Notes from Math 538: Ricci Flow and the Poincare Conjecture

Notes from Math 538: Ricci ‡ow and the Poincare conjecture

David Glickenstein

Spring 2009 ii Contents

1 Introduction 1

2 Three-manifolds and the Poincaré conjecture 3 2.1 Introduction ...... 3 2.2 Examples of 3-manifolds ...... 3 2.2.1 Spherical manifolds ...... 4 2.2.2 Sphere bundles over S1 ...... 4 2.2.3 Connected sum ...... 5 2.3 Idea of the Hamilton-Perelman proof ...... 5 2.3.1 Hamilton’s…rst result ...... 6 2.3.2 2D case ...... 7 2.3.3 General case ...... 7

3 Background in di¤erential geometry 9 3.1 Introduction ...... 9 3.2 Basics of tangent bundles and tensor bundles ...... 9 3.3 Connections and covariant derivatives ...... 12 3.3.1 What is a connection? ...... 12 3.3.2 Torsion, compatibility with the metric, and Levi-Civita connection ...... 14 3.3.3 Higher derivatives of functions and tensors ...... 16 3.4 Curvature ...... 17

4 Basics of geometric evolutions 21 4.1 Introduction ...... 21 4.2 Ricci ‡ow ...... 24 4.3 Existence/Uniqueness ...... 25

5 Basics of PDE techniques 29 5.1 Introduction ...... 29 5.2 The maximum principle ...... 29 5.3 Maximum principle on tensors ...... 33

iii iv CONTENTS

6 Singularities of Ricci Flow 35 6.1 Introduction ...... 35 6.2 Finite time singularities ...... 35 6.3 Blow ups ...... 36 6.4 Convergence and collapsing ...... 38 6.5 -noncollapse ...... 39

7 Ricci ‡ow from energies 43 7.1 Gradient ‡ow ...... 43 7.2 Ricci ‡ow as a gradient ‡ow ...... 44 7.3 Perelman entropy ...... 50 7.4 Log-Sobolev inequalities ...... 52 7.5 Noncollapsing ...... 55

8 Ricci ‡ow for attacking geometrization 59 8.1 Introduction ...... 59 8.2 Finding canonical neighborhoods ...... 60 8.3 Canonical neighborhoods ...... 60 8.4 How surgery works ...... 63 8.5 Some things to prove ...... 63

9 Reduced distance 65 9.1 Introduction ...... 65 9.2 Short discussion of W vs reduced distance ...... 65 9.3 -length and reduced distance ...... 66 9.4 VariationsL of length and the distance function ...... 70 9.5 Variations of the reduced distance ...... 74

10 Problems 79 Preface

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

v vi PREFACE Chapter 1

Introduction

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

1 2 CHAPTER 1. INTRODUCTION Chapter 2

Three-manifolds and the Poincaré conjecture

2.1 Introduction

This lecture is mostly taken from Tao’slecture 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 this proof in varying amounts of careful detail. All manifolds will be assumed to be without boundary unless otherwise speci…ed. The Poincaré conjecture is this:

Theorem 1 (Poincare conjecture) Let M be a compact 3-manifold which is connected 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’sgeometrization conjecture, which is quite a bit more complicated, but is roughly the following:

Theorem 3 (Thurston’sgeometrization conjecture) Every 3-manifold M can be cut along spheres and 1-essential tori such that each piece can be given one of 8 geometries (E3; S3; H3; S2 R; H2 R; SL (2; R) ; Nil; Sol). We may go into this conjecture a little moref if we have time, but certainly elements of this will come up in the process of these lectures. We will take a quick 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¤eomorphism, 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 in several 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 a point, and so S3 is simply connected. One can also look at the …rst de…nition and see 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 structure of the unit quaternions, and it is not too hard to see that this group is isomorphic to 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 action is nontrivial, then these new spaces are not simply connected. One can see this in several ways. The direct method is that since there must be g and x S3 such that gx = x: A path from x to gx in S3 descends to a loop2 in S3=2: If there is a homotopy6 of that loop to a point, then one can lift the homotopy to a homotopy H : [0; 1] [0; 1] S3 such that H (t; 0) = (t) : But then, looking at ! H (1; s) ; we see that H (1; 0) = gx and H (1; 1) = x and H (1; s) = g0x for some g0 for all s: But since is discrete, this is impossible. A more high level approach2 shows that the map S3 S3= is a covering map, and, in fact, the !3 universal covering map and so 1 S = = if acts e¤ectively. Hence each of these spaces S3= are di¤erent manifolds than S3: They are called spherical manifolds. The elliptization conjecture states that spherical manifolds are the only manifolds 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 as S2 [0; 1] with S2 0 identi…ed with S2 1 by a homeomorphism. Recall that a homeomorphism f g f g : S2 S2 ! induces an isomorphism on homology (or cohomology),

2 2 : H2 S H2 S ! and since 2 H2 S = Z 2.3. IDEA OF THE HAMILTON-PERELMAN PROOF 5 there are only two possibilities for : In fact, these classify the possible maps up to continuous deformation, and so there is an orientation preserving and an orientation 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 ! 1 S = 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 simply connected, this map is an isomorphism. Note that these manifolds are thus not simply-connected.

2.2.3 Connected sum One can also form new manifolds via the connected sum operation. Given two 3-manifolds, M and M 0; one forms the connected sum by removing a disk from each manifold and then identifying the boundary of the removed disks. We denote this as M#M 0: Recall that in 2D, all manifolds can be formed from the sphere 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 theorem of Kneser). We call such a decomposition a prime decomposition and such manifolds Mj prime manifolds. The proof is very similar to the fundamental theorem of arithmetic which gives the prime decomposition of positive integers.

Proposition 4 Suppose M and M 0 are connected manifolds of the same dimension. 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 simply connected.

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…ned more precisely in successive lectures. 6CHAPTER 2. THREE-MANIFOLDS AND THE POINCARÉ CONJECTURE

Any smooth manifold can be given a Riemannian metric, denoted gij or g ( ; ) ; which is essentially an inner product (i.e., symmetric, positive-de…nite bilinear form) at each tangent space which varies smoothly as the basepoint of the tangent space changes. The Riemannian metric allows one to de…ne angles between two curves and also to measure lengths of (piecewise C1-) curves by integrating the tangent vectors of a curve. That is, if : [0; a] M is a curve, we can calculate its length as ! a ` ( ) = g (_ (t) ; _ (t))1=2 dt; Z0 where _ (t) is the tangent to the curve at t: A Riemannian metric induces a metric space structure on M; as the distance between two points is given by the in…mum of lengths of all piecewise smooth curves from one point to the other. It is 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 partial di¤erential equation de…ned by @ gij = 2Rij = 2 Rc (gij) @t 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 that the Ricci ‡ow equation is a partial di¤erential equation on the Riemannian metric. In coordinates, it roughly looks like 2 @ k` @ gij = 2g gij + F (gij; @gij) @t @xk@x` k` where g is the inverse matrix of gij and F is a function depending only on the metric 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 more uniform. It was shown in Hamilton’slandmark 1982 paper that

Theorem 5 (Hamilton) Given a Riemannian 3-manifold (M; g0) with positive Ricci curvature, then the Ricci ‡ow with g (0) = g0 exists on a maximal time interval [0; t ): Furthermore, the Ricci curvature of the metrics g (t) become increasingly 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 to the 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 manifold converges under a renormalized Ricci ‡ow (renormalized by rescaling the metric and rescaling time) to a constant curvature metric. This is primarily due to Hamilton, with one case …nished by B. Chow. It is possible to use this method to prove the uniformization theorem, which states that any compact, orientable Riemannian manifold can be conformally deformed to a metric with constant curvature. (The original proofs of Hamilton and Chow use the uniformization theorem, but a recent article by Chen, Lu, and Tian shows how to avoid 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 even in the case of a sphere if the Ricci curvature is not positive. An example is the so-called neck pinch singularity. Hamilton’sidea was to do surgery at these singularities, then continue the ‡ow and continue to do this until no more surgeries are necessary. Perelman’s work describes what happens to the Ricci ‡ow near a singularity and also how to perform the surgery. The new ‡ow is called Ricci Flow with surgery. The main result of Perelman is the following.

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

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

2. (Ricci ‡ow) If I is any connected component of [0; ) T (and thus an interval), then t (M (t) ; g (t)) is the Ricci ‡ow on1I (youn can close this interval on the left! endpoint if you wish).

3. (Topological compatibility) If t T and " > 0 is su¢ ciently small, then we know the topological relationship2 M (t ") and M (t) : 4. (Geometric compatibility) For each t T; the metric g (t) on M (t) is related to a certain limit of the metric 2g (t ") on M (t ") by a certain surgery procedure.

Note, we can express the topological compatibility more precisely. We have that M (t ") is homeomorphic to the connected sum of …nitely many connected components of M (t) together with a …nite number of spherical space forms 3 3 (spherical manifolds), RP #RP , and S2 S1: Furthermore, each connected component of M (t) is used in the connected sum decomposition of exactly one component of M (t ") : 8CHAPTER 2. THREE-MANIFOLDS AND THE POINCARÉ CONJECTURE

3 3 Remark 7 The case of RP #RP is interesting in that it is apparently the only nonprime 3-manifold which admits a geometric structure (i.e., is covered by a model geometry; it is a quotient of S2 R; this does not contradict our argument above because it is not a sphere bundle over S1). I have seen this mentioned several 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 will avoid in this class.

The existence needs something more to show the Poincaré conjecture. One needs that the surgeries are only discrete and that the ‡ow shrinks everything in …nite time.

Theorem 9 (Discrete surgery times) Let t (M (t) ; g (t)) be a Ricci ‡ow with surgery starting with an orientable manifold! M (0) : Then the set T of surgery 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-manifold which is simply connected and let t (M (t) ; g (t)) be an associated Ricci ‡ow with surgery. Then M (t) is empty for! su¢ ciently large t:

With these theorems, one can conclude the Poincaré conjecture in the following way. Given M a simply connected, connected, compact Riemannian manifold, associate a Ricci ‡ow with surgery. It has …nite extinction time, and hence …nite surgery times. Now one can use the topological decomposition to work backwards and build the manifold backwards, which says that the manifold M is the connected sum of …nitely many spherical space forms, copies 3 3 of RP #RP , and S2 S1: But since M is the simply connected, everything in the connected sum must be simply connected, and hence every piece of the connected sum must be simply connected, so M must be a sphere. Chapter 3

Background in di¤erential geometry

3.1 Introduction

We will try to get as quickly as possible to a point where we can do some geometric 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 bundles

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

n n n TM = (Ui R ) = where for (x; v) Ui R ; (y; w) Uj R we i 2 2 1 1 have (x;F v) (y; w) if and only i¤ y = j (x) and w = d j (v) : i i x

TpM = paths :( "; ") M such that (0) = p = where if f ! g (i )0 (0) = (i )0 (0) for every i such that p Ui:TM = TpM: 2 p M 2 F TpM to be the set of derivations of germs at p; i.e., the set of linear functionals X on the germs at p such that X (fg) = X (f) g (p) + f (p) X (g) for germs f; g at p: T M = TpM: p M 2 F On can de…ne the cotangent bundle by essentially taking the dual of TpM at each point, which we call TpM; and taking the disjoint union of these to get the cotangent 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 back rather than pushes forward. Both of these bundles are vector bundles. One can also take a tensor bundle of two vector bundles by replacing the …ber over a point by the tensor product of the …bers over the same point, e.g.,

