1 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS FOR EVOLVING RIEMANNIAN MANIFOLDS KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Abstract. We erive ietities for geeral ows of Riemaia metrics that may be regare as local mea-value, mootoicity, or Lyapuov formulae. These geeralize previous work of the rst author for mea curvature ow a other oliear i usios. Our results apply i particular to Ricci ow, where they yiel a local mootoe quatity irectly aalogous to Perelma s reuce volume ~ V a a local ietity relate to Perelma s average eergy F.. Itrouctio To motivate the local formulas we erive i this paper, cosier the followig simple but quite geeral strategy for ig mootoe quatities i geometric ows, whose core iea is simply itegratio by parts. Let (M ; g(t)) be a smooth oeparameter family of complete Riemaia maifols evolvig for t [a; b] by (.) t g = h: Observe that the formal cojugate of the time-epeet heat operator t o the evolvig maifol (M ; g(t)) is ( t + + tr gh). If '; : M [a; b]! R are smooth fuctios for which the ivergece theorem is vali (e.g. if M is compact or if ' a a their erivatives ecay rapily eough at i ity), oe has (.) ' = f [( )'] + '[( t M M t t + + tr gh) ]g : If ' solves the heat equatio a solves the ajoit heat equatio, it follows that the itegral R ' is iepeet of time. More geerally, if [( M t )'] a '[( t + + tr gh) ] both have the same sig, the R ' will be mootoe i M t. If the prouct ' is geometrically meaigful, this ca yiel useful results. Here are but a few examples. Example. The simplest example uses the heat equatio o Eucliea space. Let (.3) (x; t) = [4(s t)] e jy xj = 4(s t) (x R ; t < s) eote the backwar heat kerel with sigularity at (y; s) R R. If ' solves the heat equatio a either it or its erivatives grow too fast at i ity, the '(y; s) = lim '(x; t) (x; t) x: t%s R R Because t '(x; t) (x; t) x = 0, oe has '(y; s) = R '(x; t) (x; t) x for all R R y R a t < s, which illustrates the averagig property of the heat operator. Here a throughout this paper, eotes the volume form associate to g(t).
2 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Example. Let F t : M,! M t R + be a oe-parameter family of hypersurfaces evolvig by mea curvature ow, t F t = H, where H is the mea curvature a the outwar uit ormal of the hypersurface M t. This correspos to h = HA i (.), where A is the seco fuametal form. De e by formula (.3) applie to x R + a t < s. Usig tr g h = H, oe calculates that ( t + H ) = (x y)? H (s t) : Hece by (.), ' = [( )'] (x y)? t t H (s t) ' : M t M t This is establishe for ' by Huiske [0, Theorem 3.] a geeralize by Huiske a the rst author [8, ] to ay smooth ' for which the itegrals are ite a itegratio by parts is permissible. Hece R is mootoe oicreasig i time a is costat precisely M t o homothetically shrikig solutios. The mootoicity implies that the esity MCF O := lim t%0 of the limit poit O = (0; 0) is well e e. Aother RM t cosequece is that sup M b ' sup M a ' if ( t )' 0 for t [a; b]. Example 3. A compact Riemaia maifol (M ; g(t)) evolvig by Ricci ow correspos to h = Rc i (.), so that tr g h = R. If ' a = (f jrfj + R) + f (4) = e f ; the Perelma s etropy may be writte as W (g(t); f(t); (t)) = R M '. If =t = a ( t + )f = jrfj R, the ( t + R) = j Rc +rrf gj (4) = e f : I this case, (.) becomes W(g(t); f(t); (t)) = j Rc +rrf t M gj (4) = e f ; which is formula (3.4) of . I particular, W is mootoe icreasig a is costat precisely o compact shrikig graiet solitos. Example 4. Agai for (M ; g(t)) evolvig smoothly by Ricci ow for t [a; b], let ` eote Perelma s reuce istace  from a origi (y; b). Take ' a choose v to be the reuce-volume esity 3 v(x; t) = M t [4(b t)] = e `(x;b t) (x M ; t < b): The Perelma s reuce volume is give by V ~ (t) = R '. By [3, 7], M ( t + R)v 0 hols i the barrier sese, hece i the istributioal sese.4 Throughout this paper, r represets the spatial covariat erivative, a = trgrr. 3 The formula use here a throughout this paper i ers from Perelma s by the costat factor (4) =. This ormalizatio is more coveiet for our applicatios. 4 It is a staar fact that a suitable barrier iequality implies a istributioal iequality. See  for relevat e itios a a proof. A irect proof for v is fou i [38, Lemma.].
