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
Perelman entropy and FL noncollapse David Glickenstein Math 538 Spring 2009 March 5 2009 1 Gradient ow Formulating an equation as a gradient ow has many advantages Consider the heat equation 3f 5 on a compact Riemannian manifold It is easy to see that if one considers the energy 1 Em gMlv gdv that if we take the time derivative of the energy when f satis es the heat equation7 we get dE 8 E f M W v if dv 8 WWW iM Af2dV 0 Thus we immediately get that the energy is decreasing and that stationary points are harmonic functions7 ie7 functions which satisfy Af 0 This monotonicity also tells us that f cannot have periodic solutions which are not xed points7 for if ft1ac ft2ac for all at then Eft1 Eft27 and the monotonicity implies that Af 0 for t 6 t1 Q The monotonicity is true in general for a gradient ow If one has an energy E 7 one de ned the gradient ow as 3f a igrad 7 where the gradient vector grad is given so that dEX9X79TadE7 for some metric g on the space of functions In our case that g is the L2 metric which uses the Riemannian metric g on It would be nice to represent Ricci ow in this way It is not at all trivial to do this 2 Ricci ow as a gradient ow An obvious choice of functional is the Einstein7Hilbert functional EH 9 RdV M To calculate its variation7 recall that if we have a variation of the metric 69 hij7 then we get 6Rh 7 ltRchgt 7 Atrg h divdiv h It is not hard to see that since 6 log det g gijhij so ixdetg dexp logdetg trg h xdetg so 6 dV try h dV Using the above formula we get 6 EH h 7 RC h Rtrghgt dV ltle7Rc7hgtdV M 2 Remark 1 Here we used the divergence theorem for a compact Riemannian manifold which says that l 7 Rc7 h 7 Atrg h divdiv h ER trg hgt dV divTSdV 7TVSgtdV for any n7tensor T and n 7 l7tensor S More explicitly 9i1j1quot399iquotjquot9ji VjTioi1 245m indVgi1jl quot39giquotjquot9ji Tioi1 inVijljz jndV So critical points of the EinsteineHilbert functional satisfy the Einstein equa tion The gradient ow would be 8 1 g 7 72 RC 7 R9gt The problem is that this ow is not parabolic and there is no existence theory for such equations Let s try a new tactic Replace dV by a xed measure dm Then the funce tional H g M Rdm satis es the variation 6H h 7 Re h e Atrg h div div h dm M You don t lose the last two terms with the divergence theorem since that only works with the volume measure However we can consider the RadoneNikodym derivative and write dm d idv m dV for a positive function 2 We can write dm 7 if dV e Then 6Hh 7Rch entrghdivdivhefdv M can be integrated by parts to get 6HhM7Rch7VtrghVfgtdivhVfe de M 7RchtrghAfitrghlVfl27lthV2fgthVfVf e de Mlt7Reev2fAfeVf2gVfohgtede Remark 2 Tao often uses to denote the Euclidean metric locally and so explicitly puts in 927 s in this case We will understand that lt requires the metric and so when quantities like this are di erentiated we also need to di erentiate the metric as we will see below 2 gt dm We need to add another term and so we get We now need that 1 0 6 dm 6 e de 7 6f 5de a try he de 76f guy h 6de Thus we have that 1 6f a try h Let Em gijvifvjf de where dm e de is xed so that f flog dmdV and if is expressed as above We can then compute 6Eh 6 gijvifvjfefdv lth7Vfogt2ltVf7V6fgt 7lVfl26f lVfl2trghgt de 7 WWW 72Af6f lVfl26f vfl2trgh de ltherVfAfglVfl29gte de Now since F H E we have 6Fh lt7Rciv2fhgte de M Thus the gradient ow of 72F is 39 2 7 72 R 7 2V at C 9 f This is almost Ricci ow but not quite The f is changing too by the equation 3f i 7A 7 R at f This is a backward heat equation which it turns out will make it useful to probe backwards Notice that 2V2f Vf97 the Lie derivative of g in the direction Vf This means that the ow above differs from Ricci ow by a diffeomorphism Instead we can consider g V9 where ab is the ow of diffeomorphisms generated by Vf and we will see that g evolves by Ricci ow Furthermore f will differ by a Lie derivative and llwfdfVf va where is the metric in here It is in Vf7 which is a vector eld gotten by raising the index on df and so under the new ow7 f f 0 ob evolves by 3i 8t 7 Example 3 thdamental important