TM T M = TpM T M : p p M G2 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 between bilinear 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 @ ;:::; @ for T M and dual basis @x1 p @xn p p 1 n dx ; : : : ; dx for T M: The generalization of the …rst de…nition above gives p jp p the idea of how one considers the trivializations of the bundle in a coordinate patch, and how the patches are linked together. Speci…cally, if x and y give di¤erent coordinates, for a point on the tensor bundle, one has

ij k @ @ @ a b c Tab c (x) i j k dx dx dx @x @x @x a b c ij k @y @y @y @x @x @x @ @ @ = T (x (y)) dy dy dy ; ab c @xi @xj @xk @y @y @y @y @y @y where technically everything should be at p (but as we shall see, one can consider this for all points in the neighborhood and this is considered as an equation of sections). Recall that a section of a bundle : E B is a function f : B E such that f is the identity on the base manifold!B: A local section may! only be 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). On @ a tensor bundle, sections are called tensors. Note that the set of @xi form a basis for the vector …elds in the coordinate x, and dxi form a basis for the local 1-forms in the coordinates. Sections in general are often written as (E) or as C1 (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

ij k Tabc : Note that if we change coordinates, we have a di¤erent representation T of the same tensor. The two are related by a b c ij k @y @y @y @x @x @x T = T (x (y)) : ab c @xi @xj @xk @y @y @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 (sections of this bundle are called di¤erential forms). In particular, we have the Riemannian 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 M

T M) which is

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

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

i j Often, we will denote the metric as gij; which is shorthand for gijdx dx ; i j 1 i j j i where dx dx = 2 dx dx + dx dx : Note that if gij = ij (the Kronecker delta) then i j 1 2 n 2 ijdx dx = dx + (dx ) : One can invariantly de…ne a trace of an endomorphism (trace of a matrix) which is independent of the coordinate change, since

n @xa @y T a = T a @y @xa a=1 a X X

= T a X

= T :

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

a Ta = 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 the Riemannian metric, we can use it to “lower an index”and then trace, so we get

ab a T gba = Ta :

In order to raise the index, we need the dual to the Riemannian metric, which ab ab a ab is g ; de…ned such that g gbc = c (so g is the inverse matrix of gab). Then we can use gab to raise indices and contract if necessary. Occasionally, extended Einstein convention is used, where all repeated indices are summed with the understanding that the Riemannian metric is used to raise or lower indices when necessary, e.g., ab Taa = Tabg : Since often we will be changing the Riemannian metric, it becomes important to understand that the metric is there when extended Einstein is used. 12 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY

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. Why do we need a new way to di¤erentiate vector …elds? Here is the idea. Suppose we want to give a notion of parallel vectors. In Rn; we know that if we take vector …elds with constant coe¢ cients, those vectors are parallel at di¤erent @ @ @ @ points. 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¢ cients of the vector …eld are constant (not functions of x1 and x2). However, this notion is not invariant under a change of coordinates. Suppose we consider the 2 new coordinates y1; y2 = x1; x2 away from x2 = 0 (where it is not a di¤eomorphism). Then the vector …eld in the new coordinates is @yi @ @yj @ @ @ @ @ + 2 = + 4x2 = + 4 y2 : @x1 @yi @x2 @xj @y1 @y2 @y1 @y2 p 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 need a notion of parallel vector …eld that is independent of coordinate changes (or covariant). Remember that we want to generalize the notion that a vector …eld has i @ constant coe¢ cients. Let X = X @xi be a vector …eld in a coordinate patch. @Xi Roughly speaking, we want to generalize the notion that @xj = 0 for all i and @ @ j: The problem occurred because @x1 @x1 is di¤erent in di¤erent coordinates. Thus we need to specify what this is. Certainly, since @ is a basis, we must @xi get a linear combination of these, so we take

@ k @ i = r @xj ij @xk

k for some functions ij: These symbols are called Christo¤el symbols. To make sense on a vector …eld, we must have

j @ i (X) = i X r r @xj @Xj @ @ = + Xjk @xi @xj ij @xk @Xk @ = + Xjk : @xi ij @xk Notice the Leibniz rule (product rule). One can now de…ne for any vector i @ r Y = Y @xi by i Y X = Y i @ X = Y ( iX) : r r @xi r 3.3. CONNECTIONS AND COVARIANT DERIVATIVES 13

This action is called the covariant derivative. k One now de…nes ij in such a way that the covariant derivative transforms appropriately under change of coordinates. This gives a global object called a connection. The connection can be de…ned axiomatically as follows.

De…nition 12 A connection on a vector bundle E B is a map ! : (TB) (E) (E) r !

(X; ) X ! r satisfying:

Tensoriality (i.e., C1 (B)-linear) in the …rst component, i.e., fX+Y = r f X + Y for any function f and vector …elds X;Y r r

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

R-linear in the second component, i.e., X (a + ) = a X () + X ( ) r r r for a R. 2 We will consider connections primarily on the tangent bundle and tensor bundles. Note that a connection on TM induces connections on all tensor bundles (also denoted ) in the followingr way: r

For a function f and vector …eld X; we de…ne X f = Xf r For vector …elds X;Y and dual form !; we use the product rule to derive

X (! (Y )) = X (! (Y )) = ( X !)(Y ) + ! ( X Y ) r r r and thus

( X !)(Y ) = X (! (Y )) ! ( X Y ) : r r In particular, the Christo¤el symbols for the connection on T M are the negative of the Christo¤el symbols of TM; i.e.,

j j k @ dx = ikdx r @xi where k are the Christo¤el symbols for the connection on TM: ij r For a tensor product, one de…nes the connection using the product rule, e.g.,

X (Y !) = ( X Y ) ! + Y X ! r r r for vector …elds X;Y and 1-form !: 14 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY

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

@ @x~` @ @x~` @ @x~k @x~` @ @ = @x~k @ = + @ i j j ` i j ` i j k ` r @x @x r @xi @x~k @x @x~ @x @x @x~ @x @x r @x~ @x~ which means that @x~p @x~` @xk @x~` @xk k = ~m + : ij p` @xi @xj @x~m @xi@xj @x~` One …nal comment. Recall that we motivated the connection by considering parallel vector …elds. The connection gives us a way of taking a vector at a point and translating it along a curve so that the induced vector …eld along the curve is parallel (i.e., _ X = 0 along ). This is called parallel translation. Parallel vector …eldsr allow one to rewrite derivatives in coordinates; that is, i @ if X = X @xi is parallel, then @Xi = Xki : @xj jk 3.3.2 Torsion, compatibility with the metric, and Levi- Civita connection There is a unique metric associated with the Riemannian metric, called the Riemannian 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 the connection to behave well with respect to di¤erentiating orthogonal vector …elds. Being compatible with the metric is the same as

X (g (Y;Z)) = g ( X Y;Z) + g (Y; X Z) : r r r

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

(X;Y ) = X Y Y X [X;Y ] r r 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

k = k k ; ij ij ji 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 x along 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 will reach in two ways: (1) go to x + tV and then ‡ow along the parallel translation of W for a distance t and (2) go to x + tW and then ‡ow along the parallel translation of V for a distance t: Note that using method (1), we get that the point is

@ (x + tV + sW ) + t (x + tV + sW ) + O t3 js=0 @s s=0 @ @ = x + tV + tW + t2 V + O t3 = x + tV + tW t2V iW jk + O t3 : @s ji @xk s=0

Note that using method (2), we get instead @ x + tV + tW t2W iV jk + O t3 ; ji @xk

3 k k Thus this vector is x+t (V + W ) up to O t only if ji = ij: 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 M; k 2 there are coordinates x around p such that ij (p) = 0:

k Proof. Suppose one can always …nd coordinates such that ij (p) = 0: Then k clearly at that point, ij = 0: However, since the torsion is a tensor, we can calculate 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

x~i (q) = xi (q) xi (p) + i (p) xj (q) xj (p) xk (q) xk (p) : jk Then notice that

@x~i = i + i (p) k x` x` (p) + i (p) xk xk (p) ` @xj j k` j k` j and so @x~i (p) = i : @xj j Thus x~ is a coordinate patch in some neighborhood of p: Moreover, we have that @2x~i = i (p) : @xj@xk jk 16 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY

One can now verify that at p; @x~p @x~` @xk @x~` @xk k (p) = ~m + ij p` @xi @xj @x~m @xi@xj @x~` ~k k = ij (p) + ij (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 derive a formula for it. In coordinates, one …nds that the Riemannian connection has the following Christo¤el symbols

k 1 k` @ @ @ = g gj` + gi` gij : ij 2 @xi @xj @x` One can easily verify that this connection has the properties expressed. Note that the gj` in the formula, etc. are not the tensors, but the functions. This k is not a tensor equation since ij is not a tensor. Also note that it is very @ @ important 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 take higher derivatives. Note that one can take the derivative of a function without a connection, and it is de…ned as

df = f r df (X) = X f = X (f) r @f df = dxi: @xi One can also raise the index to get the gradient, which is @ @f @ grad (f) = if = gij : r @xi @xj @xi However, to take the next derivative, one needs a connection. The second derivative, or Hessian, of a function is

Hess (f) = 2f = df r r 2 i f = ( idf) dx r r @f j i = i dx dx r @xj @2f @f = dxj j dxk dxi @xi@xj @xj ik @2f @f = k dxj dxi: @xi@xj @xk ij 3.4. CURVATURE 17

Often one will write the Hessian as 2 2 @ f k @f f = i jf = : rij r r @xi@xj ij @xk Note that if the connection is symmetric, then the Hessian of a function is ij 2 symmetric in the usual sense. The trace of the Hessian, f = g ijf; is called the Laplacian, and we will use it quite a bit. 4 r 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 compute the acceleration, _ ;_ we need the connection (one also sometimes sees the equivalent notationr 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 T; which has an extra down index. r 3.4 Curvature

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

R : (TM) (TM) (E) (E) ! R (X;Y ) = X Y Y X : r r r r r[X;Y ] We will consider the curvature of the Riemannian connection on the tangent bundle. One can easily see that in coordinates, the curvature is a tensor denoted as @ @ ` @ i j j i = R r r @xk r r @xk ijk @x` which gives us that

` @ ` @ @ ` @ ` m @ @ ` @ ` m @ i j = + r jk @x` r ik @x` @xi jk @x` jk i` @xm @xj ik @x` ik j` @xm @ @ @ = ` ` + m ` m` @xi jk @xj ik jk im ik jm @x` So the curvature tensor is @ @ R` = ` ` + m ` m` : ijk @xi jk @xj ik jk im ik jm Often we will lower the index, and consider instead the curvature tensor

m Rijk` = Rijkgm`: The Riemannian curvature tensor has the following symmetries:

Rijk` = Rjik` = Rij`k = Rk`ij (These imply that R can be viewed as a self-adjoint (symmetric) operator mapping 2-forms to 2-forms if one raises the …rst two or last two indices). 18 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY

(Algebraic Bianchi) Rijk` + Rjki` + Rkij` = 0: (Di¤erential Bianchi) iRjk`m + jRki`m + kRij`m: r r r

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 will ` sometimes refer to this tensor as Rm : The tensor Rijk is usually denoted by R (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 of the 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 Ricci curvature of the sphere should be positive de…nite.

k Remark 17 Note that ij involved …rst derivatives of the metric, so Riemannian curvature 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

` `m Rij = R`ij = R`ijmg :

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

De…nition 19 The scalar curvature R is the function

ij R = g Rij

De…nition 20 The sectional curvature of a plane spanned by vectors X and Y is given by R (X;Y;Y;X) K (X;Y ) = : 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 curvature tensor, i.e., if one can calculate all sectional curvatures, then one can calculate the entire tensor. 2. The sectional curvature K (X;Y ) is the Gaussian curvature of the surface generated 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 volumes of 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 does determine 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 nonnegative curvature.

