Notes from Math 538: Ricci ow and the Poincare conjecture. David Glickenstein

Transcription

1 Notes from Math 538: Ricci ow an the Poincare conjecture Davi Glickenstein Spring 9

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3 Contents Introuction Three-manifols an the Poincaré conjecture 3. Introuction Examples of 3-manifols Spherical manifols Sphere bunles over S Connecte sum Iea of the Hamilton-Perelman proof Hamilton s rst result D case General case Backgroun in i erential geometry 9 3. Introuction Basics of tangent bunles an tensor bunles Connections an covariant erivatives What is a connection? Torsion, compatibility with the metric, an Levi-Civita connection Higher erivatives of functions an tensors Curvature Basics of geometric evolutions 4. Introuction Ricci ow Existence/Uniqueness Basics of PDE techniques 9 5. Introuction The maximum principle Maximum principle on tensors iii

4 iv CONTENTS 6 Singularities of Ricci Flow Introuction Finite time singularities Blow ups Convergence an collapsing noncollapse Ricci ow from energies Graient ow Ricci ow as a graient ow Perelman entropy Log-Sobolev inequalities Noncollapsing Ricci ow for attacking geometrization Introuction Fining canonical neighborhoos Canonical neighborhoos How surgery works Some things to prove Reuce istance Introuction Short iscussion of W vs reuce istance L-length an reuce istance Variations of length an the istance function Variations of the reuce istance Problems 79

5 Preface These are notes from a topics course on Ricci ow an the Poincaré Conjecture from Spring 8. v

6 vi PREFACE

7 Chapter Introuction These are notes from a topics course on Ricci ow an the Poincaré Conjecture from Spring 8.

8 CHAPTER. INTRODUCTION

9 Chapter Three-manifols an the Poincaré conjecture. Introuction This lecture is mostly taken from Tao s lecture. In this lecture we are going to introuce the Poincaré conjecture an the iea of the proof. The rest of the class will be going into i erent etails of this proof in varying amounts of careful etail. All manifols will be assume to be without bounary unless otherwise speci e. The Poincaré conjecture is this: Theorem (Poincare conjecture) Let M be a compact 3-manifol which is connecte an simply connecte. Then M is homeomorphic to the 3-sphere S 3. Remark A simply connecte manifol is necessarily orientable. In fact, one can prove a stronger statement calle Thurston s geometrization conjecture, which is quite a bit more complicate, but is roughly the following: Theorem 3 (Thurston s geometrization conjecture) Every 3-manifol M can be cut along spheres an -essential tori such that each piece can be given one of 8 geometries (E 3 ; S 3 ; H 3 ; S R; H R; f SL (; R) ; Nil; Sol). We may go into this conjecture a little more if we have time, but certainly elements of this will come up in the process of these lectures. We will take a quick look. Examples of 3-manifols Here we look in the topological category. It turns out that in three imensions, the smooth category, the piecewise linear category, an the topological category 3

10 4CHAPTER. THREE-MANIFOLDS AND THE POINCARÉ CONJECTURE are the same (e.g., any homeomorphism can be approximate by a i eomorphism, any topological manifol can be given a smooth structure, etc.).. Spherical manifols The most basic 3-manifol is the 3-sphere, S 3 ; which can be constructe in several ways, such as: The set of unit vectors in R 4 The one point compacti cation of R 3 : Using the secon e nition, it is clear that any loop can be contracte to a point, an so S 3 is simply connecte. One can also look at the rst e nition an see that the rotations SO(4) of R 4 act transitively on S 3 ; with stabilizer SO (3) ; an so S 3 is a homogeneous space escribe by the quotient SO (4) =SO (3) : One can also notice that the unit vectors in R 4 can be given the group structure of the unit quaternions, an it is not too har to see that this group is isomorphic to SU () ; which is the ouble cover of SO (3) : As more examples of 3-manifols, it is possible to n nite groups acting freely on S 3 ; an consier quotients of S 3 by the action. Note that if the action is nontrivial, then these new spaces are not simply connecte. One can see this in several ways. The irect metho is that since there must be g an x S 3 such that gx 6= x: A path from x to gx in S 3 escens to a loop in S 3 = : If there is a homotopy of that loop to a point, then one can lift the homotopy to a homotopy H : [; ][; ]! S 3 such that H (t; ) = (t) : But then, looking at H (; s) ; we see that H (; ) = gx an H (; ) = x an H (; s) = g x for some g for all s: But since is iscrete, this is impossible. A more high level approach shows that the map S 3! S 3 = is a covering map, an, in fact, the universal covering map an so S 3 = = if acts e ectively. Hence each of these spaces S 3 = are i erent manifols than S 3 : They are calle spherical manifols. The elliptization conjecture states that spherical manifols are the only manifols with nite funamental group... Sphere bunles over S The next example is to consier S bunles over S ; which is the same as S [; ] with S fg ienti e with S fg by a homeomorphism. Recall that a homeomorphism : S! S inuces an isomorphism on homology (or cohomology), : H S! H S an since H S =