3 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 3 Thus oe obtais mootoicity of the reuce volume if M is compact or if its Ricci curvature is boue. More geerally, oe gets mootoicity of R ' for ay oegative supersolutio ' of the heat equatio. I particular, takig '(x; t) = R(x; t) R mi (0) M o a compact maifol a otig that ( t )' 0 hols poitwise, oe veri es that R [R R M mi (0)]v is oecreasig i time. I , Felma, Ilmae, a the thir author itrouce a expaig etropy a a forwar reuce volume for compact maifols evolvig by Ricci ow. Mootoicity of these quatities may also be erive from (.) with '. Similar ieas play importat roles i Perelma s proofs of i eretial Harack estimates [3, 9] a pseuolocality [3, 0]. The strategy of itegratio by parts ca be aapte to yiel local mootoe quatities for geometric ows. We shall preset a rigorous erivatio i Sectio whe we prove our mai result, Theorem 7. Before oig so, however, we will explai the uerlyig motivatios by a purely formal argumet. Suppose for the purposes of this argumet that = S atb t is a smooth, precompact subset of M [a; b]. Assume that t is smooth with outwar uit ormal, a let eote the measure o t iuce by g(t). If the prouct ' vaishes o, the (.4) [( f )'] + '[( t t + + tr gh) ]g t b = ' t + (' hr ; i hr'; i) t: a t t This formula may be regare as a space-time aalog of Gree s seco ietity. I the special case that is the super-level set f(x; t) : (x; t) > 0g a both a a b are empty, the = jr j r, whece (.4) reuces to (.5) [( f >0gf )']+'[( t t ++tr gh) ]g t+ 'jr j t = 0: f =0g Formula (.5) eables a strategy for the costructio of local mootoe quatities. Here is the strategy, agai presete as a purely formal argumet. Let ' a > 0 be give. De e = log, a for r > 0, let (r) = log(r ). Notice that r (r) = r for all r > 0. Take to be the set e e for r > 0 by (.6) := f(x; t) : (x; t) > r g = f(x; t) : (r) > 0g: (Whe is a fuametal solutio 5 of a backwar heat equatio, the set is ofte calle a heatball.) Assume for the sake of this formal argumet that the outwar uit ormal to the time slice (t) := \(M ftg) is = jr j r. Observe that (.7) ( t + ) = ( t + + tr gh) jr j tr g h: Applyig the coarea formula to each time slice (t), followe by a itegratio i t, oe obtais (.8) jr j ' t = jr j' t: r r 5 See Sectio 5 below.
4 4 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Similarly, oe has (.9a) (tr g h) (r) ' t = (tr g h)[ r r log(r )]' t = (.9b) (tr g h)' t; r because the bouary itegral vaishes i this case. Now by rearragig (.5) a usig (.7) (.9), oe gets f (r) [( )'] + [( t t + + tr gh) ]'g t = jr j (tr g h) (r) ' t 'jr j t + (tr g h)' t E r = jr j r (tr g h) (r) ' t jr j (tr g h) (r) ' t: r De ig (.0) P '; (r) := jr log j (tr g h) log(r ) ' t a applyig a itegratig factor, oe obtais the followig formal ietity. Sice log(r ) = (r) > 0 i, this ietity prouces a local mootoe quatity wheever ( t )' a '( t + + tr gh) have the same sig. Proto-theorem. Wheever the steps above ca be rigorously justi e a all itegrals i sight make sese, the ietity (.) r P'; (r) = r r + log(r will hol i a appropriate sese. )( t )' + [( t + + tr gh) ]' t I spirit, (.) is a parabolic aalogue of the formula! r r '(x) = jx yj<r r + (r jx yj )' ; jx yj<r which for harmoic ' (i.e. ' = 0) leas to the classical local mea-value represetatio formulae '(y) =! r '(x) = jx yj<r! r '(x) : jx yj=r The mai result of this paper, Theorem 7, is a rigorous versio of the motivatioal proto-theorem above. We establish Theorem 7 i a su cietly robust framework to provie ew proofs of some classical mea-value formulae (Examples 5 8), to geerate several ew results (Corollaries 3, 5, 8, 9, 3, 6, 7) a to permit geeralizatios for future applicatios. Our immeiate origial results are orgaize as follows: i Sectio 4, we stuy Perelma s reuce volume for maifols evolvig by Ricci ow; i Sectio 5, we iscuss heat kerels o evolvig Riemaia maifols (icluig xe maifols as a iterestig special case); a i Sectio 6, we cosier Perelma s average eergy for maifols evolvig by Ricci ow. I , the thir author applies some of these results to obtai local regularity theorems