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; i j 2. nonnegative Ricci curvature at x if Rc (X;X) = RijX X 0 for every vector X TxM; 2 3. nonnegative sectional curvature if R (X;Y;Y;X) = g (R (X;Y ) Y;X) 0 for all vectors X;Y TxM; 2 4. nonnegative Riemann curvature (or nonnegative curvature operator) if 2 ij k` Rm 0 as a quadratic form on (M) ; i.e., if Rijk`! ! 0 for i j all 2-forms ! = !ijdx dx (where the raised indices are done using the metric g). ^

It is not too hard to see that 4 implies 3 implies 2 implies 1: Also, in 3 dimensions, 3 and 4 are equivalent. In dimension 4 and higher, these are all distinct. 20 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY Chapter 4

Basics of geometric evolutions

4.1 Introduction

This lecture roughly follows Tao’sLecture 1. We will talk in general about ‡ows or 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 …xed Riemannian manifold M. There are more ingenious ways to de…ne such a ‡ow using spacetimes (called generalized Ricci ‡ows). However, at present I do not think that they give a signi…cant savings over the more classical idea, since one still 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 @ g (t + dt) g (t) g (t) = lim @t dt 0 dt ! since g (t) and g (t + dt) are both sections of the same vector bundle, so the di¤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 @ g =g _ @t ij ij for some prescribed g_ij: The evolution of the metric induces an evolution of the metric on the cotangent bundle, using @ @ gijg = i @t jk @t j @ ij ik `j g = g g_k`g : @t

21 22 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS

The Riemannian connection is also changing if the metric is changing. Thus for a …xed vector …eld X; we have

j @ @ @X @ k j @ iX = + X @tr @t @xi @xj ij @xk @ = Xj_ k : ij @xk We can use the fact that the connection is torsion-free and compatible with the _ k metric to derive the formula for ij:

@ @ @ ` ` 0 = ( igjk) = gjk g`k gj` @t r @t @xi ij ik ` ` = ig_jk _ g`k _ gj`; r ij ik and 0 = _ k _ k ij ji _ k so we can solve for ij as

` ` ig_jk = _ g`k + _ gj` r ij ik ` ` jg_ki = _ g`k + _ gk` r jk ji ` ` kg_ij = _ g`j + _ gi` r ki kj to get

k 1 k` _ = g ( ig_j` + jg_i` `g_ij) : (4.1) ij 2 r r r

Remark 23 This mimics the proof of the formula for the Riemannian connection given that it is torsion-free and compatible with the metric. There are other ways to derive this formula, for instance by computing in normal coor- k @ k dinates and using the fact that although ij is not a tensor, @t ij comes from the di¤erence of two connections and is thus a tensor. We will use this method below.

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

` @ ` @ ` @ R = i j : ijk @x` r jk @x` r ik @x` @ ` _ ` Since we are interested in the derivative of a tensor, @t Rijk = Rijk; we can compute this in any coordinate system we want. Recall that there is a coordinate system around p such that all Christo¤el symbols vanish at p: Doing this reduces 4.1. INTRODUCTION 23 the equation to

` @ @ ` @ ` @ R_ = i j ijk @x` @t r jk @x` r ik @x` @ @ @ @ @ = ` ` @t @xi jk @x` @xj ik @x` @ @ @ @ = _ ` _ ` @xi jk @x` @xj ik @x` ` @ ` @ = i_ j_ : r jk @x` r ik @x` _ k This last piece is tensorial (recall that ij is a tensor), and thus only depends on the point, not the coordinate patch, so we must have that

` ` ` R_ = i_ j_ : ijk r jk r ik We can now use the (4.1) to get

` 1 `m 1 `m R_ = i g ( jg_km + kg_jm mg_jk) j g ( ig_km + kg_im mg_ik) ijk r 2 r r r r 2 r r r 1 `m = g ( i jg_km + i kg_jm i mg_jk j ig_km j kg_im + j mg_ik) 2 r r r r r r r r r r r r 1 `m = g ( i jg_km j ig_km + i kg_jm i mg_jk j kg_im + j mg_ik) 2 r r r r r r r r r r r r 1 `m p p = g ( R g_mp R g_kp + i kg_jm i mg_jk j kg_im + j mg_ik): 2 ijk ijm r r r r r r r r _ _ i We can take the trace Rjk = Rijk to get

1 im p p R_ jk = g ( R g_mp R g_kp + i kg_jm i mg_jk j kg_im + j mg_ik) 2 ijk ijm r r r r r r r r 1 im p 1 pq 1 im 1 im = g R g_mp + g Rjpg_kp g i mg_jk g j kg_im 2 ijk 2 2 r r 2 r r 1 im p p + g ( k ig_jm R g_pm R g_jp + j mg_ik) 2 r r ikj ikm r r im p 1 pq 1 pq 1 im = g R g_mp + g Rjpg_kp + g Rkqg_jp g i mg_jk ijk 2 2 2 r r 1 im 1 im g j kg_im + g ( k ig_jm + j mg_ik) 2 r r 2 r r r r 1 1 im 1 im = Lg_jk g j kg_im + g ( k ig_jm + j mg_ik) 24 2 r r 2 r r r r where

im im p pq pq Lg_jk = g i mg_jk + 2g R g_mp g Rjpg_kp g Rkqg_jp 4 r r ijk is the Lichnerowitz Laplacian (notice only the …rst term has two derivatives of g_jk). (Note: I think that T. Tao has an error in this formula with the sign of the last term of R_ jk: 24 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS

Finally, we may take a trace to get jk pj qk R_ = g R_ jk g g_pqg Rjk pq jk im jk = g g Rjpg_kp g i mg g_jk r r im jk + g g k ig_jm r r = Rc; g_ trg (_g) + div divg _ h i 4 where

k` (div h) = g kh`j j r 4.2 Ricci ‡ow

Note that in the evolution of Ricci curvature, if one considers g_ = 2 Rc; one gets

1 im 1 im g j kg_im + g ( k ig_jm + j mg_ik) 2 r r 2 r r r r im im im = g j kRim g k iRjm g j mRik r r r r r r im im = j kR g j mRik g k iRjm r r r r r r Note that the di¤erential Bianchi identity implies jm k` 0 = g g ( iRjk`m + jRki`m + kRij`m) r r r jm k` = iR g jRim g kRij`m r r r so jm iR = 2g jRim r r so

1 im 1 im 1 1 g j kg_im+ g ( k ig_jm+ j mg_ik) = j kR j kR k jR = 0: 2 r r 2 r r r r r r 2r r 2r r Thus under Ricci ‡ow, @ Rc = L Rc : @t 4 Furthermore, We see that

@R i` jk = 2 Rc; Rc + 2 R 2g g i jRk` @t h i 4 r r i` = 2 Rc; Rc + 2 R g i `R h i 4 r r = R + 2 Rc 2 4 j j The important notion to get right now is that this looks very much like a heat equation with a reaction term. We will see how to make use of this in the near future. 4.3. EXISTENCE/UNIQUENESS 25

4.3 Existence/Uniqueness

Note that the Ricci ‡ow equation,

@ g = 2 Rc @t is a second order partial di¤erential equation, since the Ricci curvature comes from 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 linearization of this operator in order to …nd the principle symbol (which is basically the coe¢ cients of the linearization of the highest derivatives). Analysis of the principle symbol will often allow us to determine that a solution exists for a short time. Here is the meta-theorem for existence of parabolic PDE: Meta-Theorem (imprecise): A semi-linear PDE of the form

@u @2u aij (x; t) + F (x; t; u; @u) = 0: @t @xi@xj on a compact manifold has a solution with initial condition u (x; 0) = f (x) ij 2 if there exists > 0 such that a ij (this condition is called strict parabolicity) for t close to 0. Similarly, if wej allowj aij to depend on u (making the equation quasilinear, the same is true if we look at the linearization (which is 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 it satis…es the inequality.

@ Remark 25 We can replace @xi with i since the di¤erence has fewer derivatives. r

Remark 26 One should be able to prove a coordinate independent version, but this is not usually done. All theory is based on theory of di¤erential equations on 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 roughly like @2 @ @ G + @ G + @ G v Hij @xi@xj Vi @xi u where G = G (x; t; u; V; H) and the operator is evaluated at some u (which is where it has been linearized. Notice this now gives a semilinear PDE. The principle symbol is @Hij Gij: 26 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS

Example 28 Note that the equation @u @2 @2 @2 = u + u = ij @t (@x1)2 (@x2)2 @xi@xj is already linear. For an equation like @u @2u = u2 ; @t @x2 the linearization is @v @2v @2u = u2 + 2u v; @t @x2 @x2 thus the principal symbol is u2 which is positive if u > 0: Now, the Ricci operator is an operator on sections, not just functions, so how do we make sense of the kind of result given above. We can make a similar de…nition in terms of the linearization, but now the principle symbol is a map from sections of the symmetric 2-tensor bundle to itself. What we need is that for any = 0; the principle symbol is a linear isomorphism. 6 Recall that the linearization of Rjk is

1 im 1 im 1 im R_ jk = g i mg_jk g j kg_im + g ( k ig_jm + j mg_ik) 2 r r 2 r r 2 r r r r so the principle symbol of 2Rjk is im im im ^ [D Rc] ()(h) = g imhjk + g jkhim g (kihjm + jkhik): 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 assume that at a point gij = ij: Then we see that

1 1 1 1 ^ [D Rc] ()(h) = hjk + (h11 + + hnn) ( hj1 + h1k): jk j k k j And so the matrix for the symbol gives

h22 + + hnn 0 0 0 0 . 1 . h B C B 0 C B C @ A where 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 in the kernel). Now we will see how to overcome this issue. Rewrite the linearization as

1 im 1 im 1 im R_ jk = g i mg_jk g j kg_im + g ( k ig_jm + j mg_ik) 2 r r 2 r r 2 r r r r 1 im 1 = g i mg_jk + kVj + jVk 2 r r 2r r 4.3. EXISTENCE/UNIQUENESS 27 if im 1 im Vj = g ig_jm j g g_im : r 2r i ij The last term is equal to the Lie derivative LV gjk (where V = V = g Vj; often # denoted V ) and so we get that the linearization of 2Rjk is im g i mg_jk LV gjk: r r Lie derivatives arise from changing by di¤eomorphisms, i.e., if t are di¤eomorphisms such that d (x) = X (x) dt t and 0 is the identity (i.e., t is the ‡ow of X), then d g = L g : dt t 0 X 0 t=0

One can pretty easily see that if we take the vector …eld V as above, we can look at the ‡ow t and tg (t) and we will see that g~ (t) = tg (t) evolves by @ g~ = 2 Rc (~g) + LV g~ @t and the linearization is im g i mg_jk: r r This is like looking at an equation roughly like

@ im h = g i mhjk; @t r r 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 the modi…ed Ricci ‡ow (the equation above on g~) has a unique solution. One can then 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 a time T > 0 and Riemannian metrics g (t) on M for each t [0;T ) such that which satisfy the initial value problem 2 @ g = 2 Rc (g) @t g (0) = g0: Moreover, given the initial condition, g (t) are uniquely determined, and there is a maximal such T:

Remark 30 One can also show that this is true for complete manifolds with bounded curvature Rm ; which was done by Shi. However, the proof is much more di¢ cult on noncompactj j manifolds. 28 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS Chapter 5

Basics of PDE techniques

5.1 Introduction

This section will roughly follow Tao’slecture 3. We will look at some basic PDE techniques and apply them to the Ricci ‡ow to obtain some important results about preservation and pinching of curvature quantities. The important fact is that the curvatures satisfy certain reaction-di¤usion equations which can be studied with the maximum principle.

5.2 The maximum principle

Recall that if a smooth function u : U R where U Rn has a local minimum ! at x0 in the interior of U; then @u (x ) = 0 @xi 0 @2u (x0) 0 @xi@xj where the second statement is that the Hessian is nonnegative de…nite (has all nonnegative eigenvalues). The same is true on a Riemannian manifold, replacing regular derivatives with covariant derivatives.