11 .3. IDEA OF THE HAMILTON-PERELMAN PROOF 5 there are only two possibilities for : In fact, these classify the possible maps up to continuous eformation, an so there is an orientation preserving an an orientation reversing homeomorphism an that is all (using instea of H ). (See, for instance, Breon, Cor. 6.4 in Chapter.) Notice that these manifols have a map : M! S : It is clear that this inuces a surjective homomorphism on funamental group : (M)! S =. The kernel of this map consists of loops which can be eforme to maps only on S (constant on the other component), an since S is simply connecte, this map is an isomorphism. Note that these manifols are thus not simply-connecte...3 Connecte sum One can also form new manifols via the connecte sum operation. Given two 3-manifols, M an M ; one forms the connecte sum by removing a isk from each manifol an then ientifying the bounary of the remove isks. We enote this as M#M : Recall that in D, all manifols can be forme from the sphere an the torus in this way. Now, we may consier the class of all compact, connecte 3-manifols (up to homeomorphism) with the connecte sum operation. These form a monoi (essentially a group without inverse), with an ientity (S 3 ). Any nontrivial (i.e., non-ientity) manifol can be ecompose into pieces by connecte sums, i.e., given any M; we can write M M #M # #M k where M j cannot be written as a connecte sum any more (this is a theorem of Kneser). We call such a ecomposition a prime ecomposition an such manifols M j prime manifols. The proof is very similar to the funamental theorem of arithmetic which gives the prime ecomposition of positive integers. Proposition 4 Suppose M an M are connecte manifols of the same imension. Then. M#M is compact if an only if both M an M are compact.. M#M is orientable if an only if both M an M are orientable. 3. M#M is simply connecte if an only if both M an M are simply connecte. We leave the proof as an exercise, but it is not too i cult..3 Iea of the Hamilton-Perelman proof In orer to give the iea, we will introuce a few concepts which will be e ne more precisely in successive lectures.

12 6CHAPTER. THREE-MANIFOLDS AND THE POINCARÉ CONJECTURE Any smooth manifol can be given a Riemannian metric, enote g ij or g (; ) ; which is essentially an inner prouct (i.e., symmetric, positive-e nite bilinear form) at each tangent space which varies smoothly as the basepoint of the tangent space changes. The Riemannian metric allows one to e ne angles between two curves an also to measure lengths of (piecewise C -) curves by integrating the tangent vectors of a curve. That is, if : [; a]! M is a curve, we can calculate its length as ` () = a g ( _ (t) ; _ (t)) = t; where _ (t) is the tangent to the curve at t: A Riemannian metric inuces a metric space structure on M; as the istance 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 inuces the original topology of the manifol. The main iea is to eform any Riemannian metric to a stanar one. This is the iea of geometrizing. How oes one choose the eformation? R. Hamilton rst propose to eform by an equation calle the Ricci ow, which is a partial i erential equation e ne by t g ij = R ij = Rc (g ij ) where t is an extra parameter (not relate to the original coorinates, so g ij = g ij (t; x) ; where x are the coorinates) an R ij = Rc (g ij ) is the Ricci curvature, a i erential operator (n orer) on the Riemannian metric. That means that the Ricci ow equation is a partial i erential equation on the Riemannian metric. In coorinates, it roughly looks like t g ij = g k` x k x` gij + F (g ij ; g ij ) where g k` is the inverse matrix of g ij an F is a function epening only on the metric an rst erivatives of the metric..3. Hamilton s rst result The iea is that as the metric evolves, its curvature becomes more an more uniform. It was shown in Hamilton s lanmark 98 paper that Theorem 5 (Hamilton) Given a Riemannian 3-manifol (M; g ) with positive Ricci curvature, then the Ricci ow with g () = g exists on a maximal time interval [; t ): Furthermore, the Ricci curvature of the metrics g (t) become increasingly uniform as t! t. More precisely, R ij (t) R (t) gij (t) ; where R is the average scalar curvature, converges uniformly to zero as t! t : From this, one can easily show that a rescaling of the metric converges to the roun sphere.

13 .3. IDEA OF THE HAMILTON-PERELMAN PROOF 7.3. D case In the D case it can be shown that any compact, orientable Riemannian manifol converges uner a renormalize Ricci ow (renormalize by rescaling the metric an rescaling time) to a constant curvature metric. This is primarily ue to Hamilton, with one case nishe by B. Chow. It is possible to use this metho to prove the uniformization theorem, which states that any compact, orientable Riemannian manifol can be conformally eforme to a metric with constant curvature. (The original proofs of Hamilton an Chow use the uniformization theorem, but a recent article by Chen, Lu, an Tian shows how to avoi that)..3.3 General case Hamilton introuce a program to stuy all 3-manifols using the Ricci ow. It was iscovere quite early that the Ricci ow may evelop singularities even in the case of a sphere if the Ricci curvature is not positive. An example is the so-calle neck pinch singularity. Hamilton s iea was to o surgery at these singularities, then continue the ow an continue to o this until no more surgeries are necessary. Perelman s work escribes what happens to the Ricci ow near a singularity an also how to perform the surgery. The new ow is calle 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-manifol. Then there exists a Ricci ow with surgery t! (M (t) ; g (t)) for all t [; ) an a close set T [; ) of surgery times such that:. (Initial ata) M () = M; g () = g:. (Ricci ow) If I is any connecte component of [; ) n T (an thus an interval), then t! (M (t) ; g (t)) is the Ricci ow on I (you can close this interval on the left enpoint if you wish). 3. (Topological compatibility) If t T an " > is su ciently small, then we know the topological relationship M (t ") an M (t) : 4. (Geometric compatibility) For each t T; the metric g (t) on M (t) is relate to a certain limit of the metric g (t ") on M (t ") by a certain surgery proceure. Note, we can express the topological compatibility more precisely. We have that M (t ") is homeomorphic to the connecte sum of nitely many connecte components of M (t) together with a nite number of spherical space forms (spherical manifols), RP 3 #RP 3, an S S : Furthermore, each connecte component of M (t) is use in the connecte sum ecomposition of exactly one component of M (t ") :

14 8CHAPTER. THREE-MANIFOLDS AND THE POINCARÉ CONJECTURE Remark 7 The case of RP 3 #RP 3 is interesting in that it is apparently the only nonprime 3-manifol which amits a geometric structure (i.e., is covere by a moel geometry; it is a quotient of S R; this oes not contraict our argument above because it is not a sphere bunle over S ). I have seen this mentione several places, but I o not have a reference. Remark 8 Morgan-Tian an Tao give a more general situation where nonorientable manifols are allowe. This as some extra technicalities which we will avoi in this class. The existence nees something more to show the Poincaré conjecture. One nees that the surgeries are only iscrete an 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 manifol M () : Then the set T of surgery times is iscrete. In particular, any compact time interval contains a nite number of surgeries. Theorem (Finite time extinction) Let (M; g) be a compact 3-manifol which is simply connecte an let t! (M (t) ; g (t)) be an associate Ricci ow with surgery. Then M (t) is empty for su ciently large t: With these theorems, one can conclue the Poincaré conjecture in the following way. Given M a simply connecte, connecte, compact Riemannian manifol, associate a Ricci ow with surgery. It has nite extinction time, an hence nite surgery times. Now one can use the topological ecomposition to work backwars an buil the manifol backwars, which says that the manifol M is the connecte sum of nitely many spherical space forms, copies of RP 3 #RP 3, an S S : But since M is the simply connecte, everything in the connecte sum must be simply connecte, an hence every piece of the connecte sum must be simply connecte, so M must be a sphere.

