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
L distance David Glickenstein Math 538 Spring 2009 March 26 2009 1 Introduction We rst wish to show that a H7solution has a limit which is a gradient7shrinking soliton To do this we will need to introduce a new notion the reduced distance 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 qul2 T 2 WMgu739 u d 7 ulogu dac 7 a log 477739 7 d and that W is minimized for u 1739 47TTTd2 exp 7 lle 47 We see that this formula essentially gives the distance function 1ac02 M2 by 2 M d 7710 u x7 7710 477739 4T g 7 2 g However we have very little control over this function One can also derive the distance function as follows a 1 10 7 101300 M 1739 7a1 This strong relationship is only true in Euclidean space In general there are two different concepts the heat kernel u 1739 and the Riemannian distance function dac are 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 1 at are 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 3 length and reduced distance Consider a curve 7 0 7 A M One can de ne the length of a curve as 1370Wslds and the energy as 1 T 2 EV MSM ds 0 Note that the length is independent of reparametrization but the energy is not These notions give rise to the function Tp which is the function representing the distance to the point p ie 7 95717 lt turns out that the distance function satis es some differential equations and inequalities in particular 1 7 1 T ATlt for Ricci nonnegative Note that if T V Z xi2 then AT d1 We will show this inequality later in this lecture but for now let s assume this and see the implications on the volume Notice that d vltBlt Rgtgt7Altslt R dR I77 7 where SpR is the geodesic sphere of radius R centered at p and A is the area 1 7 1 dimensional volume measure function By Gauss lemma we have that lVTl 1 and also that we can decompose the metric to be dTZ 93pw where 93 is the metric on the geodesic sphere Denote the measure on the sphere of radius T as dAsmT We see that ATdV VT TL dAsmr 3P R 3P R dASp T 3P R ASp7R The decomposition above implies that the volume measure can be decomposed as dV dASp dT and so d d R 7 ATdV 7 ATdA T dr 13 3mm 13 0 SW SW Rdil Sp R ATdAsw R Thus d A5 17713 1 d71 d d72 E Rdil iRmdil R E ArdV 7 d 7 1 R ASpR Emit 1 c171 d72 R2d71 R ACNE ATdASW d 1 R A S 17713 1 S TWA d 7 1 RH dASmR 7 d 7 1Rd 2ASpr 3P R 0 This implies an inequality on volume ratios Theorem 1 BishopGromov theorem simpli ed version Ich 2 0 then VB pr 7 d is a nonincreasing function ofr Proof We have already showed that d 14607713 7 lt dR RC 1 0 We now consider 0 S 7 1 S 72 We have V B 17772 VB 17 If ASP7Tdr d d T 7 2 771 f 1 rd 1dr f AS0TTd71dr n Wquot frildrd ldr lt Altsltp7ngtgt 7 drilil Furthermore VB 17771 foT1 ASP77d7 Til fun 1 Td ldr 0T1 Wrd ldr f0 1 Td ldr A S 17771 d Tilil I 2 Thus we have which implies a r 7r rgv 3 mm 7 ra39v 3 pm 731037 Til T W mm 7 W m T51 T5 T5 g I We now consider Ricci ows Let Mg be a solution to backwards Ricci ow ie Rcw and let 7 07 A M be a curve in M We de ne the Ledistance to be T 7 2 L w a w am Ram da This may need a bit of explanation First 7 g is the tangent vector to the curve 7 Note that it is measured at 039 by the metric g 039 so at different 3 it is measured using different metrics The rst term looks almost like the term in the energy f WV 1039 which can be used to derive geodesics on a Riemannian manifold However the addition of the term 3 makes the integral scale like length not energy think about this Remark 2 Tao uses 7739 in some places because of the understanding that 739 it and he wants 9 t to be a solution to Ricci flow and V t de ned later to be de ned on t We will allow 9 to be parametrized by 739 and so there is no need for this In the smooth xed manifold case one considers the length functional Ln w wade and then one can nd the distance 1350 at between points by taking the in e mum of length over all curves connecting those two points One can de ne the distance function r10 which is the function which returns the distance to a xed point do The analogue of this in the Ricci ow case is the distance also called reduced length We a x info 7 where the inf is over all paths 7 from 0 to ac Remark 3 Note there is also the Lidistance which is 2 Z One can nally de ne the reduced volume 4010 739 Cl2 exp 7Z0Io7 71 WW M Our main goal will be to show the following Theorem 4 8 2 d gimme Amazon Wozog R g 2 0 As a corollary we get monotonicity of the reduced volume if 9 2 RC 9 Corollary 5 Reduced Volume is monotone 8 N EVchO 739 S 0 8739 a a 78 VozoT T dZexpklmo x dng 739 M d 812 0 7 til010 T 0 T dZGXPl owo xlldng M 2739 8739 T dZ exp Zmeo 7390 RdVyU M d N 7W010T Fm exp Z lro a 96 Rdng 739 M d M iAgU w xo lVZlZU 7 R TidZ exp 720 10 739 dVgU 0 I Of course all of this assumes suf cient regularity on I which is not in general true However this argument can be made rigorous in some generality including past the conjugate radius 4 Variations of length and the distance function We start with the energy functional given above We can do calculus of varia tions as follows Consider a variation l t s such that rltzsogtvltzgt 8 ampFts0Xt