsDs

@�

@t;@�

@t

�+ g

�Ds

@�

@t;Ds

@�

@t

�dt

=

Z �

0

g

�DsDt

@�

@s;@�

@t

�+ g

�Dt@�

@s;Dt

@�

@s

�dt

=

Z �

0

g

�DtDs

@�

@s;@�

@t

�+ g

�R

�@�

@s;@�

@t

�@�

@s;@�

@t

�+ g (DtX;DtX) dt:

= g (rXX; )j�0 +Z �

0

g (R (X; _ )X; _ ) + g (DtX;DtX) dt

The �rst term is zero if the variation is a variation of geodesics at the endpoints(or �xed endpoints). Note our only assumptions are this and that is a geodesic.


Let�s again parametrize our geodesic between 0 and 1 and �x (0) and varythe other endpoint x (s) so that X (t) = @�

@s (t; 0) �varies through geodesics.�Then

1

2

d2

ds2d (p; x (s))

2=

Z 1

0

g (R (X; _ )X; _ ) + g (DtX;DtX) dt:

Furthermore, we can take X (t) = t@x@s ; where@x@s is the parallel translation along

of @x@s :

Remark 84 The parallel translation of a vector v along a curve is the solutionto the system of ODE

Dtv = r 0v =@vk

@t+ �kij (

0)ivj = 0:

Then we get

1

2

d2

ds2d (p; x (s))

2=

Z 1

0

t2g

�R

�@x

@s; _

�@x

@s; _

�dt+ g

�@x

@s;@x

@s

�:

Thus if we sum over an orthonormal frame and assume that the sectional cur-vature is nonnegative, we have that

1

24d (p; x)2 � d:

And so1

24�f2�= f4f + jrf j2

and so

4d (p; x) � d� 1d (p; x)

Here�s another derivation: We now know that for a curve x (s) ;

1

2d (x0; x (s))

2= E ( s)

where s (t) are curves such that

Dt 0s = r 0s

0s = 0

for all s:We may parametrize all curves between 0 and 1: Furthermore, we knowthat if we consider the variation � (s; t) = s (t) ; then

@

@s

��s=0

s (t) = X (t) ;

X (0) = 0;

X (1) =dx

ds:

9.4. VARIATIONS OF LENGTH AND THE DISTANCE FUNCTION 73

E ( s) = E ( 0) + s�E 0 (X) +1

2s2�2E 0 (X;X) +O

�s3�;

where

�2E 0 (X;X) =d2

ds2E ( s) :

Thus the �rst derivative is

d

ds

��s=0

1

2d (x0; x (s))

2= lim

s!0

12d (x0; x (s))

2 � 12d (x0; x (0))

2

s

= �E 0 (X) :

The key fact is that under such a variation, we have

�E 0 (X) = g ( 00 (1) ; X (1)) = g

� 00 (1) ;

dx

ds(0)

�so we did not need to know much about X (s) in general!Now, to compute the second derivative, we look at

d2

ds2

��s=0

1

2d (x0; x (s))

2= lim

s!0

12d (x0; x (s))

2+ 1

2d (x0; x (�s))2 � 2 12d (x0; x (0))

2

s2

= �2E 0 (X;X) :

�2E (X;X) =1

2

@2

@s2

��s=0

Z 1

0

g

�@�

@t;@�

@t

�dt

=

Z 1

0

g

�DsDs

@�

@t;@�

@t

�+ g

�Ds

@�

@t;Ds

@�

@t

�dt

=

Z 1

0

g

�DsDt

@�

@s;@�

@t

�+ g

�Dt@�

@s;Dt

@�

@s

�dt

=

Z 1

0

g

�DtDs

@�

@s;@�

@t

�+ g

�R

�@�

@s;@�

@t

�@�

@s;@�

@t

�+ g (DtX;DtX) dt:

= g

�Ds

@�

@s;@�

@t

��10

+

Z 1

0

g (R (X; 0)X; 0) + g (DtX;DtX) dt

= �Z 1

0

K (X; 0) jXj2 j 0j2 +Z 1

0

g (DtX;DtX) dt

Since I can also choose the variation so that Ds@�@s (s; t = 1) = 0:

Fact: The vector �eld X = tv where v is the parallel transport of X (1) = @x@s

along the curve : This implies that

�2E (X;X) = �Z 1

0

K (X; 0) jXj2 j 0j2 +Z 1

0

g (DtX;DtX) dt

= �Z 1

0

K (X; 0) jXj2 j 0j2 +��@x@s

��2 :


Now if we take�� @x@s

��s=0

�� = 1; then we haved2

ds2

��s=0

1

2d (x0; x (s))

2 ��@x@s

��2 = 1if the sectional curvatures are positive. Now take normal coordinates at x: Atthe center of normal coordinates,

4f (0) =�

@

@x1

�2f + � � �

�@

@xd

�2f:

So, taking x (s) = expx�s @@xi

�; we conclude that

4�1

2d (x0; x)

2

�� d:

Furthermore, we have

4�1

2d (x0; x)

2

�= r � (d (x0; x)rd (x0; x))

= d (x0; x)4d (x0; x) + jrd (x0; x)j2

= d (x0; x)4d (x0; x) + 1:

Thus, we get

d (x0; x)4d (x0; x) + 1 � d

4d (x0; x) �d� 1d (x0; x)

:

9.5 Variations of the reduced distance

We will do the same thing to the reduced distance. Consider

L ( ) =Z �

0

p��j 0 (�)j2g(�) +Rg(�)

�d�:

Actually, we want to think of this as a functional on spacetime paths. A space-time path is a map

: [0; � ]!M � Iwhere I is an interval. We will only consider paths that look like

(s) = (~ (s) ; s) : (9.1)

We can naturally de�ne R ( (s)) = Rg(s) (~ (s)) :We then have the formulationof the functional as

L ( ) =Z �

0

p��j~ 0 (�)j2g(�) +R ( (�))

�d�:

9.5. VARIATIONS OF THE REDUCED DISTANCE 75

We will de�ne the reduced length as

`(0;x0) (�; x) =1

2p�inf fL ( ) : are of the form (9.1)g :

Note that the in�mum can also be considered as the in�mum over all paths ~ on M:Note that the tangent space of M � I splits as TM � TI and TI is spanned

by @@� : There is a notion of a horizontal vector �eld, which is a vector �eld

X such that d� (X) = 0: Since variations of paths of the form (9.1) must behorizontal vector �elds, we �rst consider the �rst variation of L with respect toa horizontal vector �eld X: Note that ~ 0 = 0 � @

@� is a horizontal vector �eld(but 0 is not).

Remark 85 Horizontal vector �elds onM�I are in one-to-one correspondencewith vector �elds onM: For this reason, we may abuse notation and use the samenotation for both vector �elds. In particular, ~ 0 will be considered both a vector�eld on M and a horizontal vector �eld on M � I:

We take a variation of � (s; �) such that

� (0; �) = (�) = (~ (�) ; �)

� (s; �) =�~� (s; �) ; �

�@

@s

��s=0

� (s; �) = X:

This is a horizontal variation. We get

�L (X) =@

@s

��s=0

�L (�) =Z �

0

p�

0@2*r @�@s

@~�

@�;@~�

@�

+g(�)

+r @�@sRg(�)

1A d�:

We do the same thing we did with the energy functional, �nding

�L (X) =Z �

0

p�

0@2*r @~�@�

@�

@s;@~�

@�

+g(�)

+r @�@sRg(�)

1A d�

��s=0

=

Z �

0

p�

0@2 dd�

*@�

@s;@~�

@�

+g(�)

� 4Rc @�

@s;@~�

@�

!� 2

*@�

@s;r @~�

@�

@~�

@�

+g(�)

+r @�@sRg(�)

1A d�

��s=0

=

Z �

0

p�

�2d

d�hX; ~ 0ig(�) � 4Rc (X; ~

0)� 2 hX;r~ 0~ 0ig(�) +rXRg(�)

�d�

since if X and Y are horizontal,

d

d�g (X;Y ) = 2Rc (X;Y ) + g (r~ 0X;Y ) + g (X;r~ 0Y ) :


Remark 86 In general, the formula is

d

d�g (X;Y ) =

@

@�g (X;Y ) +r~ 0g (X;Y ) ;

which has more terms since r~ 0X 6= 0; etc.