15 Chapter 3 Backgroun in i erential geometry 3. Introuction We will try to get as quickly as possible to a point where we can o some geometric analysis on Riemannian spaces. One shoul look at Tao s lecture, though I will not follow it too closely. 3. Basics of tangent bunles an tensor bunles Recall that for a smooth manifol M; the tangent bunle can be e ne in essentially 3 i erent ways ((U i ; i ) are coorinates) T M = F (U i R n ) = where for (x; v) U i R n ; (y; w) U j R n we i have (x; v) (y; w) if an only i y = j i (x) an w = j i (v) : x T p M = fpaths : ( "; ")! M such that () = pg = where if ( i ) () = ( i ) () for every i such that p U i : T M = F T p M: pm T p M to be the set of erivations 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 = F T p M: pm On can e ne the cotangent bunle by essentially taking the ual of T p M at each point, which we call T p M; an taking the isjoint union of these to get the cotangent bunle T M: One coul also use an analogue of the rst e nition, where the only i erence is that instea of using the vector space R n ; one uses 9

16 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY its ual an the equivalence takes into account that the ual space pulls back rather than pushes forwar. Both of these bunles are vector bunles. One can also take a tensor bunle of two vector bunles by replacing the ber over a point by the tensor prouct of the bers over the same point, e.g., T M T M = G T p M Tp M : pm Note that there are canonical isomorphisms of tensor proucts of vector spaces, such as V V is isomorphic to enomorphisms of V: Note the i erence between bilinear forms (V V ), enomorphisms (V V ), an bivectors (V V ). It is important to unerstan that these bunles are global objects, but will often be consiere in coorinates. Given a coorinate x = x i an a point p in the coorinate patch, there is a basis p x ; : : : ; p x for T n p M an ual basis x p ; : : : ; x n j p for Tp M: The generalization of the rst e nition above gives the iea of how one consiers the trivializations of the bunle in a coorinate patch, an how the patches are linke together. Speci cally, if x an y give i erent coorinates, for a point on the tensor bunle, one has T ijk abc (x) x i x j x k xa x b x c = T ijk y abc (x (y)) y x i x j y x a x b x k y y xc y y y where technically everything shoul be at p (but as we shall see, one can consier this for all points in the neighborhoo an this is consiere as an equation of sections). Recall that a section of a bunle : E! B is a function f : B! E such that f is the ientity on the base manifol B: A local section may only be e ne on an open set in B: On the tangent space, sections are calle vector els an on the cotangent space, sections are calle forms (or -forms). On a tensor bunle, sections are calle tensors. Note that the set of x form a i basis for the vector els in the coorinate x, an x i form a basis for the local -forms in the coorinates. Sections in general are often written as (E) or as C (E) (if we are consiering smooth sections). Now the equation above makes sense as an equation of tensors (sections of a tensor bunle). Often, a tensor will be enote as simply T ijk abc : Note that if we change coorinates, we have a i erent representation T of the same tensor. The two are relate by T = T ijk y abc (x (y)) y x i x j y x a x b x k y y xc y One can also take subsets or quotients of a tensor bunle. In particular, we may consier the set of symmetric -tensors or anti-symmetric tensors (sections of this bunle are calle i erential forms). In particular, we have the Riemannian metric tensor. : y y y y ;

17 3.. BASICS OF TANGENT BUNDLES AND TENSOR BUNDLES De nition 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 T p M; an positive e nite, i.e., g (X; X) all X T p M an g (X; X) = if an only if X =. Often, we will enote the metric as g ij ; which is shorthan for g ij x i x j ; where x i x j = xi x j + x j x i : Note that if g ij = ij (the Kronecker elta) then ij x i x j = x + (x n ) : One can invariantly e ne a trace of an enomorphism (trace of a matrix) which is inepenent of the coorinate change, since nx Ta a = X a= a = X a = X T T T : x a y y x a In fact for any complicate tensor, one can take the trace in one up inex an one own inex. This is calle contraction. Usually, when there is a repeate inex of one up an one own, we o not write the sum. This is calle Einstein summation convention. The above sum woul be written T a a = T : It is unerstoo that this is an equation of functions. We cannot contract two inices up or two inices own, since this is not inepenent of coorinate change (try it!) However, now that we have the Riemannian metric, we can use it to lower an inex an then trace, so we get T ab g ba = T a a : In orer to raise the inex, we nee the ual to the Riemannian metric, which is g ab ; e ne such that g ab g bc = a c (so g ab is the inverse matrix of g ab ). Then we can use g ab to raise inices an contract if necessary. Occasionally, extene Einstein convention is use, where all repeate inices are summe with the unerstaning that the Riemannian metric is use to raise or lower inices when necessary, e.g., T aa = T ab g ab : Since often we will be changing the Riemannian metric, it becomes important to unerstan that the metric is there when extene Einstein is use.