5 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 5 for Ricci ow. Potetial future geeralizatios that we have i mi cocer varifol (Brakke) solutios of mea curvature ow, solutios of Ricci ow with surgery, a fuametal solutios i the cotext of weak (Bakry Émery) Ricci curvature, e.g. . As ote above, Theorem 7 allows ew proofs of several previously kow local mootoicity formulae, all of which shoul be compare with (.). To wit: Example 5. Cosier the Eucliea metric o M = R with h = 0. If is the backwars heat kerel (.3) cetere at (y; s) a the heatball (y; s) is e e by (.6), the (.0) becomes P '; (r) = '(x; t) jy xj t: 4(s t) Thus (.) reuces to P'; (r) (.) r r = jy xj (y;s) r + log(r (y;s) )( t )' t: Sice R (y;s) 4(s t) t =, this implies the mea value ietity (.3) '(y; s) = r '(x; t) jy xj t 4(s t) (y;s) for all ' satisfyig ( t )' = 0. This localizes Example. To our kowlege, Pii [8, 9, 30] was the rst to prove (.3) i the case =. This was later geeralize to > by Watso . The geeral formula (.) appears i Evas Gariepy . There are may similar mea-value represetatio formulae for more geeral parabolic operators. For example, see Fabes Garofalo  a Garofalo Lacoelli . (Also see Corollaries 3 a 6, below.) Example 6. Surface itegrals over heatballs rst appear i the work of Fulks , who proves that a cotiuous fuctio ' o R (a; b) satis es '(y; s) = jy xj r '(x; t) p 4jy xj (s t) + [jy xj (s t)] for all su cietly small r > 0 if a oly if ' is a solutio of the heat equatio. (Compare to Corollary 5 below.) Example 7. Previous results of the rst author  localize Example for mea curvature ow. O R + ( ; 0), e e (x; t) := ( 4t) = e jxj =4t. Substitute ( t + + tr gh) = jr? + Hj a tr g h = H ito (.0) a (.). If the space-time track M = S t<0 M t of a solutio to mea curvature ow is well e e i the cylier B(0; p r =) ( r =4; 0), the  proves that formula (.), with the itegrals take over \ M, hols i the istributioal sese for ay r (0; r) a ay ' for which all itegral expressios are ite. I particular, P ; (r)=r is mootoe icreasig i r. The esity MCF O := lim t%0 (x; t) of the limit poit O = (0; 0) ca thus be calculate RM t locally by MCF P ; (r) O = lim r&0 r :
6 6 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING (Compare to Corollary 8 below.) Relate work of the rst author for other oliear i usios is fou i . Example 8. Perelma s scale etropy W a the forwar reuce volume + are localize by the thir author [5, Propositios 5., 5.3, 5.4]. Although oly state there for Kähler Ricci ow, these localizatios remai vali for Ricci ow i geeral. They are motivate by the rst author s work o mea curvature ow  a arise from (.) by takig ' to be a suitable cuto fuctio e e with respect to ` a `+, respectively. The remaier of this paper is orgaize as follows. I Sectio, we rigorously erive Theorem 7: the geeral local mootoicity formula motivate by formula (.) above. I Sectio 3, we erive a local graiet estimate for solutios of the cojugate heat equatio. I Sectios 4 5, we apply this machiery to obtai ew results i some special cases where our assumptios ca be checke a i which (.) simpli es a becomes more familiar. The Appeix (Sectio 7) reviews some relevat properties of Perelma s reuce istace a volume. Ackowlegmets. K.E. was partially supporte by SFB 647. D.K. was partially supporte by NSF grats DMS-0584, DMS , a a Uiversity of Texas Summer Research Assigmet. L.N. was partially supporte by NSF grats a a Alfre P. Sloa Fellowship. P.T. was partially supporte by a EPSRC Avace Research Fellowship. L.N. thaks both Professor Beett Chow a Professor Peter Ebefelt for brigig Watso s mea-value equality to his attetio. This motivate him to stuy heatball costructios a i particular  a . He also thaks Professor Peter Li for may helpful iscussios.. The rigorous erivatio Let < a < b <, a let (M ; g(t)) be a smooth oe-parameter family of complete Riemaia maifols evolvig by (.) for t [a; b]. As ote above, the formal cojugate of the heat operator t o (M ; g(t)) is ( t + + tr gh). For R, we aopt the staar otatio  + := maxf; 0g. Let be a give positive fuctio o M [a; b). As i Sectio, it is coveiet to work with (.) := log a the fuctio e e for each r > 0 by (.) (r) := + log r: For r > 0, we e e the space-time super-level set ( heatball ) (.3a) (.3b) = f(x; t) M [a; b) : > r g = f(x; t) M [a; b) : (r) > 0g: We woul like to allow to blow up as we approach time t = b; i particular, we have i mi various fuctios which have a sigularity that agrees asymptotically with a (backwars) heat kerel cetere at some poit i M at time t = b. (See Sectios 6 5.) I this cotext, we make, for the momet, the followig three assumptios about.