Remark 87 I used the notation of total derivative since there is the variationof the metric with respect to the time parameter of Ricci �ow and also thevariation with respect to 0: I have also used 0 instead of dot to denote derivativewith respect to � or �:

Now we need to take the dd� out, so we need that

d

d�

�2p� hX; ~ 0ig(�)

�=

1p�hX; ~ 0ig(�) + 2

p�d

d�hX; ~ 0ig(�) ;

so

�L (X) =Z �

0

d

d�

�2p� hX; ~ 0ig(�)

�� 1p

�hX; ~ 0ig(�)

+p��4Rc (X; ~ 0)� 2 hX;r~ 0~

0ig(�) +rXRg(�)�d�

= 2p� hX; ~ 0ig(�)

��0� 2p�

Z �

0

hG;Xig(�) d�

where G is the vector �eld

G (�) = r~ 0~ 0 +

1

2�~ 0 + 2Rc (~ 0)� 1

2rRg(�)

where Rc (X) is the vector �eld on M such that

hRc (X) ; Y i = Rc (X;Y )

for all for all vector �elds Y on M: Notice that G does not depend on thevariation X: Note that if is a minimizer with �xed endpoints (i.e., for allvariations X such that X (0) = X (�) = 0), then we must have that G = 0: Thisis the L-geodesic equation.

Problem 88 If we are in Euclidean space, what are the L-geodesics?

Now supposing there is a unique minimizer and that ` is a smooth function,we can consider variations through L-geodesics such that X (0) is �xed to get

�L (X) = 2p� hX; ~ 0ig(�) ;

which implies that

@

@s`(0;x0) (�; x (s)) =

�@x

@s; ~ 0�g(�)

9.5. VARIATIONS OF THE REDUCED DISTANCE 77

(recall that ` divides byp�). This can also be written as

r`(0;x0) = ~ 0:

Note that the reduced length also depends on time, so let�s compute the timederivative as well:

�L �d

d�

�=p��j~ 0 (�)j2g(�) +Rg(�)

�:

Note that this need not be zero since we are not minimizing in this direction.We �nd that

d

d�

�2p�`(0;x0)

�=p��j~ 0 (�)j2g(�) +Rg(�)

�:

Now, the total derivative decomposes as

d

d�`(0;x0) =

@

@�`(0;x0) +r~ 0`(0;x0);

and so

@

@�`(0;x0) (�; x) =

1

2

�j~ 0 (�)j2g(�) +Rg(�)

�� 1

2�`(0;x0) (�; x)�

1

2p�j~ 0 (�)j2g(�) :

In order to look at second derivatives of `(0;x0); we consider variations amongminimizers. In particular, we let be a minimizer with G = 0:We compute thesecond variation of L. It will be convenient to assume the variation is throughL-geodesics, that is

r @�@s

@�

@s= 0:

This implies that@2

@s2

��s=0

R (� (s; �)) = rXrXR

where rXrXR means the Hessian r2R (X;X) : Then we have,

�2L (X;X) =d2

ds2

��s=0

L (� (s; �))

=d

ds

��s=0

Z �

0

p�

0@2*r @�@s

@~�

@�;@~�

@�

+g(�)

+r @�@sRg(�)

1A d�

=

Z �

0

p�

0@2*r @�@sr @�

@s

@~�

@�;@~�

@�

+g(�)