18 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY 3.3 Connections an covariant erivatives 3.3. What is a connection? A covariant erivative is a particular way of i erentiating vector els. Why o we nee a new way to i erentiate vector els? Here is the iea. Suppose we want to give a notion of parallel vectors. In R n ; we know that if we take vector els with constant coe cients, those vectors are parallel ati erent points. That is, the vectors (;) x + (;) x an (; x ) + (; x ) are parallel. In fact, we coul say that the vector el x + x is parallel since vectors at any two points are parallel. One might say it is because the coe cients of the vector el are constant (not functions of x an x ). However, this notion is not invariant uner a change of coorinates. Suppose we consier the new coorinates y ; y = x ; x away from x = (where it is not a i eomorphism). Then the vector el in the new coorinates is y i x y i + yj x x j = y + 4x y = y + 4p y y : The coe cients are not constant, but the vector el shoul still be parallel (we have only change coorinates, so it is the same vector el)! So we nee a notion of parallel vector el that is inepenent of coorinate changes (or covariant). Remember that we want to generalize the notion that a vector el has constant coe cients. Let X = X i be a vector el in a coorinate patch. x i Roughly speaking, we want to generalize the notion that Xi x = for all i an j j: The problem occurre because x x is i erent in i erent coorinates. Thus we nee to specify what this is. Certainly, since x is a basis, we must i get a linear combination of these, so we take r i x j = k ij x k k for some functions ij : These symbols are calle Christo el symbols. To make sense on a vector el, we must have r i (X) = r i X j x j = Xj x i x j + Xj k ij x k X k = x i + X j k ij x k : Notice the Leibniz rule (prouct rule). One can now e ne r for any vector Y = Y i x by i r Y X = r X = Y i Y i (r i X) : x i

19 3.3. CONNECTIONS AND COVARIANT DERIVATIVES 3 This action is calle the covariant erivative. k One now e nes ij in such a way that the covariant erivative transforms appropriately uner change of coorinates. This gives a global object calle a connection. The connection can be e ne axiomatically as follows. De nition A connection on a vector bunle E! B is a map satisfying: r : (T B) (E)! (E) (X; )! r X Tensoriality (i.e., C (B)-linear) in the rst component, i.e., r fx+y = fr X + r Y for any function f an vector els X; Y Derivation in the secon component, i.e., r X (f) = X (f) + fr X : R-linear in the secon component, i.e., r X (a + ) = ar X () + r X ( ) for a R. We will consier connections primarily on the tangent bunle an tensor bunles. Note that a connection r on T M inuces connections on all tensor bunles (also enote r) in the following way: For a function f an vector el X; we e ne r X f = Xf For vector els X; Y an ual form!; we use the prouct rule to erive r X (! (Y )) = X (! (Y )) = (r X!) (Y ) +! (r X Y ) an thus (r X!) (Y ) = X (! (Y ))! (r X Y ) : In particular, the Christo el symbols for the connection on T M are the negative of the Christo el symbols of T M; i.e., r x i xj = j ik xk where k ij are the Christo el symbols for the connection r on T M: For a tensor prouct, one e nes the connection using the prouct rule, e.g., r X (Y!) = (r X Y )! + Y r X! for vector els X; Y an -form!:

20 4 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY Remark 3 The Christo el symbols are not tensors. Note that if we change coorinates from x to ~x; we have r x i x j = r ~x k x i which means that ~x` ~x k x j = ~x` ~x` x i x j k ij = ~ m ~x p ~x` x k ~x` p` x i x j ~x m + x k x i x j ~x` : ~x` + ~xk x i ~x` x j r ~x k ~x` One nal comment. Recall that we motivate the connection by consiering parallel vector els. The connection gives us a way of taking a vector at a point an translating it along a curve so that the inuce vector el along the curve is parallel (i.e., r _ X = along ). This is calle parallel translation. Parallel vector els allow one to rewrite erivatives in coorinates; that is, if X = X i is parallel, then x i X i x j = Xk i jk: 3.3. Torsion, compatibility with the metric, an Levi- Civita connection There is a unique metric associate with the Riemannian metric, calle the Riemannian connection or Levi-Civita connection. It satis es two properties: Torsion-free (also calle symmetric) Compatible with the metric. Compatibility with the metric is the easy one to unerstan. We want the connection to behave well with respect to i erentiating orthogonal vector els. Being compatible with the metric is the same as r X (g (Y; )) = g (r X Y; ) + g (Y; r X ) : Note that normally there woul be an extra term, (r X g) (Y; ) ; so compatibility with the metric means that this term is zero, i.e., rg = ; where g is consiere as a -tensor. Torsion free means that the torsion tensor ; given by (X; Y ) = r X Y r Y X [X; Y ] vanishes. (One can check that this is a tensor by verifying that (fx; Y ) = (X; fy ) = f (X; Y ) for any function f). It is easy to see that in coorinates, the torsion tensor is given by k ij = k ij k ji;

21 3.3. CONNECTIONS AND COVARIANT DERIVATIVES 5 which inicates why torsion-free is also calle symmetric. Tao gives a short motivation for the concept of torsion-free. Consier an in nitesimal parallelogram in the plane consisting of a point x; the ow of x along a vector el V to a point we will call x + tv; the ow of X along a vector el W to a point we will call x + tw; an then a fourth point which we will reach in two ways: () go to x + tv an then ow along the parallel translation of W for a istance t an () go to x + tw an then ow along the parallel translation of V for a istance t: Note that using metho (), we get that the point is (x + tv + sw )j s= + t s (x + tv + sw ) + O t 3 s= = x + tv + tw + t s V + O t 3 = x + tv + tw t V i W j k ji s= x k + O t3 : Note that using metho (), we get instea x + tv + tw t W i V j k ji x k + O t3 ; Thus this vector is x+t (V + W ) up to O t 3 k only if ji = k ij : Doing this aroun every in nitesimal parallelogram gives the equivalence of these two viewpoints. Here is another: Proposition 4 A connection is torsion-free if an only if for any point p M; there are coorinates x aroun p such that k ij (p) = : k Proof. Suppose one can always n coorinates such that ij (p) = : Then clearly at that point, ij k = : However, since the torsion is a tensor, we can calculate it in any coorinate, so at each point, we have that the torsion vanishes. Now suppose the torsion tensor vanishes an let x be a coorinate aroun p: Consier the new coorinates ~x i (q) = x i (q) x i (p) + i jk (p) x j (q) x j (p) x k (q) x k (p) : Then notice that ~x i x j = i j + i k` (p) k j x` x` (p) + i k` (p) x k x k (p) `j an so ~x i x j (p) = i j: Thus ~x is a coorinate patch in some neighborhoo of p: Moreover, we have that ~x i x j x k = i jk (p) :