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

iu (x0) = 0 r i ju (x0) 0 r r ij u (x0) = g (x0) i ju (x0) 0: 4 r r @u Proof. In a coordinate patch, the …rst statement is clear since iu = @xi : The second statement is that the Hessian is positive de…nite. Recallr that in

29 30 CHAPTER 5. BASICS OF PDE TECHNIQUES coordinates, the Hessian is

2 @ u k @u i ju = ; r r @xi@xj ij @xk but at a minimum, the second term is zero and the positive de…niteness follows from the case in Rn: The last statement is true since both g and the Hessian are 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 manifolds for t [0;T ]: Let u : [0;T ] M R be a C2 function such that 2 ! u (0; x) 0 for all x M: Also let A R. Then exactly one of the following is true: 2 2 1. u (t; x) 0 for all (t; x) [0;T ] M; or 2

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

u (t0; x0) < 0

iu (t0; x0) = 0; r u (t0; x0) 0; 4g(t0) @u (x ; t ) < 0: @t 0 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 M such that u (t0; x0)+"t0 = 0: 2 At the …rst such time, we must have that x0 is a spatial minimum for this function, and thus

u (t0; x0) = "t0 < 0 u (t0; x0) = 0 r u (t0; x0) 0 4 @u (t0; x0) " < 0: @t 5.2. THE MAXIMUM PRINCIPLE 31

Corollary 34 Let (M; g (t)) be a smooth family of compact Riemannian manifolds for t [0;T ]: Let u; v : [0;T ] M R be C2 functions such that 2 ! u (0; x) v (0; x) for all x M: Also let A R. Then exactly one of the following is true: 2 2 1. u (t; x) v (t; x) for all (t; x) [0;T ] M; or 2 2. There exists a (t0; x0) (0;T ] such that all of the following are true: 2

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

iu (t0; x0) = iv (t0; x0) ; r r u (t0; x0) v (t0; x0) ; 4g(t0) 4g(t0) @u @v (x0; t0) < (t0; x0) + A [u (t0; x0) v (t0; x0)] : @t @t

At Proof. Replace u with e (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 u + u + F (t; u) @t 4g(t) rX(t) and v is a subsolution of the same equation, i.e., @v u + v + F (t; v) @t 4g(t) rX(t) for all (t; x) [0;T ] M; where X (t) is a vector …eld for each t and F (t; w) is Lipschitz in 2w; i.e., there is A > 0 such that

F (t; w) F (t; w0) A w w0 : j j j j Then u (t; x) v (t; x) for all t [0;T ] : 2 Proof. Consider @ (u v) (u v) + (u v) + F (t; u) F (t; v) @t 4g(t) rX(t) (u v) + (u v) A u v : 4g(t) rX(t) j j 32 CHAPTER 5. BASICS OF PDE TECHNIQUES

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

u v < 0 (u v) = 0 4 (u v) = 0 r @ (u v) A0 (u v) = A0 u v @t j j for any A0: But the inequality above says that at that same point @ (u v) A u v ; @t j j which is a contradiction if A0 < A: Usually, instead of making v a subsolution, we will just make v the subsolution to the ODE dv F (t; v) ; dt 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 M and the metric g satis…es the Ricci ‡ow for t [0;T ), then R (t; x) 0 for2 all x M and t [0;T ]. 2 2 2 Proof. Recall that R satis…es the evolution equation @R = R + 2 Rc 2 ; @t 4g(t) j j thus it is a supersolution to the heat equation (with changing metric), i.e., @R R: @t 4 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

2 1 ij 2 T g Tij j j n since 2 1 k` Tij g Tk` gij 0 n

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

1 Rc 2 R2 j j n 5.3. MAXIMUM PRINCIPLE ON TENSORS 33 and so scalar curvature satis…es @R 2 R + R2: @t 4 n The maximum principle implies that R (t; x) f (t) for all x M; where f (t) is the solution to the ODE 2 df 2 = f 2 dt n f (0) = min R (x; 0) : x M 2 This equation can be solved explicitly as

1 2 df = dt f 2 n Z Z 1 2 1 = t f n f (0) f (0) f (t) = 1 2 f (0) t n as long as f (0) = 0: Notice that if f (0) > 0 then this says that R (t; x) goes to in…nity in …nite6 time T n : If f (0) < 0; then this says that if the ‡ow 2f(0) exists 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 apply! the 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 be a vector bundle over M with connection : Let K be a closed, …berwise convex subset of V which is parallel with respectr to the connection. Let u (V ) be a section such that 2

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

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

(This is the notion that u attains a maximum at x:) Then X u (x) is tangent ij r to Kx at u (x) and the Laplacian u (x) = g (x) i ju (x) is an inward or 4 r r tangential pointing vector to Kx at u (x) :

Here are the relevant de…nitions. 34 CHAPTER 5. BASICS OF PDE TECHNIQUES

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

De…nition 39 A subset K is parallel to the connection if it is preserved by r parallel translation, i.e., if Px;y is parallel translation along a curve from x to y; then P Ky Kx (this is if the tensors are all contravariant). x;y Example 40 The set of positive de…nite two-tensors is …berwise convex and parallel 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 dimensions.

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

Remark 41 Why do we need convex? Consider the scalar case where we replace u (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 singularities of Ricci ‡ow.

6.2 Finite time singularities

Recall that we proved short time existence for Ricci ‡ow, which means the ‡ow exists until it reaches a singularity. The …rst, most basic result on singularities is the following result of Hamilton:

Theorem 42 Let (M; g (t)) be a solution to the Ricci ‡ow on a compact manifold on a maximal time interval [0;T ): If T < ; then 1 2 lim max Rm (t) g(t) = : t T x M j j 1 ! 2 Remark 43 Uniqueness of the Ricci ‡ow is necessary for there to exist a maximal time interval. The idea is that if there is a solution on [0;T1) and another solution on [T1 "; T2); then they must agree on the overlap, so one can consider the ‡ow on [0;T2) which extends both the ‡ows. Now simply take the sup of all T2 for which the ‡ow exists and this forms the maximal time interval.

Remark 44 N. Sesum was able to replace Rm with Rc : j j j j Proof (idea). Suppose T < and Rm remains bounded. Then one can show that all derivatives k Rm1are uniformlyj j bounded for t ["; T ): One can use this to derive uniformr bounds on the metric and its derivatives2 and then to extract a smooth limit metric at T (of a subsequence using an Arzela-Ascoli type compactness theorem). The existence/uniqueness result tells us that we can extend the ‡ow for a short time, contradicting the fact that [0;T ) was maximal.

35 36 CHAPTER 6. SINGULARITIES OF RICCI FLOW

This is quite a useful theorem, but it is still possible to develop singularities in a …nite time (for instance, the sphere or a neck pinch). It will be very important to analyze what is happening at the singular times so that we can do surgery to remove the problems and then continue the ‡ow. We will do this using a PDE technique called blowing up around the singularity, which uses scaling to allow us 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 Rm 2 is going to in…nity. If we scale the metric g to cg; then we get j j

1 Rm (cg) 2 = Rm (g) 2 j j c2 j j or 1 Rm (cg) = Rm (g) ; j j c j j so if we want to prevent Rm from going to in…nity, we choose a scaling j j 2 Ln = max Rm (g (tn)) x M 2 j j 2 and rescale the metric g (tn) by Ln : We can actually rescale to a sequence of solutions of the di¤erential equation (Ricci ‡ow) by looking at

1 2 gn (t) = 2 g tn + tLn ; Ln where tn T; the singular time. Notice that ! @ @ g = L 2g t + tL2 @t n @t n n n 2 = 2 Rc g tn + tL n = 2 Rc [gn (t)] so gn is a sequence of solutions to the Ricci ‡ow whose initial value is getting closer to the singular time. Furthermore, the initial curvatures Rm (gn (0)) are j j all bounded by 1: On the down side, Ln 0 and so the metric is being multiplied by larger and larger scaling factors and! thus it is quite likely that a limit will become noncompact. We will need to have a good notion of convergence which allows convergence to noncompact manifolds.

Remark 45 We do not necessarily have to choose Ln as above. The fact that 2 2 Ln is why this is called a blow-up. If we take Ln 0 then we have what! is called1 a blow-down. ! 6.3. BLOW UPS 37

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

tn T tn ; : L2 L2 n n Thus if we can extract a limit and tn T and Ln 0; then the limit metrics will be ancient, i.e., will start at t = ! (the …nal! endpoint depends a little 1 more on how we choose the tn with respect to how we pick the Ln; allowing it to be 0; positive, or + ; we may have reason to choose any of these). The main goal is to1 …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) = dVg; ZM we see that 1 1 V M; g = V (M; g) L2 2=d n Ln and so if Ln 0; this quantity does not persist to the limit. Tao calls this behavior! supercritical. The total scalar curvature. If we look at F (M; g) = RdV; ZM we see that 1 F M; g = L2 dF (M; g) : L2 n n 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) = min R (x) ; x M 2 then we see that 1 R M; g = L2 R (M; g) min L2 n min n 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). 38 CHAPTER 6. SINGULARITIES OF RICCI FLOW

Lowest eigenvalue of the operator 4 u+R: Notice that this is subcritical since if we call this functional H (M;4 g)

1 H M; g = L2H (M; g) : L2 n It turns out that this quantity obeys a monotonicity and has some nice coercivity, though not quite enough. We will soon look at a particular critical (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 going to zero. Recall that the exponential map is the map from the tangent space at one point to the manifold where a vector v is taken to the point one unit from the origin along a geodesic starting with velocity v: This map is a local di¤eomorphism, and there is an r > 0 such that the ball B(0; r) is mapped di¤eomorphically to a ball on the manifold. The largest such r is called the injectivity 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. If all manifolds and their limit have a uniform diameter bound, then one does not need 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 collapsing if inj 0 as n . pn ! ! 1 We may wish to rescale the manifolds (Mn; gn) to be of some uniform size, say by making Rm (pn) = 1: Then one can consider a rescaled collapsing if 1=2 j j Rm (pn) injpn 0: j Whenj one assumes! that the sectional curvature is bounded, then the collapse is restricted. Let V (U) = Vg (U) denote the Riemannian volume of the Borel subset U M. 2 d Theorem 47 (Cheeger) Suppose that Rm g Cr0 on B (p; r0) M and that j j d V (B (p; r0)) r 0 for some > 0: Then the injectivity radius of p; injp; is at least

inj cr0 p for some constant c = c (C; ; d) > 0:

Let’s think about this theorem for a minute, to see if it is ever applicable since it seems the assumptions are quite strong. Note that as r0 0; the ball ! looks more and more Euclidean. That means that for very small r0 inj ; p d V (B (p; r0)) !r ; where ! = ! (d) is the correct constant for a Euclidean ball, and

Rm Rm (p) : j jg j jg So as r0 0; we see that !

2 lim r0 sup Rm (x) = 0 r0 0 x B(p;r0) j j ! 2 ! V (B (p; r0)) lim d = 1: r0 0 !r ! In particular, for any C > 0 and 0 < < 1; there is a r > 0 such that the assumptions are satis…ed if r0 r : Note that the converse is also true from more classical results.

Theorem 48 If Rm g C and injp ; then there is a = (C; ; d) such that j j V (B (p; r)) rd for all r : 40 CHAPTER 6. SINGULARITIES OF RICCI FLOW

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

De…nition 49 A Riemannian manifold M d; g is -collapsed at p M at 2 scale r0 if 2 1. (Bounded normalized curvature) Rm r for all x B (p; r0) and j jg 0 2 d 2. (Volume collapsed) V (B (p; r0)) r : 0 If these are not satis…ed, then we say the manifold is -noncollapsed at p at the scale r0: Is this a reasonable de…nition? Here are some observations:

By Cheeger’s theorem, this would imply a lower bound on injectivity radius.

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

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

This de…nition is scale independent in the following sense. If we consider 2 g = r0 g; then the conditions are

Rm (g) 1 j jg V (Bg (p; 1)) :

n The sphere S is -noncollapsed at scales r0 less than the diameter (for a suitable choice of ). The bounded normalized curvature assumption of the de…nition are satis…ed for r0 1; although one might argue that there is really no local collapsing at scales less than : Apparently this will not be important for our argument. At large scales, the curvature assumption fails.