+ 2

*r @�

@s

@~�

@�;r @�

@s

@~�

@�

+g(�)

+r @�@sr @�

@sRg(�) +r @�

@s

@�

@s(R)

1A d�

=

Z �

0

p��2 hrXrX~ 0; ~ 0ig(�) + 2 hrX~

0;rX~ 0ig(�) +rXrXRg(�)�d�

=

Z �

0

p�

0@2*r @�@sr @�

@s

@~�

@�;@~�

@�

+g(�)

+ 2

*r @�

@s

@~�

@�;r @�

@s

@~�

@�

+g(�)

+r @�@sr @�

@sRg(�) +r @�

@s

@�

@s(R)

1A d�


�L (X) =Z �

0

d

d�

�2p� hX; ~ 0ig(�)

�� 1p

�hX; ~ 0ig(�)

+p��4Rc (X; ~ 0)� 2 hX;r~ 0~

0ig(�) +rXRg(�)�d�

= 2p� hX; ~ 0ig(�)

��0� 2p�

Z �

0

hG;Xig(�) d�

Chapter 10

Problems

1) Show that if (M; g) is �-noncollapsed at x0 at the scale ofp� ; then it is

�-noncollapsed at x0 at all scales smaller thanp� :

2) Recall the functionals

F (M; g; f) =

Z �R+ jrf j2

�e�fdV

W (M; g; f; �) =

Z h��R+ jrf j2

�+ f � d

i(4��)

�d=2e�fdV

and their corresponding

� (M; g) = inf

�F (M; g; f) :

ZM

e�fdV = 1

�� (M; g; �) = inf

�W (M; g; f; �) :

ZM

(4��)�d=2

e�fe�fdV = 1

�:

By considering variations of the function f (with M; g; � �xed), show that theminimizers f� for � and f# for � satisfy the di¤erential equations

24f� � jrf�j2 +R = �;

��R+ 24f# � jrf#j2

�+ f# � d = �:

Hint: you must use Lagrange multipliers to enforce the constraint.3) Suppose (M; g) = ([0; a]� [0; b] = �; gflat) is a �at torus gotten by iden-

tifying the interval [0; a] � [0; b] : Find constants c� and c# such that f� = c�and f# = c# satisfy both the di¤erential equations and the constraint equations(the constants c� and c# should depend on the volume, which equals ab). Infact, you could do this for any closed manifold with R = 0:

4) On the same torus, suppose a � b: For which scalesp� is (M; g) a-

noncollapsed? How does this compare with the estimate one might get frompart 3?

79

80 CHAPTER 10. PROBLEMS

Then we have

W (M; g; f; �) =

Z h� jrf j2 + f � d

i(4��)

�d=2e�fdV

Now consider

� (M; g; �) = inf

�Z h� jrf j2 + f

i(4��)

�d=2e�fdV � d

�And so a minimizer satis�es

��R+ 24f � jrf j2

�+ f � d = �

If we restrict to R = 0; then

��24f � jrf j2

�+ f � d = �

Notice that if f (x) = c; thenZ(4��)

�d=2e�fdV = (4��)

�d=2e�cV (M)

and so the constraint is

(4��)�d=2

e�cV (M) = 1

c = logh(4��)

�d=2V (M)

i;

which satis�es

��24f � jrf j2

�+ f � d = log

h(4��)

�d=2V (M)

i� d = �

or

� = logab

�d=2� d� d

2log (4�)

�F =

Z ��Rh� jrf j2 h+ 2rf � rh

�e�fdV

=

Z ��Rh� jrf j2 h� 24f + 2 jrf j2

�e�fdV

�W =

Z h� (2rf � rh) + h� h

h��R+ jrf j2

�+ f � d

ii(4��)

�d=2e�fdV

= �Z h

hh��R+ 24f � jrf j2

�+ f � d

ii(4��)

�d=2e�fdV

Notes from Math 538: Ricci ⁄ow and the Poincare...

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