22 6 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY One can now verify that at p; k ij (p) = ~ m ~x p ~x` x k ~x` p` x i x j ~x m + x k x i x j ~x` = ~ k ij (p) + k ij (p) : The Riemannian connection is the unique connection which is both torsionfree an compatible with the metric. One can use these two properties to erive a formula for it. In coorinates, one ns that the Riemannian connection has the following Christo el symbols k ij = gk` x i g j` + x j g i` x` gij One can easily verify that this connection has the properties expresse. Note that the g j` 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 coorinates (i.e., that x ; i x = ). j Higher erivatives of functions an tensors One of the important reasons for having a connection is it allows us to take higher erivatives. Note that one can take the erivative of a function without a connection, an it is e ne as : f = rf f (X) = r X f = X (f) f = f x i xi : One can also raise the inex to get the graient, which is gra (f) = r i f x i = f x j gij x i : However, to take the next erivative, one nees a connection. erivative, or Hessian, of a function is The secon Hess (f) = r f = rf r f = (r i f) x i f = r i x j xj f = x i x j xj f = x i x j x i f x k f x j k ij j ik xk x i x j x i :

23 3.4. CURVATURE 7 Often one will write the Hessian as r ijf = r i r j f = f x i x j k ij f x k : Note that if the connection is symmetric, then the Hessian of a function is symmetric in the usual sense. The trace of the Hessian, 4f = g ij r ijf; is calle the Laplacian, an we will use it quite a bit. We also may use the connection to compute acceleration of a curve. The velocity of a curve is _; which oes not nee a connection, but to compute the acceleration, r ; we nee the connection (one also sometimes sees the equivalent notation D _=t). A curve with zero acceleration is calle a geoesic. Finally, given any tensor T; one can use the connection to form a new tensor rt; which has an extra own inex. 3.4 Curvature One can e ne the curvature of any connection on a bunle E! B in the following way R : (T M) (T M) (E)! (E) R (X; Y ) = r X r Y r Y r X r [X;Y ] : We will consier the curvature of the Riemannian connection on the tangent bunle. One can easily see that in coorinates, the curvature is a tensor enote as r i r j x k r j r i x k = Rìjk x` which gives us that ` r i jk r j x` ` ik So the curvature tensor is Rìjk = = x` x i = `jk x m x j x` + `jk m i` `jk x i x j ìk + m jk ìm m ik `jm `jk x i x j ìk + m jk ìm m ik `jm : Often we will lower the inex, an consier instea the curvature tensor R ijk` = R m ijkg m`: The Riemannian curvature tensor has the following symmetries: ìk x` x` R ijk` = R jik` = R ij`k = R k`ij (These imply that R can be viewe as a self-ajoint (symmetric) operator mapping -forms to -forms if one raises the rst two or last two inices). ` ik m j` x m

24 8 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY (Algebraic Bianchi) R ijk` + R jki` + R kij` = : (Di erential Bianchi) r i R jk`m + r j R ki`m + r k R ij`m : Remark 5 The tensor R ijk` can also be written as a tensor R (X; Y; ; W ) ; which is a function when vector els X; Y; ; W are plugge in. We will sometimes refer to this tensor as Rm : The tensor Rìjk is usually enote by R (X; Y ) ; which is a vector el when vector els X; Y; are plugge in. Remark 6 Sometimes, the up inex is lowere into the 3r spot instea of the 4th, This will change the e nitions of Ricci an sectional curvature below, but the sectional curvature of the sphere shoul always be positive an the Ricci curvature of the sphere shoul be positive e nite. k Remark 7 Note that ij involve rst erivatives of the metric, so Riemannian curvature tensor involves rst an secon erivatives of the metric. From these one can erive all the curvatures we will nee: De nition 8 The Ricci curvature tensor R ij is e ne as R ij = R``ij = R`ijm g`m : Note that R ij = R ji by the symmetries of the curvature tensor. Ricci will sometimes be enote Rc (g) ; or Rc (X; Y ) : De nition 9 The scalar curvature R is the function R = g ij R ij De nition The sectional curvature of a plane spanne by vectors X an Y is given by R (X; Y; Y; X) K (X; Y ) = g (X; X) g (Y; Y ) g (X; Y ) : Here are some facts about the curvatures: Proposition. The sectional curvatures etermine the entire curvature tensor, i.e., if one can calculate all sectional curvatures, then one can calculate the entire tensor.. The sectional curvature K (X; Y ) is the Gaussian curvature of the surface generate by geoesics in the plane spanne 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 i erence between the volumes of small metric balls an the volumes of Eucliean balls of the same raius.

25 3.4. CURVATURE 9 6. In imensions, each curvature etermines the others. 7. In 3 imensions, scalar curvature oes not etermine Ricci, but Ricci oes etermine the curvature tensor. 8. In imensions larger than 3, Ricci oes not etermine the curvature tensor; there is an aitional piece calle the Weyl tensor. With this in min, we can talk about several i erent kins of nonnegative curvature. De nition Let x be a point on a Riemannian manifol (M; g) : Then x has. nonnegative scalar curvature if R (x) ;. nonnegative Ricci curvature at x if Rc (X; X) = R ij X i X j for every vector X T x M; 3. nonnegative sectional curvature if R (X; Y; Y; X) = g (R (X; Y ) Y; X) for all vectors X; Y T x M; 4. nonnegative Riemann curvature (or nonnegative curvature operator) if Rm as a quaratic form on (M) ; i.e., if R ijk`! ij! k` for all -forms! =! ij x i ^ x j (where the raise inices are one using the metric g). It is not too har to see that 4 implies 3 implies implies : Also, in 3 imensions, 3 an 4 are equivalent. In imension 4 an higher, these are all istinct.