so we consider the variation to be X Furthermore we consider variations which x the endpoints ie X 0 X 739 0 Compute 1 T 5E7X5 0 9mvdi 13 BEBE dz 28530 0 9 are T Br ar 09 Wald Remark 6 The notion DtX means DtX VBLX It only depends on the t values along the curve 7 At a critical point for the energy we must have 6E7 X 0 for all X so we get that Df39y 0 if the endpoints are xed ie if X 0 X 739 0 This is the geodesic equation Notice that if we restrict to a geodesic curve ie Dy39y 0 and x the initial point X 0 0 then 5E7 X 9X WWW On a geodesic d a la W2 29 Dmv 0 so 7 has constant velocity So along a geodesic we have 1 T 2 1 2 E 7 d 7 7 20 W 5 M T T 1370WldsWT and so if y is a geodesic then and so we have that if y is the gives the distance d 17 ac then 1 2 E 7d 7 2 p x If we want to compute a variation d gums away from the cut locus so variations in geodesics stay minimizing we have dac wnuefwwmyw l where X is a variation of geodesics one must show that these exist but they do Note that we can always reparametrize so that 739 1 and then we get that 2 wwm1mwmi or MW mum MOW MW and so 1W p 1H 1 Now compute the second variation of energy when 7 is a geodesic We get a T 6 X7X i gamma 85 0 gXw3 7 9XDndt 0 1 82 T 8P 8P 62E X X 7 7 i i dt VT 2 852 300 glt82 82gt 739 8P 8P 8P 8P 0 9 Dst575gt 9 D357D35gt di T 8P 8P 8P 8P 9 DSDtgv 9 DtgyDtEgt dt T 3P 3P 3P 3P 3P 3P 0 9 DtDsE75gt 9 R gDXDXdt 9VwalS 9RXampXmgDX7DXdt 0 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 y is a geodesic Let s again parametrize our geodesic between 0 and 1 and x 7 0 and vary the other endpoint 15 so that X t 271 LO varies through geodesics Then 1 d2 i E Furthermore we can take X t t where is the parallel translation along y of 27 Then we get 1d2 2 12 890890 1890 ggdm s 70 i9R any in dtgltE7Egt wwew gmmmxmwaxamw Thus if we sum over an orthonormal frame and assume that the sectional cure vature is nonnegative7 we have that l iAd 177 x2 S d Andso 1 iA f2 fAf W2 andso d 1 Ad 3 7 7 10736 5 Variations of the reduced distance We will do the same thing to the reduced distance Consider L w T E w am Ram da Actually7 we want to think of this as a functional on spacetime paths A space time path 7 is a map 7 OJ A M X I where I is an interval We will only consider paths that look like 75 7 57S 1 We can naturally de ne R 7 1393 We then have the formulation of the functional as 127 T E w wit R v on do We will de ne the reduced length as 0350 739 at 23F inf L y 7 are of the form Note that the in mum can also be considered as the in mum over all paths Ny on M Note that the tangent space of Mgtlt I splits as TMgtlt TI and TI is spanned by There is a notion of a horizontal vector eld7 which is a vector eld X such that 1739 X 0 Since variations of paths of the form 1 must be horizontal vector elds7 we rst consider the rst variation of L with respect to a horizontal vector eld X Note that W y 7 33 is a horizontal vector eld but 7 is not T Remark 7 Horizontal vector elds on M X I are in oneitoione correspondence with vector elds on M For this reason we may abuse notation and use the same notation for both vector elds In particular if will be considered both a vector eld on M and a horizontal vector eld on M X I We get 6137X 0 VE2ltWW gtWVngmda We do the same thing we did with the energy functional7 nding 7 6137X a 2 ltV7Xw gtgm 1wa T 7 d N N N 0 W 2 Kngt907 7 4R0 X y 7 2 X vwwgm VX1390 do since if X and Y are horizontal7 1 E9 XY 2Rc XY 9 DyXY 9 XDzY Remark 8 In general the formula is d XY8 XYV XY dTg 7 8T9 7 7 9 7 7 which has more terms since foX 7E 07 etc Remark 9 I used the notation of total derivative since there is the variation of the metric with respect to the time parameter of Ricci flow and also the variation with respect to y I have also used instead of dot to denote derivative with respect to 739 or 039 Now we need to take the out7 so we need that 1 d N N d N E lt2 ltX7vlgtg0 W XXVgt907 2W3 XXVgt907 7 SO 7 T d 7 N 1 N 6137X 7 3 we ltX7 gm 7 W ltX7 gt907 E 74Rc X y 7 2 X vwwgm vXRgm do Namegt907 0 7 g O ltGXgtg0da where G is the vector eld 2039 where Rc X is the vector eld on M such that RCXY RcXY for all for all vector elds Y on M Notice that G does not depend on the variation X Note that if y is a minimizer with xed endpoints ie7 for all variations X such that X 0 X 739 07 then we must have that G 0 This is the Legeodesic equation N l N N l G a V377 77 2R0 39y 7 EVRQW Problem 10 If we are in Euclidean space what are the Legeodesics Now supposing there is a unique minimizer and that Z is a smooth function we can consider variations through Ligeodesics such that X 0 is xed to get 6amp7 X mum which implies that 8 81 714010 m 5 lt7wgt 85 85 9 recall that Z divides by This can also be written as VZw IO 3 Note that the reduced length also depends on time so let s compute the time derivative as well d 7 6L7 w w wit mam Note that this need not be zero since we are not minimizing in this direction We nd that d N 2 E 2mm w w mm Rm Now the total derivative decomposes as d 8 Eggnog gimme szm o and so 8 1 N 2 1 1 N 2 gimme i W Tllgm R9T gimme WigU 10
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