Class Note for MATH 538 with Professor Glickenstein at UA
Class Note for MATH 538 with Professor Glickenstein at UA
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Date Created: 02/06/15
Maximum principle and pinching David Glickenstein Math 538 Spring 2009 February 10 2009 1 Introduction This section Will roughly follow Tao s lecture 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 reactionediffusion equations Which can be studied With the maximum principle 2 The maximum principle Recall that if a smooth function u U gt R where U C R has a local minimum at 0 in the interior of U then Bu 835i 82a f at gt 0 8x281 0 7 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 0 0 Lemma 1 Let Mg be a Riemannian manifold and u M A R be a smooth or at least CZ function that has a local minimum at 10 E M Then via 10 0 Vivi14350 2 0 Aim 92710 Vivi14350 2 0 Proof In a coordinate patch the rst statement is clear since Vin 331 The second statement is that the Hessian is positive de nite Recall that in coordinates the Hessian is 82a k Bu vivju 812817 7 W but at a minimum the second term is zero and the positive de niteness follows from the case in R The last statement is true since both g and the Hessian are positive de nite l Remark 2 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 3 Let Mg be a smooth family of compact Riemannian manifolds for t E 0T Let u 0T X M A R be a 02 function such that u 0 x 2 0 for all x E M Also let A E R Then exactly one of the following is true 1 utx 2 Ofor all tx E 0T X M or 2 There exists a to x0 6 0T such that all of the following are true Proof Certainly both cannot hold Now suppose 1 fails Then there must exist a gt 0 and to x0 such that u to x0 lt 0 We may move this to the minimum point at which all of the rst three must hold If we take this to be the rst time that such a point occurs the last must hold as well I Corollary 4 Let Mg be a smooth family of compact Riemannian mania folds for t E 0T Let uv 0T X M A R be 02 functions such that wow 2M0 for all x E M Also let A E R Then exactly one of the following is true 1 utx 2 v tx for all tx E 0T X M or 2 There exists a to x0 6 0T such that all of the following are true Viv to x0 2 Agtov to 0 81 lti 8t t0x0 Aut0x0 7 v t0x0 Proof Replace u With e At u 7 v I This Will allow us to estimate subsolutions of a heat equation by supersolue tions of the same heat equation Corollary 5 Let the assumptions be the same as in Corollary 4 including u0x 2v0x Suppose u is a supersolutioh of a reactiohidi usioh equation ie Bu 5g21 muvxmuF l and v is a subsolutioh of the same equation ie 81 a S Amnu VXU F LU for all t7 at E 07 T X M7 where X t is a vector eld for each t and F t7 w is Lipschitz in w7 ie there is A gt 0 such that WW 7 PM Alw 7 wquot Then u t a 2 U Lac for allt E QT Proof Consider 8 a u 7v 2 A90 u 7v VXO u 7v Ftu 7Ftv 2 Agtuivvxmuiv iAluivl The dichotomy in Corollary 4 says that either u 7 v 2 0 for all t7 at or else there is a point to7 are such that at that point7 u7vlt0 Au7v0 Vu7v0 uiv AluiviAlluivl for any A But the inequality above says that at that same point 8 Ewiv 2 iAluivl Which is a contradiction if 7A lt 7A l Usually7 instead of making 1 a subsolution7 we Will just make u the subsolue tion to the ODE ltFhm dt Where v v t is independent of ac and so this is also a subsolution to the PDE Here is an easy application Proposition 6 Nonhegative scalar curvature is preserved by the Ricci flow ie if R0x 2 0 for all at E M and the metric 9 satis es the Ricci flow for t E 0T theh Rtx 2 0 for all at E M ahdt E QT Proof Recall that R satis es the evolution equation 8R 2 E AMOR l 2 lRCl thus it is a supersolution to the heat equation With changing metric7 ie7 8R 7 gt AB 8t 7 By Corollary 57 we must have that R 2 0 for all t I We can actually do better Notice that if Tij is a Qetensor on an nedimensional Riemannian manifold M797 then 1 lTl2 2 QZJTij2 since 2 1 T2quot QMTM 92739 2 0 expand that out and see it implies the previous inequality Thus 1 chl2 2 7R2 n and so scalar curvature satis es 8R 2 7 gt AB 7R2 at 7 n The maximum principle implies that R t7 at 2 f t for all at 6 M7 Where f t is the solution to the ODE df i 2 2 a 2f f0 gggRW l This equation can be solved explicitly as df in 75 fin f0 as long as f 0 7E 0 Notice that if f 0 gt 0 then this says that Rtx goes to in nity in nite time T S If f 0 lt 07 then this says that if the ow exists for all time7 then the scalar curvature becomes nonnegative in the limit 3 Maximum principle on tensors Sometimes it may be useful to use a tensor variant7 for a function u 0T A I V Where I V are sections of a tensor bundle such as if we Wish to apply the maximum principle to the Ricci tensor7 for instance Here is the theorem possibly due to Hamilton Lemma 7 Let M7g be a dedimensional Riemannian manifold and let V be a vector bundle over M with connection V Let K be a closed berwise convex subset of V which is parallel with respect to the connection Let u E I V be a section such that 1 E 8K1 at some point x 6 M7 and 2 E Ky for all y in a neighborhood ofx This is the notion that u attains a maximum at Then qu is tangent to KI at and the Laplacian Aux gij ViVJu is an inward or tangential pointing vector to KI at u Here are the relevant de nitions De nition 8 A subset K of a tensor bundle 7T E A M is berwise convex if the ber KI K N Ex where Ex TF1 is a convex subset of the vector space Ex De nition 9 A subset K is parallel to the connection V if it is preserved by parallel translation ie if Paw is parallel translation along a curve from x to y then P yKy C KI this is if the tensors are all contravariant Example 10 The set of positive de nite twoetensors is berwise convex and parallel with respect to the LevieCivita 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 dimene sions We Will go into this in more detail in future lectures
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