26 CHAPTER 3. BACKGROUND IN DIFFERENTIAL GEOMETRY

27 Chapter 4 Basics of geometric evolutions 4. Introuction This lecture roughly follows Tao s Lecture. We will talk in general about ows or Riemannian metrics an Ricci ow. We will consier a ow of Riemannian metrics to be a one-parameter family of Riemannian metrics, usually enote g (t) or g ij (t) or g ij (x; t) on a xe Riemannian manifol M. There are more ingenious ways to e ne such a ow using spacetimes (calle generalize Ricci ows). However, at present I o not think that they give a signi cant savings over the more classical iea, since one still nees to consier singular spacetimes. For more on generalize Ricci ows, consult the book by Morgan-Tian. The family g (t) is a one-parameter family of sections of a vector bunle, an one can take its erivative as g (t) = lim t t! g (t + t) t g (t) since g (t) an g (t + t) are both sections of the same vector bunle, so the i erence makes sense. In fact, we can i erentiate any tensor in this way. Similarly, we can try to solve i erential equations of the form t g ij = _g ij for some prescribe _g ij : The evolution of the metric inuces an evolution of the metric on the cotangent bunle, using t g ij g jk = t i j t gij = g ik _g k`g`j :

28 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS The Riemannian connection is also changing if the metric is changing. Thus for a xe vector el X; we have t r ix = X j t x i x j + k ijx j x k = X j _ k ij x k : We can use the fact that the connection is torsion-free an compatible with the metric to erive the formula for _ k ij : = t (r ig jk ) = t x i g jk = r i _g jk _ ìj g`k _ ìk g j`; ` ijg`k ìk g j` an = _ k ij _ k ji so we can solve for _ k ij as r i _g jk = _ ìjg`k + _ ìkg j` r j _g ki = _ `jk g`k + _ `ji g k` r k _g ij = _ `ki g`j + _ `kj g i` to get _ k ij = gk` (r i _g j` + r j _g i` r` _g ij ) : (4.) Remark 3 This mimics the proof of the formula for the Riemannian connection given that it is torsion-free an compatible with the metric. There are other ways to erive this formula, for instance by computing in normal coorinates an using the fact that although k ij is not a tensor, k t ij comes from the i erence of two connections an is thus a tensor. We will use this metho below. We may now look at the inuce formula for evolution of the Riemannian curvature tensor. Recall that, in coorinates, Rìjk x` = r i ` jk x` r j ` ik x` Since we are intereste in the erivative of a tensor, trìjk = Rìjk; _ we can compute this in any coorinate system we want. Recall that there is a coorinate system aroun p such that all Christo el symbols vanish at p: Doing this reuces :

29 4.. INTRODUCTION 3 the equation to _Rìjk x` = t r i ` jk x i `jk x` = t = x _ `jk i x` = r i _ `jk x` ` r j ik x` x` x j ìk x` x _ j ìk x` r j _ ìk x` : This last piece is tensorial (recall that _ k ij is a tensor), an thus only epens on the point, not the coorinate patch, so we must have that _Rìjk = r i _ `jk r j _ ìk : We can now use the (4.) to get _Rìjk = r g`m i (r j _g km + r k _g jm r m _g jk ) r g`m j (r i _g km + r k _g im r m _g ik ) = g`m (r i r j _g km + r i r k _g jm r i r m _g jk r j r i _g km r j r k _g im + r j r m _g ik ) = g`m (r i r j _g km r j r i _g km + r i r k _g jm r i r m _g jk r j r k _g im + r j r m _g ik ) = g`m ( R p ijk _g mp R p ijm _g kp + r i r k _g jm r i r m _g jk r j r k _g im + r j r m _g ik ): We can take the trace R _ jk = R _ ijk i to get _R jk = gim ( R p ijk _g mp R p ijm _g kp + r i r k _g jm r i r m _g jk r j r k _g im + r j r m _g ik ) where = gim R p ijk _g mp + gpq R jp _g kp gim r i r m _g jk gim r j r k _g im + gim (r k r i _g jm R p ikj _g pm R p ikm _g jp + r j r m _g ik ) = g im R p ijk _g mp + gpq R jp _g kp + gpq R kq _g jp gim r i r m _g jk gim r j r k _g im + gim (r k r i _g jm + r j r m _g ik ) = 4 L _g jk gim r j r k _g im + gim (r k r i _g jm + r j r m _g ik ) 4 L _g jk = g im r i r m _g jk + g im R p ijk _g mp g pq R jp _g kp g pq R kq _g jp is the Lichnerowitz Laplacian (notice only the rst term has two erivatives 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 :

30 4 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS Finally, we may take a trace to get _R = g jk _R jk g pj _g pq g qk R jk = g pq g jk R jp _g kp g im r i r m g jk _g jk + g im g jk r k r i _g jm = hrc; _gi 4 tr g ( _g) + iv iv _g where (iv h) j = g k`r k h`j 4. Ricci ow Note that in the evolution of Ricci curvature, if one consiers _g = Rc; one gets gim r j r k _g im + gim (r k r i _g jm + r j r m _g ik ) = g im r j r k R im g im r k r i R jm g im r j r m R ik = r j r k R g im r j r m R ik g im r k r i R jm Note that the i erential Bianchi ientity implies so = g jm g k` (r i R jk`m + r j R ki`m + r k R ij`m ) = r i R g jm r j R im g k`r k R ij`m r i R = g jm r j R im so gim r j r k _g im + gim (r k r i _g jm +r j r m _g ik ) = r j r k R r jr k R r kr j R = : Thus uner Ricci ow, Furthermore, We see that R t t Rc = 4 L Rc : = hrc; Rci + 4R gi`g jk r i r j R k` = hrc; Rci + 4R g i`r i r`r = 4R + jrcj 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.