7 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 7 Assumptio. The fuctio is locally Lipschitz o M [a; s] for ay s (a; b). Assumptio. There exists a compact subset M such that outsie [a; b). Assumptio 3. There exists r > 0 such that! j j = 0 a lim s%b \(M fsg) jr j t < : is boue Remark 4. By the cotiuity of from Assumptio a its boueess from Assumptio, we ca be sure, after reucig r > 0 if ecessary, that r outsie some compact subset of M (a; b]. I particular, we the kow that the super-level sets lie isie this compact subset for r (0; r]. Remark 5. By Assumptio 3 a compactess of, oe has R j j t <. Remark 6. We make o irect assumptios about the regularity of the sets themselves. Let ' be a arbitrary smooth fuctio o M (a; b]. By Assumptio 3, the quatity (.4) P '; (r) := [jr j (r)(tr g h)]' t is ite for r (0; r]. Our mai result is as follows: Theorem 7. Suppose that (M ; g(t)) is a smooth oe-parameter family of complete Riemaia maifols evolvig by (.) for t [a; b], that : M [a; b)! (0; ) satis es Assumptios 3, that r > 0 is chose accorig to Assumptio 3 a Remark 4, a that 0 < r 0 < r r. If is smooth a the fuctio belogs to L ( ), the (.5) P '; (r ) r r r 0 r + If, istea, iequality P '; (r 0 ) r 0 [ ( t + + tr gh) = ( t + + jr j + tr g h)' t + + jr j + tr g h ( + log r)( ' t ')] t r: is merely locally Lipschitz i the sese of Assumptio, a the ( t + + tr gh) 0 hols i the istributioal sese, a ' 0, the P '; (r ) P '; (r 0 ) r (.6) r r0 r + ( + log r)( ' t r 0 ') t r:
8 8 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Remark 8. If ' solves the heat equatio a solves the cojugate heat equatio, the (.5) implies that P '; (r)=r is iepeet of r. See Example 5 (above) a Corollary 3 (below). Proof of Theorem 7. We begi by assumig that is smooth. I the proof, we write P () P '; (). For most of the proof, we will work with a moi e fuctio, amely (.7) P (r; s) := [jr j \(M [a;s]) (r)(tr g h)]' t; arisig from restrictio to the time iterval [a; s], for some s (a; b). As a result, we will oly be workig o omais o which a its erivatives are boue, a the covergece of itegrals will ot be i oubt. A limit s % b will be take at the e. Let : R! [0; ] be a smooth fuctio with the properties that (y) = 0 for y 0 a 0 (y) 0. Let : R! [0; ) eote the primitive of e e by (y) = R (x) x. Oe shoul keep i mi that ca be mae very close to y the Heavisie fuctio, i which case (y) will lie a little below [y] +. For r (0; r] a s (a; b), we e e (.8) Q(r; s) := [jr j ( (r) ) M [a;s] ( (r) )(tr g h)]' t; which shoul be regare as a perturbatio of P (r; s), a will relieve us of some techical problems arisig from the fact that we have o cotrol o the regularity of. Note that ( (r) ) a ( (r) ) have support i. Therefore, the covergece of the itegrals is guaratee. I the followig computatios, we suppress the epeece of Q o s a assume that each itegral is over the space-time regio M [a; s] uless otherwise state. Oe has (.9) r + r Q(r) r = r Q0 (r) Q(r) = [jr j 0 ( (r) ) 0 ( (r) )(tr g h)]' t Q(r) = [jr j 0 ( (r) )]' t [( (r) )(tr g h)]' t [jr j ( (r) )]' t + [( (r) )(tr g h)]' t: The rst itegral a the last itegral i the last equality o the right-ha sie require further attetio.