example Let M79 be Euclidean space M Rd and let 7A f7Rvf 2 d f Lac E flog 477739 flog 4777 d2 e lIlZMq 4739 2 where 739 t0 7 t7 Notice that e fdac is the Gaussian measure which solves the backward heat equation the fundamental solution to the heat equation If t lt to7 this choice ofg and f satisfy the equations 39 54mw m 8f 2 EiiAfiRlth 2 We can ch eck so it works Notice that when we pull back by ab the measure Ndm is not static Thus it makes sense to rewrite the functional as F Migyf M R v Z 6de Notice that this functional is invariant under dil feornorphisrn7 ie7 FM7 97f0lt15 FM797lt15 for any diffeornorphisrns ab We also have that under the coupled ows 1 and 2 887 Mg f 2 RcV2f2e de Thus we have that F is monotone increasing under Ricci ow Unfortunately now we have to explicitly deal with this quantity f To eliminate this we take the in mum AMg infFMg f M e de 1 the in mum is over Using the following exercise we can show that A is nite Exercise 4 Show that A Mg is the smallest number for which one has the inequality Anvil Ru2 dV 2 A quV M M where u is in H1 Wl 2 M the Soboleu space offunctions with 1 deriuai tiue in L2 so it has norm f lVfl f2 dV for 01 functions Hint show that we can assume u is positive and then write u e fZ Thus A is the smallest eigenvalue of the operator 74Agu R Using the exercise one sees that the fact that every compact manifold sate is es a Poincare inequality qulZdV 2cd u2dv M M implies that A is bounded below basically by the best constant in the Poincare inequality plus min B Note that the Poincare inequality constant depends on the dimension We will see later a similar inequality for which the constant does not depend on dimension Furthermore one can prove that A is realized by a positive function u e f2 with Hulluwm 1 Note that H1 embeds compactly into L2 since M is compact Thus if we take a minimizing sequence in H1 there is a subsequence which we also denote by which converges in L2 to a function u Now consider ltqunl2 Rui dV quml2 Ruin dV lwu fuml2Runium2dVltlVunuml2Runum2dV ltlVun fuml2Run 7 dVltqunl2Ru dVltquml2RuZ dV i ltlVunuml2Runum2 dV The right side goes to zero in the limit since the terms go to A A and 72A Also we know that mianl 7 umHLZ S Run 7 um2 dV S mianl 7 umHLZ and each term goes to zero Thus we know that the sequence is Cauchy in H1 and since H1 is complete it must converge to a function in H1 Since A is attained at a function we can sometimes prove inequalities like the following taken from the notes of Kleiner and Lott Let h Si at be a two7parameter family of functions such that h57i0 f to 5 8h 2 7 7Ah7R Vh at l l There is a solution to this for t S to since the equation is backwards elliptic Note we have only shown that f is in H1 so we mean a weak solution to the parabolic equation We could also show that f as a minimizer satis es a particular elliptic equation which implies that f is smooth by elliptic regularity theory Mia S FM79t07h57i0 7 5 F Mg to sh 5 7 23 Rc g to 0 V2h 5J0 a2amp hsquot 7dV do We then have s we 3 A to 372 ch g to 0 V2hsto a2e squot dV d0 0 M and so 8A 7 At0s ewe a 00gt 7 1231 f 2 s 2 lim 7 chgt0 0 V2hsto a2e squot dV do 370 S 0 M 2 IRC 9 U0 V2h 07t02 eih0quot dV M 2 IRC9 00 V2f to2e fdv M We can now derive 8A E EQIRchVZ IZe fdV 2 gRJrAfQZe dV 2 g R Af e dvr g 13 VMZ e dvr A2 3 Perelman entropy We now wish to make our functional scale invariant so that we get a critical quantity7 not just subcritical ln particular7 we know that dF Timmy 2RcV2fZe de is xed under the gradient ow if Rc 7v f i e7 if M79 is a gradient Ricci soliton We wish to have a new functional which is xed if M79 is a gradient shrinking soliton7 ie7 2 1 RC 7V f 79 2739 for some 739 gt 0 A round sphere is a gradient shrinking soliton7 so it makes sense that we would want something like this Under the Ricci ow7 this structure is preserved except that 739 decreases at a constant rate First note that if we consider the Nash entropy dm dm dm N M 71 idv 1 7 d 7 d m 9 MdV lt3ng MOgdvgt m fm de 7 dt 7 then M7AfiRdm M 3 V Q dm Fm My This will come in handy Now7 suppose we want a quantity W M79 such that 2 dW dm 