31 4.3. EXISTENCE/UNIQUENESS Existence/Uniqueness Note that the Ricci ow equation, t g = Rc is a secon orer partial i erential equation, since the Ricci curvature comes from secon erivatives of the metric. To truly look at existence/uniqueness, one must write this as an equation in coorinates. We will look at the linearization of this operator in orer to n the principle symbol (which is basically the coe cients of the linearization of the highest erivatives). Analysis of the principle symbol will often allow us to etermine 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 t a ij (x; t) u x i + F (x; t; u; u) = : xj on a compact manifol has a solution with initial conition u (x; ) = f (x) if there exists > such that a ij i j jj (this conition is calle strict parabolicity) for t close to. Similarly, if we allow a ij to epen on u (making the equation quasilinear, the same is true if we look at the linearization (which is then semilinear). Remark 4 a ij is calle the principal symbol of the parabolic i erential operator. If one takes out the t; the i erential operator is sai to be elliptic if it satis es the inequality. with r i since the i erence has fewer eriva- Remark 5 We can replace tives. x i Remark 6 One shoul be able to prove a coorinate inepenent version, but this is not usually one. All theory is base on theory of i erential equations on omains in the plane. Remark 7 For an arbitrary, nonlinear secon orer PDE of the form G x; t; u; u; u = ; one can consier the linearization of G with respect to u: This will look roughly like Hij G x i x j + V i G x i + ug v where G = G (x; t; u; V; H) an the operator is evaluate at some u (which is where it has been linearize. Notice this now gives a semilinear PDE. The principle symbol is Hij G i j :

32 6 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS Example 8 Note that the equation is alreay linear. For an equation like u t = (x ) u + (x ) u = ij x i x j u t = u u x ; the linearization is v t = u v x + u u x v; thus the principal symbol is u which is positive if u > : Now, the Ricci operator is an operator on sections, not just functions, so how o we make sense of the kin of result given above. We can make a similar e nition in terms of the linearization, but now the principle symbol is a map from sections of the symmetric -tensor bunle to itself. What we nee is that for any 6= ; the principle symbol is a linear isomorphism. Recall that the linearization of R jk is _R jk = gim r i r m _g jk gim r j r k _g im + gim (r k r i _g jm + r j r m _g ik ) so the principle symbol of R jk is ^ [D Rc] () (h) = g im i m h jk + g im j k h im g im ( k i h jm + j k h ik ): In orer to see if this is an isomorphism, we can rotate so that > an = = n = an by scaling we can assume = : We can also assume that at a point g ij = ij : Then we see that ^ [D Rc] () (h) jk = h jk + j k (h + + h nn ) ( kh j + j h k ): An so the matrix for the symbol gives h + + h nn B. h C A where ; n: We see immeiately that there is an n-imensional kernel (we can let h k equal anything we want an if everything else is zero, we are in the kernel). Now we will see how to overcome this issue. Rewrite the linearization as _R jk = gim r i r m _g jk gim r j r k _g im + gim (r k r i _g jm + r j r m _g ik ) = gim r i r m _g jk + r kv j + r j V k

33 4.3. EXISTENCE/UNIQUENESS 7 if V j = g im r i _g jm r j g im _g im : The last term is equal to the Lie erivative L V g jk (where V = V i = g ij V j ; often enote V # ) an so we get that the linearization of R jk is g im r i r m _g jk L V g jk : Lie erivatives arise from changing by i eomorphisms, i.e., if t are i eomorphisms such that t t (x) = X (x) an is the ientity (i.e., t is the ow of X), then t t g = L X g : t= One can pretty easily see that if we take the vector el V as above, we can look at the ow t an t g (t) an we will see that ~g (t) = t g (t) evolves by an the linearization is t ~g = Rc (~g) + L V ~g g im r i r m _g jk : This is like looking at an equation roughly like t h = gim r i r m h jk ; which is a heat equation with a unique solution. This has principal symbol g ij ; which is strictly positive e nite. It can be shown that this implies that the moi e 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 9 Given an initial close Riemannian manifol (M; g ) ; there is a time T > an Riemannian metrics g (t) on M for each t [; T ) such that which satisfy the initial value problem t g = Rc (g) g () = g : Moreover, given the initial conition, g (t) are uniquely etermine, an there is a maximal such T: Remark 3 One can also show that this is true for complete manifols with boune curvature jrmj ; which was one by Shi. However, the proof is much more i cult on noncompact manifols.

34 8 CHAPTER 4. BASICS OF GEOMETRIC EVOLUTIONS

35 Chapter 5 Basics of PDE techniques 5. Introuction This section will roughly follow Tao s lecture 3. We will look at some basic PDE techniques an apply them to the Ricci ow to obtain some important results about preservation an pinching of curvature quantities. The important fact is that the curvatures satisfy certain reaction-i usion equations which can be stuie with the maximum principle. 5. The maximum principle Recall that if a smooth function u : U! R where U R n has a local minimum at x in the interior of U; then u x i (x ) = u x i x j (x ) where the secon statement is that the Hessian is nonnegative e nite (has all nonnegative eigenvalues). The same is true on a Riemannian manifol, replacing regular erivatives with covariant erivatives. Lemma 3 Let (M; g) be a Riemannian manifol an u : M! R be a smooth (or at least C ) function that has a local minimum at x M: Then r i u (x ) = r i r j u (x ) 4u (x ) = g ij (x ) r i r j u (x ) : Proof. In a coorinate patch, the rst statement is clear since r i u = u x : i The secon statement is that the Hessian is positive e nite. Recall that in 9