The ‡at torus has Rm = 0; and so it is noncollapsed at scales less than the injectivity radius.j Thej 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 small enough (say, less than the constant for the area of a Euclidean ball) so that 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) as follows.

De…nition 50 Let M d; 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)

2 Rm (t; x) r j jg(t) 0 2 for all (t; x) t0 r ; t0 B (x0; r0) ; and 2 0 g(t0) 2. (Collapsed volume) d V B (x0; r0) r : g(t0) 0

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 time 2 interval [t0 r ; t0]: 0 Remark 52 As remarked above, for each smaller than the volume of a unit ball in Euclidean space, there is a r so that M is -noncollapsed at scales less than r : For Ricci ‡ow, one can still …nd r ; but it will, in general depend on t0: As t0 goes to a singular time, it may be possible that r 0: This is what we 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 that t 1=2 Kg t + is -noncollapsed at (K (t0 t ) ; x0) at the scale of K r0: K Here is a typical noncollapsing theorem along the lines of Perelman.

Theorem 55 (Perelman’snoncollapsing theorem, …rst version) Let (M; g (t)) be a solution to the Ricci ‡ow on compact 3-manifolds for t [0;T0] such that at t = 0 we have 2

Rm (p) 1 j jg(0) V B (p; 1) ! g(0) for all p M and ! > 0 …xed. Then there exists = (!;T0) > 0 such 2 that the Ricci ‡ow is -noncollapsed for all (t0; x0) [0;T0] M and scales 2 0 < r0 < pt0.

A big point of this theorem is that it rules out a limit of R where is the cigar soliton solution of Hamilton. The manifold R is essentially a …xed point of Ricci ‡ow (when we consider it a ‡ow on metric spaces, not Riemannian metrics). is a positive curvature metric on R2 which has maximum curvature at the origin and is asymptotic to a cylinder as one moves away from the origin. 42 CHAPTER 6. SINGULARITIES OF RICCI FLOW

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 to 2 Cr; not Cr :) 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

1 tn T T0 Ln r0 for the time interval L2 ; L2 : As n becomes large, we see that gn (t) n n becomes -noncollapsed ath 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 the heat equation @f = f @t 4 on a compact Riemannian manifold. It is easy to see that if one considers the energy 1 E (f) = f dV; 2 jr jg ZM that if we take the time derivative of the energy when f satis…es the heat equation, we get

dE @ (f) = f f dV dt r r @t ZM @ = f f dV 4 @t ZM = ( f)2 dV 0: 4 ZM Thus we immediately get that the energy is decreasing and that stationary points are harmonic functions, i.e., functions which satisfy f = 0: This monotonicity also tells us that f cannot have periodic solutions which4 are not …xed points, for if f (t1; x) = f (t2; x) for all x; then E (f (t1; )) = E (f (t2; )), and the monotonicity implies that f = 0 for t [t1; t2]: The monotonicity is true4 in general for2 a gradient ‡ow. If one has an energy E (f) ; one de…ned the gradient ‡ow as

@f = grad (E) ; @t 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) = RdV: ZM

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

R (h) = Rc; h trg (h) + div div h: h i 4 It is not hard to see that since

ij log det g = g hij so 1 det g = exp log det g 2 p 1 = (tr h) det g 2 g so p 1 (dV ) = (tr h) dV: 2 g Using the above formula we get 1 (EH)(h) = Rc; h trg (h) + div div h + R (trg h) dV h i 4 2 ZM 1 = Rc; h + R (trg h) dV h i 2 ZM 1 = Rg Rc; h dV: 2 ZM Remark 56 Here we used the divergence theorem for a compact Riemannian manifold, which says that

div T;S dV = T; S dV; h i h r i Z Z 7.2. RICCI FLOW AS A GRADIENT FLOW 45 for any n-tensor T and (n 1)-tensor S: More explicitly,

i1j1 injn ji0 i1j1 injn ji0 g g g jTi0i1 in Sj1j2 jn dV = g g g Ti0i1 in jSj1j2 jn dV: r r Z Z So critical points of the Einstein-Hilbert functional satisfy the Einstein equation. The gradient ‡ow would be

@ 1 g = 2 Rc Rg : @t 2 The problem is that this ‡ow is not parabolic and there is no existence theory for such equations. Let’s try a new tactic. Replace dV by a …xed measure dm: Then the functional H (g) = Rdm ZM satis…es the variation

H (h) = ( Rc; h trg (h) + div div h) dm: h i 4 ZM You don’tlose the last two terms with the divergence theorem, since that only works with the volume measure. However, we can consider the Radon-Nikodym derivative and write dm dm = dV dV dm for a positive function dV : We can write dm = e f : dV Then, f H (h) = ( Rc; h trg (h) + div div h) e dV h i 4 ZM can be integrated by parts to get

f H (h) = ( Rc; h trg (h) ; f + div h f) e dV h i hr r i r ZM 2 2 f = Rc; h + trg (h) f trg (h) f h; f + h ( f; f) e dV M h i 4 jr j r r r Z 2 2 f = Rc f + f f g + f f; h e dV: M r 4 jr j r r Z D E Remark 57 Tao often uses ; to denote the Euclidean metric locally, and so explicitly puts in gij’s in thish i case. We will understand that ; requires the metric, and so when quantities like this are di¤erentiated, weh alsoi need to di¤erentiate the metric, as we will see below. 46 CHAPTER 7. RICCI FLOW FROM ENERGIES

We need to add another term, and so we get

2 2 f dm F (g) = R + f e dV = R + log dm: jr j r dV ZM ZM !

We now need that

f f 1 f 0 = (dm) = e dV = (f) e dV + trg he dV 2 1 f = f + trg h e dV 2 Thus we have that 1 f = tr h 2 g Let ij f E (g) = g if jf e dV r r Z f where dm = e dV is …xed (so that f = log dm=dV and f is expressed as above). We can then compute

ij f E (h) = g if jf e dV r r Z 2 1 2 f = h; f f + 2 f; (f) f f + f trg h e dV h r r i hr r i jr j 2 jr j Z 2 1 2 f = h; f f 2 f (f) + f f + f trg h e dV h r r i 4 jr j 2 jr j Z 2 f = h; f f ( f) g + f g e dV: r r 4 jr j Z D E Now, since F = H + E; we have

2 f F (h) = Rc f; h e dV: r ZM

Thus the gradient ‡ow of 2F is @g = 2 Rc (g) 2 2f: @t r This is almost Ricci ‡ow, but not quite. The f is changing, too, by the equation @f = f R: @t 4 This is a backward heat equation, which it turns out will make it useful to probe backwards. Notice that 2 2 f = f g; r Lr 7.2. RICCI FLOW AS A GRADIENT FLOW 47 the Lie derivative of g in the direction f: This means that the ‡ow above r di¤ers from Ricci ‡ow by a di¤eomorphism. Instead, we can consider g = g; where is the ‡ow of di¤eomorphisms generated by f; and we will see that g evolves by Ricci ‡ow. Furthermore, f will di¤er by ar Lie derivative, and

2 f f = df ( f) = f Lr r jr j (where is the metric in here? It is in f; which is a vector …eld gotten by raising the index on df) and so under the newr ‡ow, f = f evolves by @f 2 = gf R + f : @t 4 r g

Example 58 (Fundamental, important example) Let (M; g) be Euclidean space M = Rd and let 2 x d d=2 x 2=(4) f (t; x) = j j + log 4 = log (4) ej j 4 2 h i f where = t0 t; Notice that e dx is the Gaussian measure, which solves the backward heat equation (the fundamental solution to the heat equation). If t < t0; this choice of g and f satisfy the equations @g = 2 Rc (g) (7.1) @t @f = f R + f 2 : (7.2) @t 4 jr j We can check:

@f @ x 2 d = j j + log 4 @t @t " 4 2 # x 2 d = j j : 4 2 2

x f = r 2 x 2 f 2 = j j jr j 4 2 d f = 4 2 so it works.

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

2 f F (M; g; f) = R + f e dV: M jr j Z 48 CHAPTER 7. RICCI FLOW FROM ENERGIES

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 2 2 f (M; g; f) = 2 Rc + f e dV: @t r Z

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 take the in…mum:

f (M; g) = inf F (M; g; f): e dV = 1 ZM (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 the inequality 2 2 2 4 u g + Ru dV u dV; M jr j M Z Z where u is in H1 (M) = W 1;2 (M), the Sobolev space of functions with 1 deriva- 2 2 2 1 tive in L (so it has norm f 1 = f + f dV for C functions). Hint: k kH jr jg f=2 show that we can assume u is positiveR and then write u = e : Thus is the smallest eigenvalue of the operator 4 gu + R: 4 Using the exercise, one sees that the fact that every compact manifold satis…es a Poincaré inequality,

u 2 dV c (d) u2dV; jr j ZM ZM implies that is bounded below, basically, by the best constant in the Poincaré inequality plus min R: Note that the Poincaré inequality constant depends on the dimension. We will see later a similar inequality for which the constant does not depend on dimension. Furthermore, one can prove that is realized by a positive function u = f=2 1 2 e with u L2(M) = 1: Note that H (M) embeds compactly into L (M) k k 1 since M is compact. Thus if we take a minimizing sequence un in H ; there f g 2 is a subsequence (which we also denote by un which converges in L to a function u: Now consider: f g

2 2 2 2 un + Ru dV + um + Ru dV jr j n jr j m Z Z 1 2 2 1 2 2 = (un um) + R (un um) dV + (un + um) + R (un + um) dV 2 jr j 2 jr j Z Z 7.2. RICCI FLOW AS A GRADIENT FLOW 49

1 2 2 2 2 2 2 (un um) + R (un um) dV = un + Ru dV + um + Ru dV 2 jr j jr j n jr j m Z Z Z 1 2 2 (un + um) + R (un + um) dV: 2 jr j Z The right side goes to zero in the limit since the terms go to ; and 2: Also, we know that

2 min R un um R (un um) dV min R un um j j k kL2 j j k kL2 Z and each term goes to zero. Thus we know that the sequence un is Cauchy in H1 and since H1 is complete, it must converge to a functionf in Hg 1: 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 = h + R + h 2 : @t 4 jr j

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

(t0) F (M; g (t0) ; h (s; t0 s)) s 2 2 h(s;t0+) = F (M; g (t0 + s) ; h (s; t0)) 2 Rc (g (t0 + )) + h (s; t0 + ) e dV d: r Z0 ZM

We then have s 2 2 h(s;t0+) (t0) (t0 + s) 2 Rc (g (t0 + )) + h (s; t0 + ) e dV d r Z0 ZM and so

@ (t0 + s) (t0) (t0) = lim @t s 0+ s ! s 2 2 2 h(s;t0+) lim Rc (g (t0 + )) + h (s; t0 + ) e dV d s 0+ s r ! Z0 ZM 2 2 h(0;t0) = 2 Rc (g (t0)) + h (0; t0) e dV r ZM 2 2 f = 2 Rc (g (t0)) + f (t0) e dV r ZM

50 CHAPTER 7. RICCI FLOW FROM ENERGIES

We can now derive

@ 2 2 f 2 Rc + f e dV @t r Z 2 2 f (R + f ) e dV 3 4 Z 2 2 f (R + f ) e dV 3 4 Z 2 2 2 f = R + f e dV 3 jr j Z 2 = 2: 3

7.3 Perelman entropy

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

dFm 2 2 f (M; g) = 2 Rc + f e dV dt r Z is …xed (under the gradient ‡ow) if

Rc = 2f; r i.e., if (M; g) is a gradient Ricci soliton. We wish to have a new functional which is …xed if (M; g) is a gradient shrinking soliton, i.e.,

1 Rc = 2f + g r 2 for some > 0: A round sphere is a gradient shrinking soliton, so it makes sense that we would want something like this. Under the Ricci ‡ow, this structure is preserved except that decreases at a constant rate. First note that if we consider the Nash entropy

dm dm dm Nm (M; g) = log dV = log dm = fdm dV dV dV ZM ZM Z then dN m = ( f R) dm dt 4 ZM 2 = R + f dm = Fm (M; g) : M jr j Z 7.3. PERELMAN ENTROPY 51

