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
Singularities of Ricci flow limits and FL noncollapse David Glickenstein Math 538 Spring 2009 February 18 2009 1 Introduction This roughly covers Lecture 7 of Tao We will look at analysis of singularities of Ricci ow 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 1 Let Mg be a solution to the Ricciflow on a compact manifold on a maximal time interval 0T If T lt 00 then 2 33 Rm lam Remark 2 Uniqueness of the Ricci flow is necessary for there to exist a maxi mal time interval The idea is that if there is a solution on 0T1 and another solution on T1 75T2 then they must agree on the overlap so one can consider the flow on 0T2 which extends both the flows Now simply take the sup of all T2 for which the flow exists and this forms the maximal time interval Remark 3 N Sesum was able to replace lle with chl Proof idea Suppose T lt 00 and lle remains bounded Then one can show that all derivatives IVC le are uniformly bounded for t E 5 T One can use this to derive uniform bounds on the metric and its derivatives and then to extract a smooth limit metric at T of a subsequence using an ArzelaeAscoli type compactness theorem The existenceuniqueness result tells us that we can extend the ow for a short time contradicting the fact that 0T was maximal l 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 3 Blow ups Here is the idea At the singular time we know that lle2 is going to in nity If we scale the metric g to cg then we get 1 1Rmltcggt12 2 mm 912 01 1 lRmcgl E Rm 91 so if we want to prevent lle from going to in nity we choose a scaling L172 2215 lRm9 MM and rescale the metric g tn by L2 We can actually rescale to a sequence of solutions of the differential equation Ricci ow by looking at 1 9n t g in iLi 7 n where tn A T the singular time Notice that 8 8 597i 5 L739 in tLi 72 Re 9 tn 2L9 2 RC 1 W so 9 is a sequence of solutions to the Ricci ow whose initial value is getting closer to the singular time Furthermore the initial curvatures Rm 9 are all bounded by 1 On the down side Ln A 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 4 We do not necessarily have to choose Ln as above The fact that 172 A 00 is why this is called a bloweup If we take L732 A 0 then we have what is called a blowedown Notice that if the original Ricci ow is de ned on an interval 0 T then the rescaled solutions 9 t exist on the interval ti T727 L3 L2 39 Thus if we can extract a limit and in A T and Ln A 0 then the limit metrics will be ancient ie will start at t 700 the nal endpoint depends a little more on how we choose the in with respect to how we pick the Ln allowing it to be 0 positive or 00 we may have reason to choose any of these The main goal is to nd quantities which 1 become better as we go to the limit Tao calls these critical or subcritical and 2 severely restrict the geometry of the limit Tao calls these coercive Examples of quantities to study or not a The volume of the manifold If we look at Vwmdw M we see that 1 1 V M7 E9 WV M79 and so if Ln A 0 this quantity does not persist to the limit Tao calls this behavior supercritical The total scalar curvature If we look at meRw M we see that 1 7 129 Thus it is preserved in dimension 2 this is the GausseBonnet theorem which implies critical behavior and does not persist in higher dimensions supercritical F M Li dFMg The minimum scalar curvature If we look at RmnM79 EIIERWL then we see that 1 2 Rmin My Lglg Lanin M79 and so this goes to zero as Ln A 0 This is subcritical behavior Unfortue nately Rmin 0 does not give suf cient information of the limit to classify not coercive enough 4 I Lowest eigenvalue of the operator 74AuR Notice that this is subcritical since if we call this functional H Mg 1 H Mi LQHM lt ugly 79 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 ie scale invariant quantity which is related 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 37sphere collapsing to a 27sphere by shrinking Hopf bers There are several issues here notably ls 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 diffeomorphism and there is an 7 gt 0 such that the ball B07 is mapped diffeomorphically to a ball on the manifold The largest such 7 is called the injectivitg 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 5 n noncollapse One de nition of a collapsing sequence is the following De nition 5 A pointed sequence Mngnpn of Riemannian manifolds is col lapsing if injpn A 0 as n A 00 We may Wish to rescale the manifolds Mmgn to be of some uniform size say by making lRm 1 Then one can consider a rescaled collapsing if Rmpn 21njp e 0 When one assumes that the sectional curvature is bounded then the collapse is restricted Let VU Vg U denote the Riemannian volume of the Borel subset U C M Theorem 6 Cheeger Suppose that lleg S 0762 on B 1770 C MCI and that V B 17713 2 673 for some 