1 7 R v2 if dt M 3 f 29 But in this case would not have scale invariance for W since t scales like distance squared so the integrand should scale like distance squared We will x this by assuming 1739 i1 dt and trying for dW 1 2 72 R v2 if d 3 dt TM 3 f 2T9 m Now to nd such a quantity consider 2 1 1 RcV2f7 g RcV2f27RAf Thus we have that 2T M 2 dFm d d 772F 7 m rdt m 1 R v2 if 3 f 279 2739 d d a TFm iNm iglogrgt This is what our Wm would be However as we did last time we wanted to reparametrize so that dm 5de where dV is evolving according to Ricci ow evolution This time we will change f f 7 glog l r so that dm e de 47w 2 5de Remark 5 This looks like the heat kernel for Euclidean space which is why this particular normalization is given Note that the preservation of dm implies that d of 1 75 trh70 Under the gradient ow 39 2 7 72R 72V at 0 f7 we have N 8f d i7 iRiA 8t 2739 f Thus we get for d WmMg739 TFm 7 Nm 7 ilogf 2 d 712 7f 7 739 R Vf f7 log4739r 477739 e dV Actually we usually renormalize this to vanish in the Euclidean case and so we can change the 1 term appropriately to N 2 N Wm Mg7 739 13 lVfl gt f 7 d 47777d257de We can also de ne the Perelman entropy as the functional 7 2 7d2 if WMgf7397 739 RlVfl f7d 477739 e dV where g is a Riemannian metric f is a function on M and 739 is a positive constant If 9 satis es the Ricci ow then we need to pull back f to get the three evolutions 897 572 4 8f d 2 57 7R7Aflvfl 5 7 dt Under these three the Perelman Entropy satis es 2 dW 47T7397d2 e de 7Mgf73927 1 R v2 77 dt M c f 279 We would like to nd a minimum over all functions f and constants 739 so that we have an invariant of the Riemannian manifold However it is not yet clear that such an in mum exists Recall that last time the existence followed from a Poincare inequality In this scale invariant setting the existence of a minimizer will follow from a log7Sobolev inequality 4 LogSobolev inequalities Let s rst consider what happens if g is the Euclidean metric We would like to switch to a function which looks like the heat kernel namely u 47T7397d2 eff 10 Recall that our model case is when f lle 4739 in which case this is precisely the backwards heat kernel The backwards heat kernel satis es Bu i A a u for 739 gt 0 and lirgli u 739 at 60 weakly where 60 is the delta function Furthermore one can check that udac 47TT7d2e lxl14Tdac 1 6 Rd W for any 739 The backwards heat kernel can be used to solve the heat equation with some given nal conditions eg to solve la 7 7A dt MTJE Hi we see that the convolution utx 4777 d2 l1yl14Tdy Rd is a solution Exercise 6 Show that all of this is true Hint to show 6 turn the integral into polar coordinates and assume the dimension is at least 2 For the dimension 1 case there is a trick involving turning it into a dimension 2 integral and separating We can check that for g Euclidean and f as above we have 2 WM797fm 7 7d 4777 d26 l1l14Tdx 739 One can show that this is zero since W Mam 7 2T va 7 A0 WW26 and integrating by parts needs to be justi ed shows this is equal to zero Now we rewrite W by replacing f with u We see that remembering still we are in Euclidean space H W142 u2 7 7 ulogu dac 7 glog 477739 7 d 11 using identities such as u 47TT7d2 eff d logu 7 log 477739 7 f W142 7 47TTleVfl26 2f W142 u2 va Tao shows that one can show that W 2 07 which implies a log7SoboleV inequality 2 T dac 2 ulogudac glog lm39 d7 u or as it is usually stated7 with J52 u7 l d 4Tqu5l2dac 2 7q5210gq52dx log4777 d 739 For the general case7 we have W142 WMgf739 739 Ru 2 7ulogu u One can show that the dV 7 glog 477739 7 d WM797f7739 2 CM7977 This implies essentially a log7SoboleV inequality7 ie7 d TRQSZdV T4anl2dv 2 7C Zlogq52dV ilog4m d ln fact7 we can take the uMg739 infWMgfT 47TT7d2e de 1 7 which is the best possible constant 7C It can be shown that u is nite7 which is what we could call a log7SoboleV inequality We can now show that if g t is a solution to Ricci ow on t E 0T0 and 739 To 7 t7 then u Mg7 is increasing The rst exercise is important Exercise 7 Show thatu Mg7 739 W My7 f 7 739 for afunction f E H1 Not quite true What is the true