9 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 9 For the rst of these, we keep i mi that r = r (r) a compute (.0) [jr j 0 ( (r) )]' t r = ; r(( (r) )) ' t = [( )( (r) )' + r (r) ; r' ( (r) )] t = [ ( )( (r) )' + (') (r) ( (r) ) + hr ; r'i (r) 0 ( (r) )] t; the calculatio beig vali o each time slice. For the fourth itegral, we compute that at each time t (a; b), oe has ( (r) )' = [ 0 ( (r) ) t t '+( (r)) ' t +( (r))'(tr g h)] ; M ftg M ftg the al term comig from i eretiatio of the volume form. Itegratig over the time iterval [a; s] a usig the facts that 0 = a that ( (r) ) = 0 at t = a (which hols because (r) 0 at t = a by Remark 4), we that (.) ( (r) )(tr g h)' t = [( (r) ) t ' + ( (r)) ' ] t t + ( (r) )' ; M fsg where the itegrals are still over M [a; s] uless otherwise iicate. We ow combie (.9) with (.0) a (.) to obtai r + Q(r) (.) r r = ( t + + jr j + tr g h)( (r) )' t ' + (') (r) ( (r) ) t t ( (r)) t + hr ; r'i (r) 0 ( (r) ) t + ( (r) )' : M fsg The etire ietity may ow be multiplie by =r + a itegrate with respect to r betwee r 0 a r, where 0 < r 0 < r r, to get a ietity for the quatity Q(r ; s)=r Q(r 0 ; s)=r0. We may simplify the resultig expressio by pickig a appropriate sequece of vali fuctios a passig to the limit. Precisely, we pick a smooth : R! [0; ] with the properties that (y) = 0 for y =, (y) = for y, a (y) 0 0. The we e e a sequece k : R! [0; ] by k (y) = ( k y). As k icreases, this sequece icreases poitwise to the characteristic fuctio of (0; ). The correspoig k coverge uiformly to the fuctio y 7! [y] +. Crucially, we also ca make use of the facts that k ( (r) ) coverges to the characteristic fuctio of i L (M [a; b]) a that (r) k 0 ( (r)) is a boue sequece of fuctios o M [a; b) with isjoit supports for each k. Iee, the support of k 0 lies withi the iterval ( k ; k ).
10 0 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING For each r (0; r], we have Q(r; s)! P (r; s) as k!. Usig the omiate covergece theorem, our expressio becomes (.3) P (r ; s) r = + P (r 0 ; s) r0 r r 0 r r 0 r r 0 r + r + r + \(M [a;s]) \(M [a;s]) M fsg ( t + + jr j + tr g h)' t r (r)( ' t [ (r) ] + ' r: ') t r Now we may take the limit as s % b. By Assumptio 3, the al term coverges to zero, a we e up with (.5) as esire. Next we tur to the case that is merely Lipschitz, i the sese of Assumptio. Give s (a; b) a fuctios a as above, there exists a sequece of smooth fuctios j o M [a; b] such that j! i both W ; a C 0 o the set \ (M [a; s]). By hypothesis o our Lipschitz, we have I := M [a;s] ( t + + jr j + tr g h)( (r) )' t 0; where we make sese of the Laplacia term via itegratio by parts, amely ( )( (r) )' t := [hr ; r'i ( (r) ) + jr j 0 ( (r) )'] t: By e itio of j, we have lim j! M [a;s] ( j t + j + jr j j + tr g h)(( j ) (r) )' t = I 0; uiformly i r (0; r]. Cosequetly, we may carry out the same calculatios that we i i the rst part of the proof to obtai a iequality for the quatity Q(r ; s)=r Q(r 0 ; s)=r0, with j i place of. We the pass to the limit as j! to obtai the iequality Q(r ; s) Q(r 0 ; s) r (.4) r r0 r + (') (r) ( (r) ) t r M [a;s] + + r 0 r r 0 r r 0 r r 0 r + r + r + M [a;s] M [a;s] M fsg ' t ( (r)) t r hr ; r'i (r) 0 ( (r) ) t r ( (r) )' r for our Lipschitz. Fially, we replace with the same sequece of cut-o fuctios k that we use before (thus approximatig the Heavisie fuctio), take the limit as k!, a the take the limit as s % b. This gives the iequality (.6).