6 gt 0 Then the injectiuity radius of p injp is at least injp 2 CT0 for some constant c CC 6 d gt 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 7 0 A 0 the ball looks more and more Euclidean That means that for very small 7 0 ltlt injp V B 177 H Lard Where w w d is the correct constant for a Euclidean ball and lleg H lRm Mg So as 7 0 A 0 we see that lim 7 sup Rm 0 T090 IEBpTo lim 30750 7 1 r090 on In particular for any C gt 0 and 0 lt 6 lt 1 there is a 7 gt 0 such that the assumptions are satis ed if 7 0 S 72 Note that the converse is also true from more classical results Theorem 7 If lleg S C and injp 2 1 then there is a 6 6 CLd such that VB pr 2 6rd for all 7 S L 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 moi tivates the following noncollapsing de nition which is not quite the de nition we will use De nition 8 A Riemannian manifold Mdg is Hicollapsed atp E M at scale To 1If 1 Bounded normalized curvature lleg S r62 for all at E B pr0 and 2 Volume collapsed V B p 713 S Hrg If these are not satis ed then we say the manifold is Henoncollapsed at p at the scale m ls this a reasonable de nition Here are some observations By Cheeger s theorem this would imply a lower bound on injectivity rae dius Note that if the one is on a ball where the curvature is large then the manifold is automatically Henoncollapsed at that scale a By the discussion above every manifold is Henoncollapsed at a small enough scale and H smaller than the constant for a Euclidean ball a This de nition is scale independent in the following sense If we consider g r629 then the conditions are lmeg 1 VBgp1 S H a The sphere S is Henoncollapsed at scales r0 less than the diameter for a suitable choice of H The bounded normalized curvature assumption of the de nition are satis ed for To S 1 although one might argue that there is really no local collapsing at scales less than 7T Apparently this will not be important for our argument At large scales the curvature assumption fails a The at torus has lle 0 and so it is noncollapsed at scales less than the injectivity radius The curvature assumption is valid for large scales but for a given H if To 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 H such that a manifold is H7 noncollapsed at large and small scales Certainly one can make H small enough say less than the constant for the area of a Euclidean ball so that a manifold is Hecollapsed at many scales but this is not of use to us Perelman adapted these ideas to Ricci ow time dependent metrics as follows De nition 9 Let Mdg be a solution to Ricci flow and let H gt 0 Then Ricci flow is H7collapsed at a point to do in spacetime at a scale r0 1 Bounded normalized curvature lRm by W90 3 732 for all t at E to 7 rate gtlt Bguo 0713 and 2 Collapsed volume V 3900 0713 S HTg Otherwise we say that the solution is H7noncollapsed at p at scale m Remark 10 Notice that the assumptions require Ricci flow to exist on the time interval to 7 rgt0 Remark 11 As remarked above for each H smaller than the volume of a unit ball in Euclidean space there is a 73 so that M is H7noncollapsed at scales less than T For Ricci flow one can still nd ri but it will in general depend on to As to goes to a singular time it may be possible that r 7gt 0 This is what we would like to rule out as we shall see Remark 12 Notice that H is dimensionless Remark 13 Ifgt is H7noncollapsed at t0aco at the scale m we see that Kg t is H7noncollapsed at K to 7 ti aco at the scale of K 12r0 Here is a typical noncollapsing theorem along the lines of Perelman Theorem 14 Perelman s noncollapsing theorem rst version Let Mg be a solution to the Ricci flow on compact 37manifolds fort E 0T0 such that at t 0 we have lRmPlg0 S 1 V 390 1771 2 w for all p E M and w gt 0 xed Then there exists H HwT0 gt 0 such that the Ricci flow is H7noncollapsed for all t0aco E 0T0 X M and scales 0 lt r0 lt Vi A big point of this theorem is that it rules out a limit of E X R Where 2 is the cigar soliton solution of Hamilton The manifold E X R is essentially a xed point of Ricci ow When we consider it a ow on metric spaces not Riemannian metrics E 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 This implies that E X R has volume V B 07 asymptotic to 072 for large 7 For a cylinder notice that large balls of radius 7 have volumes asymptotic to CT not 072 Thus E X R is not Henoncollapsed at large scales 7 0 large Consider the blowups Mgn de ned above Note that we have a H such that the Ricci ow 9 t is Henoncollapsed for time interval 0 To for any To lt T Thus by Remark 13 we must have that 9 t is Henoncollapsed at the scale of 1er for the time interval 72775 TEZTO becomes Henoncollapsed at all scales which is not true for E X 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 As n becomes large we see that 9 t
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