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

dW 1 2 = Rc + 2f g dm: dt r 2 ZM

But in this case, would not have scale invariance for W; since t scales like distance squared, so the integrand should scale like distance squared. We will …x this by assuming d = 1 dt and trying for dW 1 2 = 2 Rc + 2f g dm: (7.3) dt r 2 ZM

Now, to …nd such a quantity, consider

2 1 2 1 d Rc + 2f g = Rc + 2f (R + f) + : r 2 r 4 4 2

Thus we have that 2 2 1 dFm d 2 Rc + f g dm = 2Fm + r 2 dt 2 ZM

d d = Fm Nm log : dt 2

This is what our Wm would be. However, as we did last time, we wanted to reparametrize so that f dm = e dV where dV is evolving according to Ricci ‡ow evolution. This time, we will change

d f~ = f log (4) 2 so that f d=2 f~ dm = e dV = (4) e dV:

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

Note that the preservation of dm implies that

d @f~ 1 + tr h = 0: 2 @t 2 Under the gradient ‡ow @g = 2 Rc 2 2f; @t r 52 CHAPTER 7. RICCI FLOW FROM ENERGIES we have @f~ d = R f: @t 2 4 Thus we get for d Wm (M; g; ) = Fm Nm log 2 2 d d=2 f~ = R + f~ + f~ log (4) (4) e dV r 2 Z

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

2 d=2 f~ Wm (M; g; ) = R + f~ + f~ d (4) e dV: r Z

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

2 d=2 f W (M; g; f; ) = R + f + f d (4) e dV jr j Z h i where g is a Riemannian metric, f is a function on M, and is a positive constant. If g satis…es the Ricci ‡ow, then we need to pull back f to get the three evolutions @g = 2 Rc (7.4) @t @f d = R f + f 2 (7.5) @t 2 4 jr j d = 1: dt Under these three, the Perelman Entropy satis…es

2 dW 2 1 d=2 f (M; g; f; ) = 2 Rc + f g (4) e dV: dt r 2 ZM

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 yet clear that such an in…mum exists. Recall that last time, the existence followed from a Poincaré inequality. In this scale invariant setting, the existence of a minimizer 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 to switch to a function which looks like the heat kernel, namely,

d=2 f u = (4) e : 7.4. LOG-SOBOLEV INEQUALITIES 53

Recall that our model case is when f = x 2 = (4) ; in which case this is precisely the backwards heat kernel. The backwardsj j heat kernel satis…es: @u = u @t 4 for > 0 and lim u (; x) = 0 (x) ; 0 ! weakly, where 0 is the delta function. Furthermore, one can check that

d=2 x 2=(4) udx = (4) ej j dx = 1 (7.6) d d ZR ZR for any : The backwards heat kernel can be used to solve the heat equation with some given …nal conditions, e.g., to solve

du = u dt 4 u (T; x) = f (x) ; we see that the convolution

d=2 x y 2=(4) u (t; x) = f (y) (4) ej j dy d ZR is a solution.

Exercise 61 Show that all of this is true. Hint: to show (7.6), turn the integral into polar coordinates and assume the dimension is at least 2: For the dimension 1 case, there is a trick involving turning it into a dimension 2 integral and separating.

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

2 x d=2 x 2=(4) W (M; g; f; ) = j j d (4) ej j dx: 2 Z " # One can show that this is zero since

2 d=2 f W (M; g; f; ) = 2 f f (4) e dx; jr j 4 Z h i 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),

u 2 d W = jr j u log u dx log (4) d u2 2 Z " # 54 CHAPTER 7. RICCI FLOW FROM ENERGIES using identities such as

d=2 f u = (4) e : d log u = log (4) f 2 2 d 2 2f u = (4) f e jr j jr j u 2 f 2 = jr j : jr j u2 Tao shows that one can show that W 0; which implies a log-Sobolev inequality u 2 d jr j dx u log udx + log (4) + d; u2 2 Z Z or as it is usually stated, with 2 = u;

1 d 4 2 dx 2 log 2dx + log (4) + d: jr j 2 Z Z

For the general case, we have

u 2 d W (M; g; f; ) = Ru + jr j u log u dV log (4) d u2 2 Z " ! # One can show that the

W (M; g; f; ) C (M; g; ) : This implies essentially a log-Sobolev inequality, i.e.,

d R2dV + 4 2 dV C + 2 log 2dV + log (4) + d: jr j 2 Z Z Z In fact, we can take the

d=2 f (M; g; ) = inf W (M; g; f; ): (4) e dV = 1 ; Z which is the best possible constant C: It can be shown that is …nite, which is what we could call a log-Sobolev inequality. We can now show that if g (t) is a solution to Ricci ‡ow on t [0;T0] and 2 = 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 H1 (M) : (Not quite true... What is the true statement? Hint: you need to change 2 to a new 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 for f in 7.4 backwards to t with initial condition f (t0; x) = f (t0; x) (since f is in H1 (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. Suppose we have a ball B (p; p) with bounded normalized curvature, i.e., 1 Rm (x) j j for x B (p; p) : Then R c (d) for some constant depending only on dimension.2 Then the log-Sobolevj j inequality can be rewritten as d c (d) 2dV + 4 2 dV (M; g; ) + 2 log 2dV + log (4) + d: jr j 2 Z Z Z 2 Suppose is a function supported on B (p; p) such that M dV = 1: Then Jensen’sinequality implies that R 1 1 1 2 log 2dV 2dV log 2dV V (B) V (B) V (B) ZB ZB ZB 1 1 = log ; V (B) V (B) where B = B (p; p) : (Recall that Jensen’s inequality requires a probability measure.) So 1 2 log 2dV log : V (B) ZM We now get, for this particular choice of ;

d=2 2 4 dV (M; g; ) + log c0 (d) : jr j V (B) Z Now we will specialize even more. Suppose d (x; p) (x) = c 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 described it). Thus (x) = c on B (p; p=2) and c is such that

2dV = 1; ZB 56 CHAPTER 7. RICCI FLOW FROM ENERGIES

1=2 so c V (B (p; p=2)) : We can choose so that c00c=p on the jr j ball (for some constant c00), and so

V (B) d=2 4c00 (M; g; ) + log c0 (d) : V B1=2 V (B) Finally, we can use a Bishop-Gromov volume comparison theorem:

Theorem 63 (Bishop-Gromov comparison) If M d; g is a complete Rie- mannian manifold with Rc (n 1) Kg for some K R; then for any p M; the volume ratio 2 2 V (B (p; r))

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

In particular, we have that

V (B) V B 1=2 ; V 1= B p 1= ; p 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 d=2 c000 (M; g; ) log ; V (B) i.e., c000 d=2 V (B) e ; which implies -noncollapsing at a scale p for = exp ( c000) : Let’sformu- late this into a proposition:

Proposition 64 There is a constant c = c (d) depending only on dimension such that if M d; g; is …nite, then for = exp ( (M; g; ) c) ; the Rie- mannian manifold (M; g) is -noncollapsed at the scale of p: 7.5. NONCOLLAPSING 57

Let’scollect the facts about : Proposition 65 The following are true about : 1. (M; g; ) > for any …xed manifold (M; g) and > 0: 1 2. If (M; g (t)) satis…es the Ricci ‡ow for t [0;T0] and (t) = T0 t; then (M; g (t) ; (t)) is increasing. 2 3. There is a constant c = c (d) depending only on dimension such that the Riemannian 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’snoncollapsing theorem, …rst version) Let (M; g (t)) be a solution to the Ricci ‡ow on compact 3-manifolds for t [0;T ) such that at t = 0 we have 2 Rm (p) 1 j jg(0) V B (p; 1) ! g(0) for all p M and ! > 0 …xed. For any > 0; there exists = (!; T; ) > 0 2 such that the Ricci ‡ow is -noncollapsed for all (t0; x0) [0;T ) M and scales 2 0 < 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 lower 2 bound. For any r0; we see by monotonicity that M; g (t) ; r2 M; g (0) ; r2 + t : 0 0 Thus we have that if

2 2 0 = inf M; g (0) ; r : r (0; + T ) 2 then 2 M; g (t) ; r 0: 0 Thus (M; g (t)) is -noncollapsed at the scale of r0 for all 2 = exp (0 c) exp M; g (t) ; r c : 0 We need to see that 0 is not : Since Tis …nite, there is no problem at the top of the interval for r2: It can1 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! the other side. Remark 67 This is a bit stronger than what I proposed in an earlier lecture. I think Tao was thinking about future incarnations of this theorem, which is why he formulated as he did. 58 CHAPTER 7. RICCI FLOW FROM ENERGIES Chapter 8

Ricci ‡ow for attacking geometrization

8.1 Introduction

I would like to go back to the general program and see what we need to learn about. 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 a singularity. We will consider a Ricci ‡ow with surgery (M; g (t)) de…ned for 0 t < T < which satis…es the following properties: 1 1. (Normalized initial conditions) We have

Rm (g (0)) 1 j j and 1 V (B (x; 1)) V (B 3 (0; 1)) 2 E for any x M: 2 2. (Curvature pinching) The curvature is pinched towards positive. This means that as the scalar curvature R + ; the ratio of the absolute value of the smallest eigenvalue of the Riemannian! 1 curvature tensor to the largest 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 canonical neighborhood.

The key is to show that these conditions both allow surgery and then persist after the surgery. In order to do this, we will need to be more precise with 3 and 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. If one takes a sequence ti T; where T is a singular time, then the blow ups ! t g (t) = M g t + : i i i M i

If Mi is comparable to sup Rm (g (ti)) ; then Mi : Furthermore, since j j ! 1 T < ; we have that gi (t) is de…ned on the interval [ Miti; (T ti) Mi): Thus the limit1 will de…nitely be de…ned on ( ; 0]; and is thus ancient. Moreover, 1 if we show that g is -noncollapsed at some scale r0, then gi (t) will be - noncollapsed at a scale r0pMi; and so the limit is -noncollapsed on all scales. Finally, one can show that any 3-dimensional ancient solution has nonnegative curvature.

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

These are the limits as you go to a …nite time singularity. The key is that Perelman 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. It turns out that there are 4 types of canonical neighborhoods:

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

1. x is contained in a C-component.