11 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS The argumet above may be compare to proofs of earlier results, especially the proof  of the local mootoicity formula for mea curvature ow. There is a alterative formula for (.4) that we useful i the sequel: Lemma 9. Suppose that (M ; g(t)) is a smooth oe-parameter family of complete Riemaia maifols evolvig by (.) for t [a; b], that : M [a; b)! (0; ) satis es Assumptios 3, that r > 0 is etermie by Assumptio 3 a Remark 4, a that 0 < r 0 < r r. If ' a t + jr j L ( ), the for all r (0; r], oe has P '; (r) = ( t + jr j ) t: Proof. I the case that ', substitutig formula (.) ito formula (.8) yiels Q(r; s) = ( t + jr j )( (r) ) t ( (r) ) : M [a;s] M fsg Although (.) was erive assumig smoothess of, oe ca verify that it hols for locally Lipschitz satisfyig Assumptio by approximatig by a sequece of smooth j (as i the proof of Theorem 7) a the passig to the limit as j!. The if t + jr j L ( ), oe may (agai as i the proof of Theorem 7) choose a sequece k alog which Q(r; s)! P (r; s) as k! a the let s % b to obtai the state formula. 3. A local graiet estimate I orer to apply Theorem 7 to a fuametal solutio of the heat equatio of a evolvig maifol i Sectio 5, we ee a local graiet estimate. Oe approach woul be to aapt existig theory of local heat kerel asymptotics. Istea, we prove a more geeral result which may be of iepeet iterest. Compare , , , the recet , , a [36, 37]. Let (M ; g(t)) be a smooth oe-parameter family of complete Riemaia maifols evolvig by (.) for t [0; t]. We shall abuse otatio by writig g() to mea g((t)), where (t) := t t: I the remaier of this sectio, we state our results solely i terms of. particular, g() satis es g = h o M [0; t]. Give x M a > 0, e e (3.) () := [ B g() (x; ) fg M [0; t]: 0t We ow prove a local a priori estimate for boue positive solutios of the cojugate heat equatio (3.) ( We will apply this i Sectio 5. tr g h)v = 0: I
12 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Theorem 0. Let (M ; g()) be a smooth oe-parameter family of complete Riemaia maifols evolvig by g = h for 0 t. Assume there exist k ; k ; k 3 0 such that h k g; Rc k g; a jr(tr g h)j k 3 i the space-time regio () give by (3.). Assume further that v() solves (3.) a satis es 0 < v A i (). The there exist a costat C epeig oly o a a absolute costat C such that at all (x; ) (), oe has jrvj v ( + log A v ) + C k + k + k 3 + p k 3 + C Proof. By scalig, we may assume that A =. We e e 6 computig that f := log v a w := jr log( f)j ; p k coth( p k ) + C ( )f = jrfj + (tr g h): The usig Bocher Weitzeböck, we calculate that ( )jrfj = h(rf; rf) Rc(rf; rf) jrrfj + r(tr g h + jrfj ); rf a ( )w = ( f) [h(rf; rf) Rc(rf; rf) + hr(tr gh); rfi] " rjrfj ( f) jrrfj ; rf # + + jrfj4 f ( f) 4 jrfj4 ( f) 4 + (tr gh)jrfj + jrfj 4 rjrfj ; rf ( f) 3 f ( f) 3 : By rewritig the last term above as rjrfj ; rf f ( f) 3 = f jrfj4 hrw; rfi + 4 f ( f) 4 4 jrfj4 ( f) 3 a cacellig terms, we obtai ( )w = ( f) [h(rf; rf) Rc(rf; rf) + hr(tr gh); rfi] " rjrfj ( f) jrrfj ; rf # + + jrfj4 f ( f) + (tr gh)jrfj jrfj 4 f ( f) 3 + hrw; rfi : f Now let (s) be a smooth oegative cuto fuctio such that (s) = whe s a (s) = 0 whe s, with 0 0, j 0 j C, ( 0 ) C, a 00 C. De e g() (x; x) u(x; ) := : 6 Note that w is use by Souplet hag [33, Theorem.] i geeralizig Hamilto s result . A similar fuctio is employe by Yau . Also see relate work of the thir author . :
13 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 3 Observe that at each xe, u is smooth i space o of the g() cut locus of x. However, for our purposes of applyig the maximum priciple, Calabi s trick allows us to procee as though u were smooth everywhere. Thus, we calculate that a a jruj Now let G := uw a compute that ( )(G) = G + u ( u C u C k u C p k coth( p k ) + C : )w + w ( )u hru; rwi : For ay > 0, cosier G o M [0; ]. At ay poit (x 0 ; 0 ) where G attais its maximum o M [0; ], we have 0 ( )( G) a ( )(G) G hru; rwi h(rf; rf) Rc(rf; rf) + hr(trg h); rfi + u ( f) + (tr gh)jrfj ( f) 3 f + u f hrw; rfi jrfj 4 ( f) 3 + w C k + C p k coth( p k ) + C : Usig the fact that rg(x 0 ; t 0 ) = 0, we ca replace urw by wru above. The multiplyig both sies of the iequality by u [0; ] a usig =( f), we obtai p o 0 G + [( + )k + k ]G + k 3 G f + Gjrujjrfj ( f)g f + G C k + C p k coth( p k ) + C : p Noticig that k 3 G k3 (G + ) a that f Gjrujjrf j G f jrfj f u + jruj u f f ( f)g + G C f f ; we estimate at (x 0 ; t 0 ) that 0 k 3 + G + C k + k + k 3 + C p k coth( p k ) + C + G C f f ( f)g :