statement Hint you need to change to a new function 12 Once we know that p is realized by a function we can show that p M g t 739 is increasing as follows Calculate p M g to 739 t0 W M g to f to 739 to for some minimizer f to For any time t S to we can solve the equation for f in 4 backwards to t with initial condition f to at f to at since f is in H1 there exists a weak solution to this parabolic ow We know that MQULTUD S WM79t7ftv739t S WM79107f10773910 M79i0 i0 5 Noncollapsing We will now show that logeSoboleV inequalities imply noncollapsing Suppose we have a ball B p with bounded normalized curvature ie 1 R lt7 1mltxgt1 for at E B p Then 1R1 739 S Cd for some constant depending only on dimension Then the logeSoboleV inequality can be rewritten as Cd 2dVT4lvq5l2dv 2 M797T Zlogq52dVglog47T7 d Suppose 15 is a function supported on B p f such that fM qSZdV 1 Then Jensen s inequality implies that quloquZdV 2 Bq d log qudv 7 1 1 VB1 gVB where B B p Recall that Jensen s inequality requires a probability measure So 152 log WV 2 log M We now get for this particular choice of 15 1 VB39 4TlWlZdV 2MgmlogVT Now we will specialize 15 even more Suppose d was cw j for some bump function 11 on the real line which is 1 on 0 12 and supported on 0 1 technically we only need half the bump function which is how 1 described it Thus c on B p and c is such that 2dV 1 B 13 so c S VB p2712 We can choose ab so that quSl S cllc on the ball for some constant c 7 and so VB TdZ quot7 2 Mg7 log7 VB12 M VB Finally7 we can use a Bishop7Gromov volume comparison theorem 4c Theorem 8 BishopGromov comparison If Mdg is a complete Rie7 mannian manifold with Rc 2 n 7 1Kg for some K E R then for anyp E M 7 the volume ratio V B PM VK B pK7T is non7increasing as a function of T7 where pK is a point in the d7dimensional simply connected space of constant sectional curvature K7 and VK B pK7r is the volume of a ball of radius r in that space In particular7 we have that v B lt v Bl2 V71T B p71T7 7 V71T B 1771739727 and thus there is a a a 739 d such that V B v Bl2 a In fact7 a is independent of 739 since V71T B p71T7 V71 B 177171 V71T B PaTMFQ V71 31771712 Thus there is a constant cm Which depends on d such that 0 Myg 210g ie7 V B 2 akaU751 Which implies H7noncollapsing at a scale W for H exp u 7 cm Let s formu7 late this into a proposition Proposition 9 There is a constant c c d depending only on dimension such that ifu Mdg7 739 is nite then for H exp u Mg7 7 c 7 the Riemannian manifold M79 is H7noncollapsed at the scale of V 14 Let s collect the facts about u Proposition 10 The following are true about u 1 uMg739 gt 700 for any xed manifold Mg and 739 gt 0 2 If Mg satis es the Ricci flow fort E 0T0 and 739 t T0 7 t then u Mg t 739 is increasing 5 There is a constant c Cd depending only on dimension such that the Riemannian manifold Mg is Hinoncollapsed at the scale of W at every point for H exp u Mg 739 7 c We can now prove Theorem 11 Perelman s noncollapsing theorem rst version Let Mg be a solution to the Ricci flow on compact 3imanifolds fort E 0T such that at t 0 we have lRmPlgo S 1 V 390 1771 2 w for all p E M and w gt 0 xed For any p gt 0 there exists H H wTp gt 0 such that the Ricci flow is Hinoncollapsed for all to 0 6 0T X M and scales 0 lt r0 lt p We could also take p p t and get a similar result as long as pt is uniformly bounded on 0 T Proof We already showed that for a given 739 and metric u M g 739 has a lower bound For any r3 we see by monotonicity that M79i3 2M7907r3i Thus we have that if 0 inf147907r2T2 E 07 T then M79i 3 2 Mo Thus Mg is Henoncollapsed at the scale of To for all H exp 0 7 C S exp II M79 i 3 Cl We need to see that no is not 700 Since T is nite there is no problem at the top of the interval for r2 It can be shown that as r2 A 0 u Mg 0 r2 A 0 in the interest of time we will not show this and so there is no problem at the other side I Remark 12 This is a bit stronger than what proposed in an earlier lecture I think Tao was thinking about future incarnations of this theorem which is why he formulated as he did 15
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