1=" 2. x is contained in an open set which is within " of round in the Cb 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

k De…nition 70 De…ne the C (X; g0) “norm” on (X; g) to be

k 2 2 ` 2 (X; g) k = sup g (x) g (x) + g (x) : C (X;g0) 0 g0 g0 g k k x X j j r 0 ( `=1 ) 2 X

Remark 71 Technically, the norm should be

k 2 2 ` 2 (X; g) k = sup g (x) + g (x) : C (X;g0) g0 g0 g jjj jjj x X j j r 0 ( `=1 ) 2 X

The function de…ned above is essentially (X; g) = (X; g g0) : With kk k k jjj jjj our current de…nition, (X; g) k = 0 if g = g0: k kC (X;g0) Problem 72 Here is something to think about. Fix X Rn; say bounded. Given a sequence of Riemannian metrics gi on X; under what conditions f g does there exist a subsequence such that (M; gi) converge to some limit (X; g ) 1 for some Riemannian metric g (i.e., (X; gi) Ck(X;g ) 0 as i : 1 k k 1 ! ! 1

Problem 73 How could one use this norm idea to compare (X; g) and (X0; g0) where X = X0? 6 De…nition 74 Let (N; g) be a Riemannian manifold and x N a point. Then an "-neck structure on (N; g) centered at x consists of a di¤eomorphism2

1 1 : S2 ; N; " " ! with x S2 0 ; such that 2 f g (N;R (x) g) < " k k 1=" 2 1 1 where the norm is with respect to Cb c S " ; " ; gstd ; where gstd is the product of the metric with curvature 1=2 onS2with the Euclidean metric on the interval. We say N is a "-neck centered at x:

De…nition 75 A compact, connected, Riemannian manifold (M; g) is called a C-component if

3 1. M is di¤eomorphic to S3 or RP ; 2. (M; g) has positive sectional curvature,

3. For every 2-plane P in TX;

1 inf K (P ) < P ; C supy M R (y) 2 62 CHAPTER 8. RICCI FLOW FOR ATTACKING GEOMETRIZATION

4. 1 1 1 C sup < diam (M) < C inf : y M R (y) y M R (y) 2 2 De…nition 76 A compact,p connected 3-manifold (M; g) isp within " of round in 1=" the Cb c-topology if there exists a constant > 0; a compact manifold (Z; g0) of constant curvature +1; and a di¤eomorphism

: Z M ! such that

(Z; (g)) 1=" ": k kCb c(Z;g0) Finally, we have the complicated de…nition of a cap. The last conditions essentially say that the diameter, volume, and curvature di¤erences are controlled and 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 ( ; g ) together with an open submanifold M with the following properties: C jC C

3 1. is di¤eomorphic to an open 3-ball or to a punctured RP : C 2. N is a "-neck. 3. Y = N is a compact submanifold with boundary. Its interior Y is called the coreC n of . C 4. The scalar curvature R (y) > 0 for every y and 2 C C diam ( ; g ) < : C jC supy R (y) 2C q 5. R (x) sup < C: x;y R (y) 2C 6. C V ( ) < : C 3=2 supy R (y) 2C

7. For any y Y; let ry de…ned by the the condition that 2 1 sup R (y0) = 2 : y B(y;ry ) ry 02

Then for each y Y the ball B (y; ry) has compact closure in and 2 C 1 V (B (y; r )) < inf y : C y Y r3 2 y 8.4. HOW SURGERY WORKS 63

8. R (y) < C sup jr 3=2j y R (y) 2C and @R @t (y) sup 2 < C y R (y) 2C 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 of Ricci ‡ow with is the union of (C2;") canonical neighborhoods. 3. We can do surgery on canonical neighborhoods if they are su¢ ciently small and positively curved.

Consider a Ricci ‡ow which becomes singular at a time T: Fix T < T so that there are no surgeries in the interval [T ;T ): By the assumptions, there is an open set M such that the curvature is bounded for all t [T ;T ); so there is a limiting metric on as t T: Every end is the end of2 a canonical neighborhood, which looks like a tube.! We call these ends "-horns. We can then …x a constant and consider the subset in which the scalar curvature 2 is bounded above by : One can then show that " horns with boundaries in are -necks. We then do surgery on these -necks by gluing in a standard solution.

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 solution 4) Show …nite time extinction. 64 CHAPTER 8. RICCI FLOW FOR ATTACKING GEOMETRIZATION Chapter 9

Reduced distance

9.1 Introduction

We …rst wish to show that a -solution has a limit which is a gradient-shrinking soliton. 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 the reduced distance (whatever that is)? Let’s …rst think a bit about the case of Euclidean space. Recall that in Euclidean space, W becomes

u 2 d W (M; g; u; ) = jr j u log u dx log (4) d u2 2 Z " # and that W is minimized for

d=2 2 u (x; ) = (4) exp x = (4) : j j We see that this formula essentially gives the distance function d (x; 0)2 = x 2 by j j x 2 d j j = log u (x; ) log (4) : 4 2 However, we have very little control over this function. One can also derive the distance function as follows: a d (x; 0) = inf _ d: (0)=0 0 j j (a)=x Z

This strong relationship is only true in Euclidean space. In general, there are two 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 a PDE 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 information about. The two things are closely related, but not the same concept. We will try to do something similar for the functional W:

9.3 -length and reduced distance L Consider a curve : [0; ] M. One can de…ne the length of a curve as ! ` ( ) = _ (s) ds j j Z0 and the energy as 1 E ( ) = _ (s) 2 ds: 2 j j Z0 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 the distance to the point p; i.e.,

rp (x) = d (x; p) : It turns out that the distance function satis…es some di¤erential equations and inequalities, in particular d 1 r 4 r i 2 d 1 for Ricci nonnegative. Note that if r (x) = (x ) ; then r = : We will 4 r show this inequality later in this lecture, butq for now, let’sassume this and see the implications on the volume. Notice that P d V (B (p; R)) = A (S (p; R)) dR where S (p; R) is the geodesic sphere of radius R centered at p; and A is the area (d 1 dimensional volume measure) function. By Gauss lemma, we have that r = 1 jr j and also that we can decompose the metric to be 2 dr + gS(p;r) where gS(p;r) is the metric on the geodesic sphere. Denote the measure on the sphere of radius r as dAS(p;r): We see that

rdV = ( r n) dA 4 r S(p;r) ZB(p;R) ZS(p;R) = r dA jr j S(p;r) ZS(p;R) = A (S (p; R)) : 9.3. -LENGTH AND REDUCED DISTANCE 67 L

The decomposition above implies that the volume measure can be decomposed as

dV = dAS(p;r)dr and so

d d R rdV = rdA dr dR 4 dR 4 S(p;r) ZB(p;R) Z0 ZS(p;r) d 1 = R rdA : 4 S(p;R) ZS(p;R) Thus,

d A (S (p; R)) 1 d 1 d d 2 = R rdV (d 1) R A (S (p; R)) dR Rd 1 R2(d 1) dR 4 " ZB(p;R) #

1 d 1 d 2 = R rdA (d 1) R A (S (p; R)) R2(d 1) 4 S(p;R) " ZS(p;R) #

1 d 2 d 2 (d 1) R dA (d 1) R A (S (p; r)) R2(d 1) S(p;R) " ZS(p;R) # = 0:

This implies an inequality on volume ratios.

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

Proof. We have already showed that

d A (S (p; R)) d 1 0: dR R

We now consider 0 r1 r2: We have r V (B (p; r )) V (B (p; r )) 2 A (S (p; r)) dr 2 1 = r1 d d r2 d 1 r2 r1 d r dr R r1 r2 A(S(p;r)) d 1 R n 1 r dr = r1 r r2 d rd 1dr R r1 A (SR (p; r1)) d 1 : d r1 68 CHAPTER 9. REDUCED DISTANCE

Furthermore,

r V (B (p; r )) 1 A (S (p; r)) dr 1 = 0 rd r1 d rd 1dr 1 R 0 r1 A(S(p;r)) d 1 R n 1 r dr = 0 r r1 d rd 1dr R 0 A (SR (p; r1)) d 1 : d r1 Thus we have

V (B (p; r1)) V (B (p; r2)) V (B (p; r1)) ; rd rd rd 1 2 1 which implies

V (B (p; r1)) V (B (p; r2)) V (B (p; r1)) 0 d d d r1 r2 r1 d d r V (B (p; r1)) r V (B (p; r2)) = 2 1 rd rd rd 1 2 1 rd V (B (p; r )) V (B (p; r )) = 2 1 2 : rd rd rd rd 2 1 1 2 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 -distance to be ! L 2 ( ) = p 0 () g() + Rg() d: L 0 j j Z d This may need a bit of explanation. First, 0 = d is the tangent vector to the curve : Note that it is measured at by the metric g () (so at di¤erent s; it is measured using di¤erent metrics!) The …rst term looks almost like the term 2 in the energy 0 d, which can be used to derive geodesics on a Riemannian manifold. However,j j the addition of the term p makes the integral scale like length, not energyR (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 no need for this. 9.3. -LENGTH AND REDUCED DISTANCE 69 L

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

1 L ( ) = 0 () d j jg Z0 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 the distance 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 (also called reduced length): 1 ` (; x) = inf ( ) (0;x0) 2p fL g where the inf is over all paths from x0 to x: Remark 80 Note there is also the -distance, which is 2p`: L One can …nally de…ne the reduced volume

d=2 V~ () = exp ` (; x) dV : (0;x0) (0;x0) g() ZM Our main goal will be to show the following: Theorem 81 @ 2 d ` ` + ` R + 0: @ (0;x0) 4g() (0;x0) r (0;x0) g() 2 @ As a corollary, we get monotonicity of the reduced volume if @ g = 2 Rc (g) : Corollary 82 (Reduced Volume is monotone) @ V~ () 0: @ (0;x0) Proof.

@ @ d=2 V~ () = exp ` (; x) dV @ (0;x0) @ (0;x0) g() ZM d @`(0;x0) d=2 = V~ () exp ` (; x) dV 2 (0;x0) @ (0;x0) g() ZM d=2 + exp ` (; x) RdV (0;x0) g() ZM d d=2 V~ () + exp ` (; x) RdV 2 (0;x0) (0;x0) g() ZM 2 d d=2 + ` + ` R + exp ` (; x) dV 4g() (0;x0) jr jg() 2 (0;x0) g() ZM = 0:

Of course, all of this assumes su¢ cient regularity on `; which is not, in general true. However, this argument can be made rigorous in some generality, including past the “conjugate radius.” 70 CHAPTER 9. REDUCED DISTANCE

9.4 Variations of length and the distance function

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

(t; s = 0) = (t) @ (t; s = 0) = X (t) ; @s so we consider the variation to be X: Furthermore, we consider variations which …x the endpoints, i.e., X (0) = X () = 0: Compute 1 E (X) = g (_ ; _ ) dt 2 Z0 1 @ @ @ = g ; dt 2 @s @t @t s=0 Z0 @ @ = g Ds ; dt 0 @t @t Z = g (DtX; _ ) dt 0 Z = g (X; _ ) g (X;Dt _ ) dt: j0 Z0 Remark 83 The notion DtX means DtX = @ X: It only depends on the r @t values along the curve : (Why?)

At a critical point for the energy, we must have E (X) = 0 for all X; so we get that Dt _ = 0 if the endpoints are …xed, i.e., if X (0) = X () = 0: This is 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 2 _ (t) = 2g (Dt ;_ _ ) = 0 dt j j so has constant velocity. So along a geodesic, we have 1 1 E ( ) = _ 2 ds = _ 2 2 0 j j 2 j j Z ` ( ) = _ ds = _ j j j j Z0 and so if is a geodesic, then 1 E ( ) = [` ( )]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

1 E ( ) = d (p; x)2 : 2 If we want to compute a variation

d d (p; x (s))2 ds away from the cut locus (so variations in geodesics stay minimizing), we have

d dx d (p; x (s))2 = 2E (X) = 2g ; _ () ; ds ds where X is a variation of geodesics (one must show that these exist, but they do!) Note that we can always reparametrize so that = 1, and then we get that

2 d (p; x)2 = 4 _ (1) 2 ; r j j or

4d (p; x)2 d (p; x) 2 = 4 _ (1) 2 = 4d (p; x)2 jr j j j and so d (p; x) = 1: jr j Now compute the second variation of energy when is a geodesic. We get

2 @ E (X;X) = g (DtX; _ ) dt @s 0 Z = g (X; _ ) g (X;Dt _ ) dt: j0 Z0

1 @2 @ @ 2E (X;X) = g ; dt 2 @s2 @t @t s=0 Z0 @ @ @ @ = g D D ; + g D ;D dt s s @t @t s @t s @t Z0 @ @ @ @ = g D D ; + g D ;D dt s t @s @t t @s t @s Z0 @ @ @ @ @ @ = g DtDs ; + g R ; ; + g (DtX;DtX) dt: 0 @s @t @s @t @s @t Z = g ( X X; ) + g (R (X; _ ) X; _ ) + g (DtX;DtX) dt r j0 Z0 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. 72 CHAPTER 9. REDUCED DISTANCE

Let’sagain parametrize our geodesic between 0 and 1 and …x (0) and vary @ the other endpoint x (s) so that X (t) = @s (t; 0) “varies through geodesics.” Then 1 d2 1 d (p; x (s))2 = g (R (X; _ ) X; _ ) + g (D X;D X) dt: 2 ds2 t t Z0 @x @x Furthermore, we can take X (t) = t @s ; where @s is the parallel translation along @x of @s : Remark 84 The parallel translation of a vector v along a curve is the solution to the system of ODE

k @v k i j Dtv = v = + ( 0) v = 0: r 0 @t ij

Then we get 1 d2 1 @x @x @x @x d (p; x (s))2 = t2g R ; _ ; _ dt + g ; : 2 ds2 @s @s @s @s Z0 Thus if we sum over an orthonormal frame and assume that the sectional curvature is nonnegative, we have that 1 d (p; x)2 d: 24 And so 1 f 2 = f f + f 2 24 4 jr j and so d 1 d (p; x) 4 d (p; x) Here’sanother derivation: We now know that for a curve x (s) ; 1 d (x ; x (s))2 = E ( ) 2 0 s where s (t) are curves such that

Dt 0 = 0 = 0 s r s0 s for all s: We may parametrize all curves between 0 and 1: Furthermore, we know that if we consider the variation (s; t) = s (t) ; then @ (t) = X (t) ; @s s s=0

X (0) = 0;

dx X (1) = : ds 9.4. VARIATIONS OF LENGTH AND THE DISTANCE FUNCTION 73

1 E ( ) = E ( ) + sE (X) + s22E (X;X) + O s3 ; s 0 0 2 0 where d2 2E (X;X) = E ( ) : 0 ds2 s Thus the …rst derivative is 1 2 1 2 d 1 2 2 d (x0; x (s)) 2 d (x0; x (0)) d (x0; x (s)) = lim ds 2 s 0 s s=0 !