14 4 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Diviig both sies by ( f) while otig that =( f) a f=( f), we get 0 k 3 + G + C k + k + k 3 + C p k coth( p k ) + C G ; from which we ca coclue that G + C k + k + k 3 + p k 3 + C p k coth( p k ) + C at (x 0 ; 0 ). Hece W ( ) := sup xbg() (x;) w(x; ) may be estimate by W ( ) 0 G(x 0 ; 0 ) + 0 C k + k + k 3 + p k 3 + C p k coth( p k ) + C + C k + k + k 3 + p k 3 + C Sice > 0 was arbitrary, the result follows. p k coth( p k ) + C : Remark. I the special case that h 0, we have jrvj v ( + log A v ) + k + C p k coth( p k ) + C at (x; ), for all times [0; t] a poits x B g() (x; ), which slightly improves a result of . 4. Reuce volume for Ricci flow Our rst applicatio of Theorem 7 is to Ricci ow. Let (M ; g(t)) be a complete solutio of Ricci ow that remais smooth for 0 t t. This correspos to h = Rc a tr g h = R i (.). 4.. Localizig Perelma s reuce volume. Perelma [3, 7] has iscovere a remarkable quatity that may be regare as a ki of parabolic istace for Ricci ow. De e (t) := t t, otig that g((t)) the satis es g = Rc for 0 t. Fix x M a regar (x; 0) (i (x; ) cooriates) as a space-time origi. The space-time actio of a smooth path with (0) = (x; 0) a () = (x; ) is p L() := j (4.a) 0 j + R p (4.b) = j s j + s R s (s = p ): 0 Takig the i mum over all such paths, Perelma e es the reuce istace from (x; 0) to (x; ) as (4.) `(x; ) = `(x;0) (x; ) := p if L(); a observes that (4.3) v(x; ) = v (x;0) (x; ) := `(x;) e (4) =
15 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 5 is a subsolutio of the cojugate heat equatio u = u Ru i the barrier sese , hece i the istributioal sese. 7 It follows that the reuce volume (essetially a Gaussia weighte volume) (4.4) V ~ (t) = V(x;0) ~ (t) := v(x; ) M is a mootoically icreasig fuctio of t which is costat precisely o shrikig graiet solitos. (Compare to Example 4 above.) The iterpretatios of ` as parabolic istace a V ~ as Gaussia weighte volume are eluciate by the followig examples. Example 9. Let (M ; g) be a Riemaia maifol of oegative Ricci curvature, a let q be ay smooth superharmoic fuctio (q 0). I their semial paper , Li a Yau e e (x; ) = if 4 0 j j + 0 q(()) ; where the i mum is take over all smooth paths from a origi (x; 0). As a special case of their more geeral results [, Theorem 4.3], they observe that (4) = e (x;) is a istributioal subsolutio of the liear parabolic equatio ( + q)u = 0. Example 0. Let (R ; g) eote Eucliea space with its staar at metric. Give R, e e X = gra( 4 jxj ). The oe has 0 = Rc = g L X g. Hece there is a Ricci solito structure (i.e. a i itesimal Ricci solito) o Eucliea space, calle the Gaussia solito. It is otrivial wheever 6= 0. Take = to give (R ; g) the structure of a graiet shrikig solito. The () = p = x is a L-geoesic from (0; 0) to (x; ). Thus the reuce istace is `(0;0) (x; ) = jxj =4 a the reuce volume itegra is exactly the heat kerel v (0;0) (x; ) = (4) = e jxj =4. Hece V ~ (0;0) (t). (Compare [, 5].) Example. Let Sr() eote the rou sphere of raius r() = p ( ). This is a positive Eistei maifol, hece a homothetically shrikig (i t) solutio of Ricci ow. Alog ay sequece (x k ; k ) of smooth origis approachig the sigularity O at = 0, oe gets a smooth fuctio `O(x; ) := lim k! `(xk ; k )(x; ) = measurig the reuce istace from O. Hece V ~ O (t) [( )=(e)] = Vol(S ) for all t < 0. (See [4, 7.].) Our rst applicatio of Theorem 7 is where is Perelma s reuce-volume esity v (4.3). Let ` eote the reuce istace (4.) from a smooth origi (x; t) a assume there exists k (0; ) such that Rc kg o M [0; t]. I what follows, we will freely use results from the Appeix (Sectio 7, below). Lemma 39 guaratees that ` is locally Lipschitz, hece that Assumptio is satis e. (Also see  or .) The estimate i Part () of Lemma 8 esures that Assumptio is satis e. Assumptio 3 follows from combiig that estimate, Corollary 3, a Lemma 40. Here we may take ay r > 0 satisfyig r < mift=c; 4g, where c = e 4kt=3 =(4). So for r (0; r], cosier P ';v (r) := [jr`j r + R( log p `)]' t: 4 7 See  for a irect proof of the istributioal iequality.
