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
Back to the program David Glickenstein Math 538 Spring 2009 March 12 2009 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 MorganeTian7 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 Mg de ned for 0 S t lt T lt 00 which satis es the following properties 1 Normalized initial conditions We have lRm90l S 1 and for any at E M 2 Curvature pinching The curvature is pinched towards positive This means that as the scalar curvature R A 007 the ratio of the absolute value of the smallest eigenvalue of the Riemannian curvature tensor to the largest positive eigenvalue goes to zero 3 Noncollapsed There is a H gt 0 so that the Ricci ow is Henoncollapsed 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 this7 we will need to be more precise with 3 and especially 4 2 Finding canonical neighborhoods The main way to nd these is to take bloweups as one goes to a singularity If one takes a sequence ti A T where T is a singular time then the blow ups t tM t 7 92 zgltzMigt lf Mi is comparable to sup Rmg then Mi A 00 Furthermore since T lt 00 we have that gi t is de ned on the interval iMith T 7 ti Thus the limit will de nitely be de ned on 7000 and is thus ancient Moreover if we show that g is Henoncollapsed at some scale r0 then gi t will be H7 noncollapsed at a scale rum and so the limit is Henoncollapsed on all scales Finally one can show that any 37dimensional ancient solution has nonnegative curvature De nition 1 A Hesolution of Ricci flow is a solution de ned fort 6 7000 which is Henoncollapsed on all scales and has nonnegative curvature 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 Hesolutions look like gradient shrinking solitons as t A 700 2 Gradient shrinking solitons in dimension 3 must have nite covers isomete ric to 37spheres or ii Zespheres cross 1R 3 Hesolutions have canonical neighborhoods Now the idea is that as one goes to a singularity the manifold is like a Hesolution and thus has canonical neighborhoods 3 Canonical neighborhoods What is a canonical neighborhood This is where the surgery should be done so it needs to be classi ed su iciently to allow us to do a careful surgery It turns out that there are 4 types of canonical neighborhoods De nition 2 A point at E M is in a 057canonical neighborhood if one of the following holds 1 at is contained in a Cecomponent 2 at is contained in an open set which is within a of round in the CUEL topology 5 at is contained in the core of a 057cap 4 at is in the center of a strong neck De nition 3 De ne the 0 Xgo norm on Xg to be c 2 Humans 32 9 lt3 90 W Z lv oMlgol39 9 21 Remark 4 Technically the norm should be c 2 m X79 H mgo s3 19 am Z lv og owls 1 21 The function de ned above is essentially HXgH Xg ego With our current de nition X79Hckx go 0 ifg go Problem 5 Here is something to think about FixX C R say bounded Given a sequence of Riemannian metrics on X under what conditions does there exist a subsequence such that 1W7 9i converge to some limit Xgoo for some Riemannian metric goo ie HXgiHckX gw A 0 as i A 00 Problem 6 How could one use this norm idea to compare Xg and X g where X 7E X De nition 7 Let Ng be a Riemannian manifold and at E N a point Then an eineck structure on Ng centered at at consists of a di eomorphism with at E b 5392 X such that HN7R 9H lt 5 where the norm is with respect to CUM 5392 X 7 gstd where gstd is the product of the metric with curvature 12 on 5392 with the Euclidean metric on the interval We say N is a eineck centered at ac De nition 8 A compact connected Riemannian manifold Mg is called a Cicomponent if 1 M is di eomorphic to S3 or RP3 2 Mg has positive sectional curvature 5 For every Qiplane P in TX 1 111pr P lt 7 C supngRy 1 0 1 sup lt diamM lt C inf 7 yeM My KM My De nition 9 A compact connected 3imanifold Mg is within a of round in the CllElitopology if there exists a constant p gt 0 a compact manifold Zgo of constant curvature 1 and a di eomorphism q5ZHM such that Maw unwell2M2 5 Finally we have the complicated de nition of a cap The last conditions ese sentially say that the diameter volume and curvature differences are controlled and are technical conditions needed in some arguments De nition 10 Let Mg be a Riemannian 3imanifold A 057cap in Mg is a noncompact submanifold C glc together with an open submanifold M C C with the following properties 1 C is di eomorphic to an open 3iball or to a punctured lRlP 3 2 N is a aineck Y CN is a compact submanifold with boundary Its interior Y is called the core of C 93 4 The scalar curuature R gt 0 for every y E C and C diam C glc lt supyethu 5 R as sup lt C zyEC RW 6 C V C lt suPyECRy32 7 For any y E Y let ry de ned by the the condition that 1 sup W 7 y eBuM Ty Then for each y E Y the ball B 31 has compact closure in C and 1 V B ilt in lt arm er ry VR y up 7 3 lt C yea Ry and LR sup 3 ml 2 lt0 yEC RM 4 5 How surgery works The key observations are this 1 For every H and every small a gt 0 there is a Cl Cl EH such that a Hesolution is the union of Cl a canonical neighborhoods 2 For every small a gt 0 there is a 02 02 a and a standard solution of Ricci ow with is the union of 025 canonical neighborhoods 3 We can do surgery on canonical neighborhoods if they are suf ciently small and positively curved Consider a Ricci ow which becomes singular at a time T Fix T lt T so that there are no surgeries in the interval T T By the assumptions there is an open set 9 C M such that the curvature is bounded for all t E T T so there is a limiting metric on Q as t A T Every end is the end of a canonical neighborhood which looks like a tube We call these ends aehorns We can then x a constant p and consider the subset Q C Q in which the scalar curvature is bounded above by p 2 One can then show that a horns with boundaries in Q are 67necks We then do surgery on these 67necks by gluing in a standard solution 6 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 Hesolutions 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
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