36 3 CHAPTER 5. BASICS OF PDE TECHNIQUES coorinates, the Hessian is r i r j u = u x i x j k ij u x k ; but at a minimum, the secon term is zero an the positive e niteness follows from the case in R n : The last statement is true since both g an the Hessian are positive e nite. Remark 3 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 xe metric. Lemma 33 Let (M; g (t)) be a smooth family of compact Riemannian manifols for t [; T ]: Let u : [; T ] M! R be a C function such that u (; x) for all x M: Also let A R. Then exactly one of the following is true:. u (t; x) for all (t; x) [; T ] M; or. There exists a (t ; x ) (; T ] such that all of the following are true: Proof. Consier the function u (t ; x ) < r i u (t ; x ) = ; 4 g(t)u (t ; x ) ; u t (x ; t ) < : u (t; x) + "t: If u (t; x) + "t > for all " > (small), then u (t; x) : Otherwise there is an " > an an initial t such that there is a x M such that u (t ; x ) + "t = : At the rst such time, we must have that x is a spatial minimum for this function, an thus u (t ; x ) = "t < ru (t ; x ) = 4u (t ; x ) u t (t ; x ) " < :

37 5.. THE MAXIMUM PRINCIPLE 3 Corollary 34 Let (M; g (t)) be a smooth family of compact Riemannian manifols for t [; T ]: Let u; v : [; T ] M! R be C functions such that u (; x) v (; x) for all x M: Also let A R. Then exactly one of the following is true:. u (t; x) v (t; x) for all (t; x) [; T ] M; or. There exists a (t ; x ) (; T ] such that all of the following are true: u (t ; x ) < v (t ; x ) r i u (t ; x ) = r i v (t ; x ) ; 4 g(t)u (t ; x ) 4 g(t)v (t ; x ) ; u t (x ; t ) < v t (t ; x ) + A [u (t ; x ) v (t ; x )] : Proof. Replace u with e At (u v) : This will allow us to estimate subsolutions of a heat equation by supersolutions of the same heat equation. Corollary 35 Let the assumptions be the same as in Corollary 34, incluing u (; x) v (; x) : Suppose u is a supersolution of a reaction-i usion equation, i.e., u t 4 g(t)u + r X(t) u + F (t; u) an v is a subsolution of the same equation, i.e., v t 4 g(t)u + r X(t) v + F (t; v) for all (t; x) [; T ] M; where X (t) is a vector el for each t an F (t; w) is Lipschitz in w; i.e., there is A > such that jf (t; w) F (t; w )j A jw w j : Then u (t; x) v (t; x) for all t [; T ] : Proof. Consier t (u v) 4 g(t) (u v) + r X(t) (u v) + F (t; u) F (t; v) 4 g(t) (u v) + r X(t) (u v) A ju vj :

38 3 CHAPTER 5. BASICS OF PDE TECHNIQUES The ichotomy in Corollary 34 says that either u is a point (t ; x ) such that at that point, v for all t; x or else there u v < 4 (u v) = r (u v) = t (u v) A (u v) = A ju vj for any A : But the inequality above says that at that same point (u v) A ju vj ; t which is a contraiction if A < A: Usually, instea of making v a subsolution, we will just make v the subsolution to the ODE v F (t; v) ; t where v = v (t) is inepenent of x an so this is also a subsolution to the PDE. Here is an easy application: Proposition 36 Nonnegative scalar curvature is preserve by the Ricci ow, i.e., if R (; x) for all x M an the metric g satis es the Ricci ow for t [; T ), then R (t; x) for all x M an t [; T ]. Proof. Recall that R satis es the evolution equation R t = 4 g(t)r + jrcj ; thus it is a supersolution to the heat equation (with changing metric), i.e., R t 4R: By Corollary 35, we must have that R for all t: We can actually o better. Notice that if T ij is a -tensor on an n-imensional Riemannian manifol (M; g), then since T ij jt j n gij T ij n gk`t k` gij (expan that out an see it implies the previous inequality). Thus jrcj n R

39 5.3. MAXIMUM PRINCIPLE ON TENSORS 33 an so scalar curvature satis es R t 4R + n R : The maximum principle implies that R (t; x) f (t) for all x M; where f (t) is the solution to the ODE f t = n f f () = min R (x; ) : xm This equation can be solve explicitly as f f = n t f = n t f () f () f (t) = n f () t as long as f () 6= : Notice that if f () > then this says that R (t; x) goes to in nity in nite time T n f() : If f () < ; then this says that if the ow 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 : [; T ]! (V ) where (V ) are sections of a tensor bunle (such as if we wish to apply the maximum principle to the Ricci tensor, for instance). Here is the theorem (possibly ue to Hamilton?) Lemma 37 Let (M; g) be a -imensional Riemannian manifol an let V be a vector bunle over M with connection r: Let K be a close, berwise convex subset of V which is parallel with respect to the connection. Let u (V ) be a section such that. u (x) K x at some point x M; an. u (y) K y for all y in a neighborhoo of x (This is the notion that u attains a maximum at x:) Then r X u (x) is tangent to K x at u (x) an the Laplacian 4u (x) = g ij (x) r i r j u (x) is an inwar or tangential pointing vector to K x at u (x) : Here are the relevant e nitions.

40 34 CHAPTER 5. BASICS OF PDE TECHNIQUES De nition 38 A subset K of a tensor bunle : E! M is berwise convex if the ber K x = K \ E x (where E x = (x)) is a convex subset of the vector space E x : De nition 39 A subset K is parallel to the connection r if it is preserve by parallel translation, i.e., if P x;y is parallel translation along a curve from x to y; then P x;yk y K x (this is if the tensors are all contravariant). Example 4 The set of positive e nite two-tensors is berwise convex an parallel with respect to the Levi-Civita connection. The maximum principle on tensors can be use to show things like:. Nonnegative Ricci curvature is preserve by Ricci ow in imension 3.. Nonnegative curvature operator is preserve by Ricci ow in all imensions. We will go into this in more etail in future lectures. Remark 4 Why o we nee convex? Consier the scalar case where we replace u (t; x) with u (t; x) x nearby x = : Consier the rst time u (t; x)