= E 0 (X) :

The key fact is that under such a variation, we have dx E (X) = g ( (1) ;X (1)) = g (1) ; (0) 0 0 0 ds so we did not need to know much about X (s) in general! Now, to compute the second derivative, we look at

2 1 2 1 2 1 2 d 1 2 2 d (x0; x (s)) + 2 d (x0; x ( s)) 2 2 d (x0; x (0)) d (x0; x (s)) = lim ds2 2 s 0 s2 s=0 ! 2 = E 0 (X;X) :

1 @2 1 @ @ 2E (X;X) = g ; dt 2 @s2 @t @t s=0 Z0 1 @ @ @ @ = g DsDs ; + g Ds ;Ds dt @t @t @t @t Z0 1 @ @ @ @ = g D D ; + g D ;D dt s t @s @t t @s t @s Z0 1 @ @ @ @ @ @ = g D D ; + g R ; ; + g (D X;D X) dt: t s @s @t @s @t @s @t t t Z0 @ @ 1 1 = g D ; + g (R (X; ) X; ) + g (D X;D X) dt s @s @t 0 0 t t 0 Z0 1 1 2 2 = K (X; 0) X 0 + g (DtX;DtX) dt j j j j Z0 Z0 @ Since I can also choose the variation so that Ds @s (s; t = 1) = 0: @x Fact: The vector …eld X = tv where v is the parallel transport of X (1) = @s along the curve : This implies that

1 1 2 2 2 E (X;X) = K (X; 0) X 0 + g (DtX;DtX) dt j j j j Z0 Z0 1 2 2 2 @x = K (X; 0) X 0 + : j j j j @s Z0

74 CHAPTER 9. REDUCED DISTANCE

@x Now if we take @s s=0 = 1; then we have

2 2 d 1 2 @x d (x0; x (s)) = 1 ds2 2 @s s=0

if the sectional curvatures are positive. Now take normal coordinates at x: At the center of normal coordinates,

@ 2 @ 2 f (0) = f + f: 4 @x1 @xd @ So, taking x (s) = expx s @xi ; we conclude that

1 2 d (x0; x) d: 4 2 Furthermore, we have

1 2 d (x0; x) = (d (x0; x) d (x0; x)) 4 2 r r 2 = d (x0; x) d (x0; x) + d (x0; x) 4 jr j = d (x0; x) d (x0; x) + 1: 4 Thus, we get

d (x0; x) d (x0; x) + 1 d 4 d 1 d (x0; x) : 4 d (x0; x) 9.5 Variations of the reduced distance

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

2 ( ) = p 0 () g() + Rg() d: L 0 j j Z Actually, we want to think of this as a functional on spacetime paths. A spacetime path is a map : [0; ] M I ! where 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 formulation of the functional as 2 ( ) = p ~0 () g() + R ( ()) d: L 0 j j Z 9.5. VARIATIONS OF THE REDUCED DISTANCE 75

We will de…ne the reduced length as 1 ` (; x) = inf ( ): are of the form (9.1) : (0;x0) 2p fL 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 be horizontal vector …elds, we …rst consider the …rst variation of with respect to @ L a horizontal vector …eld X: Note that ~0 = 0 is a horizontal vector …eld @ (but 0 is not).

Remark 85 Horizontal vector …elds on M I are in one-to-one correspondence with vector …elds on M: For this reason, we may abuse notation and use the same notation 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; ) = X: @s s=0

This is a horizontal variation. We get

@ @~ @~ (X) = () = p 2 @ ; + @ Rg() d: L @s L 0 0 r @s @ @ r @s 1 s=0 Z * +g()

@ A We do the same thing we did with the energy functional, …nding

@ @~ (X) = p 2 @~ ; + @ Rg() d L 0 0 r @ @s @ r @s 1 Z * +g() s=0 @ A ~ ~ ~ d @ @ @ @ @ @ = p 2 ; 4 Rc ; 2 ; @~ + @ Rg() d 0 0 d @s @ @s @ @s r @ @ r @s 1 Z * +g() ! * +g() s=0 @ A d = p 2 X; ~0 4 Rc (X; ~0) 2 X; ~ ~0 + X R d d h ig() h r 0 ig() r g() Z0 since if X and Y are horizontal, d g (X;Y ) = 2 Rc (X;Y ) + g ( ~ X;Y ) + g (X; ~ Y ) : d r 0 r 0 76 CHAPTER 9. REDUCED DISTANCE

Remark 86 In general, the formula is d @ g (X;Y ) = g (X;Y ) + ~ g (X;Y ) ; d @ r 0 which has more terms since ~ X = 0; etc. r 0 6 Remark 87 I used the notation of total derivative since there is the variation of the metric with respect to the time parameter of Ricci ‡ow and also the variation with respect to 0: I have also used 0 instead of dot to denote derivative with respect to or :

d Now we need to take the d out, so we need that d 1 d 2p X; ~0 = X; ~0 + 2p X; ~0 ; d h ig() p h ig() d h ig() so d 1 (X) = 2p X; ~0 g() X; ~0 g() L 0 d h i p h i Z + p 4 Rc (X; ~0) 2 X; ~ ~0 + X R d h r 0 ig() r g() = 2p X; ~0 g() 2p G; X g() d h i 0 0 h i Z where G is the vector …eld 1 1 G () = ~ ~0 + ~0 + 2 Rc (~ 0) R r 0 2 2r g() where Rc (X) is the vector …eld on M such that

Rc (X) ;Y = Rc (X;Y ) h i for all for all vector …elds Y on M: Notice that G does not depend on the variation X: Note that if is a minimizer with …xed endpoints (i.e., for all variations X such that X (0) = X () = 0), then we must have that G = 0: This is the -geodesic equation. L Problem 88 If we are in Euclidean space, what are the -geodesics? L Now supposing there is a unique minimizer and that ` is a smooth function, we can consider variations through -geodesics such that X (0) is …xed to get L

(X) = 2p X; ~0 ; L h ig() which implies that

@ @x ` (; x (s)) = ; ~ @s (0;x0) @s 0 g() 9.5. VARIATIONS OF THE REDUCED DISTANCE 77

(recall that ` divides by p). This can also be written as

` = ~0: r (0;x0) Note that the reduced length also depends on time, so let’s compute the time derivative as well:

d 2 = p ~0 () + R : L d j jg() g() Note that this need not be zero since we are not minimizing in this direction. We …nd that d 2 2p` = p ~0 () + R : d (0;x0) j jg() g() Now, the total derivative decomposes as d @ ` = ` + ~ ` ; d (0;x0) @ (0;x0) r 0 (0;x0) and so

@ 1 2 1 1 2 ` (; x) = ~0 () + R ` (; x) ~0 () : @ (0;x0) 2 j jg() g() 2 (0;x0) 2p j jg()

In order to look at second derivatives of `(0;x0); we consider variations among minimizers. In particular, we let be a minimizer with G = 0: We compute the second variation of . It will be convenient to assume the variation is through -geodesics, that is L L @ @ = 0: r @s @s This implies that @2 R ( (s; )) = X X R @s2 r r s=0 2 where X X R means the Hessian R (X;X) : Then we have, r r r 2 2 d (X;X) = ( (s; )) L ds2 L s=0

~ ~ d @ @ = p 2 @ ; + @ Rg() d ds 0 0 r @s @ @ r @s 1 s=0 Z * +g()

@ A @~ @~ @~ @~ @ = p 2 @ @ ; + 2 @ ; @ + @ @ Rg() + @ (R) d 0 0 r @s r @s @ @ r @s @ r @s @ r @s r @s r @s @s 1 Z * +g() * +g() @ A = p 2 X X ~0; ~0 g() + 2 X ~0; X ~0 g() + X X Rg() d 0 hr r i hr r i r r Z @~ @~ @~ @~ @ = p 2 @ @ ; + 2 @ ; @ + @ @ Rg() + @ (R) d 0 0 r @s r @s @ @ r @s @ r @s @ r @s r @s r @s @s 1 Z * +g() * +g() @ A 78 CHAPTER 9. REDUCED DISTANCE

d 1 (X) = 2p X; ~0 g() X; ~0 g() L 0 d h i p h i Z + p 4 Rc (X; ~0) 2 X; ~ ~0 + X R d h r 0 ig() r g() = 2p X; ~0 g() 2p G; X g() d h i 0 0 h i Z

Chapter 10

Problems

1) Show that if (M; g) is -noncollapsed at x0 at the scale of p; then it is -noncollapsed at x0 at all scales smaller than p:

2) Recall the functionals

2 f F (M; g; f) = R + f e dV jr j Z 2 d=2 f W (M; g; f; ) = R + f + f d (4) e dV jr j Z h i and their corresponding

f (M; g) = inf F (M; g; f): e dV = 1 ZM d=2 f f (M; g; ) = inf W (M; g; f; ): (4) e e dV = 1 : ZM By considering variations of the function f (with M; g; …xed), show that the minimizers f for and f# for satisfy the di¤erential equations 2 f f 2 + R = ; 4 jr j 2 R + 2 f# f# + f# d = : 4 jr j 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 identifying 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). In fact, you could do this for any closed manifold with R = 0:

4) On the same torus, suppose a b: For which scales p is (M; g) a- noncollapsed? How does this compare with the estimate one might get from part 3?

79 80 CHAPTER 10. PROBLEMS

Then we have

2 d=2 f W (M; g; f; ) = f + f d (4) e dV jr j Z h i Now consider

2 d=2 f (M; g; ) = inf f + f (4) e dV d jr j Z h i And so a minimizer satis…es

R + 2 f f 2 + f d = 4 jr j If we restrict to R = 0; then

2 f f 2 + f d = 4 jr j Notice that if f (x) = c; then

d=2 f d=2 c (4) e dV = (4) e V (M) Z and so the constraint is

d=2 c (4) e V (M) = 1 d=2 c = log (4) V (M) ; h i which satis…es

2 d=2 2 f f + f d = log (4) V (M) d = 4 jr j h i or ab d = log d log (4) d=2 2

2 f F = Rh f h + 2 f h e dV jr j r r Z 2 2 f = Rh f h 2 f + 2 f e dV jr j 4 jr j Z

2 d=2 f W = (2 f h) + h h R + f + f d (4) e dV r r jr j Z h h ii 2 d=2 f = h R + 2 f f + f d (4) e dV 4 jr j Z h h ii