16 6 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Notice that jr`j replaces the term jx xj 4 i the heatball formulas for Eucliea space a solutios of mea curvature ow. See Examples 5 a 7, respectively. Remark. For r (0; r], oe may write P ;v (r) i either alterative form (4.5a) P ;v (r) = ( + ` + jr`j ) t = ( (4.5b) 3= K) t: Here K(x; ) = R 0 3= H(=) is compute alog a miimizig L-geoesic, where H(X) = Rc(X; X) (R + hrr; Xi+R=) is Hamilto s trace i eretial Harack expressio. If R 0 a ' 0 o, the for all r (0; r], oe has (4.6) P ';v (r) = [jr`j + R (r) ]' t jr`j ' t 0: If (M ; g(0)) has oegative curvature operator a r < 4t( some C >, the for all r (0; r], = + C` (4.7) P ;v (r) t: =C) for Proof. By Part () of Lemma 39, the argumets of Lemma 40 apply to show that t + jr j = + ` + jr`j L ( ). Hece Lemma 9 a ietities (7.5) a (7.6) of  imply formulae (4.5). Sice (r) > 0 i, the iequalities i (4.6) are clear. If (M ; g(0)) has oegative curvature operator, Hamilto s trace i eretial Harack iequality  implies that alog a miimizig L-geoesic. Hece H( ) R( + t ) = t R t 3= t K 3= t 0 p (R + j j ) = By Lemma 3, oe has < r =4, which gives estimate (4.7). t t ` : Our mai result i this sectio is as follows. Recall that (r) := log( p 4 ) `. Corollary 3. Let (M ; g(t)) be a complete solutio of Ricci ow that remais smooth for 0 t t a satis es Rc kg. Let ' be ay smooth oegative fuctio of (x; t) a let c = e 4kt=3 =(4). The wheever 0 < r 0 < r < mif p t=c; p g, oe has (4.8) Furthermore, P ';v (r ) r P ';v (r 0 ) r 0 r r 0 r + (4.9) '(x; t) = lim r&0 P ';v (r) r : (r)( ' t ') t r: r
17 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 7 I particular, (4.0) '(x; t) P ';v(r ) r r + 0 r + (r)[( t )'] t r: Proof. The quatity = v satis es v t + v Rv 0 as a istributio. (This is implie by Perelma s barrier iequality [3, (7.3)]; see [38, Lemma.] for a irect proof.) Hece we may apply Theorem 7 i the form (.6) to obtai (4.8). Formula (7.6) of Perelma  implies that P ';v (r) = [ ` + R (r) R 3= K]' t: By Corollary 3, there is a precompact eighborhoo U of x with U [0; cr ] for all r > 0 uer cosieratio. By Lemma 35, there exists a precompact set V such that the images of all miimizig L-geoesics from (x; 0) to poits i U [0; cr ] are cotaie i the set V [0; cr ], i which oe has uiform bous o all curvatures a their erivatives. So by Lemma 8, oe has ` 4 + O( ) a r R (r) = R( log p 4 `) = O( ) as & 0. By Corollary 37, 3= K is also O( ) as & 0. Aaptig the argumets i the proof of Lemma 40, oe coclues that P ';v (r) lim r&0 r = lim 0(x; x) r&0 r 4 ' t = '(x; t); exactly as i the calculatio for Eucliea space. (Also see Corollary 3, below.) = 0 (x;x) A example of how this result may be applie is the followig local Harack iequality, which follows irectly from (4.0). Remark 4. Assume the hypotheses of Corollary 3 hol. If R 0 o, the R(x; t) r r [jr`j + R (r) ]R t + 0 r + (r) jrcj t r: The iequality (4.8) is sharp i the followig sese. Corollary 5. Let (M ; g(t)) be a complete solutio of Ricci ow that is smooth for 0 t t, with Rc kg. If equality hols i (4.8) for ', the ( ; g(t)) is isometric to a shrikig graiet solito for all r < mif p t=c; p g. Proof. From the proof of Theorem 7, it is easy to see that P;v (r) ( r r = r Er t + + tr gh)v + t v for almost all r < mif p t=c; p g. Therefore, equality i (4.8) implies that v is a istributioal solutio of the parabolic equatio ( + R)v = 0 i for almost all small r. By parabolic regularity, v is actually smooth. This implies that oe has equality i the chai of iequalities ` jr`j + R ` = ( R + 3= K) + ` ` + ` that follow from equatios (7.3), (7.5), a (7.0) of . Hece oe has u := (` jr`j + R) + ` = 0:
18 8 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING By equatio (9.) of  (where the roles of u a v are reverse), this implies that 0 = ( + R)(uv) = j Rc +rr` gj v: This is possible oly if ( ; g(t)) has the structure of a shrikig graiet solito with potetial fuctio `. Remark 6. For applicatios of Corollary 5 to regularity theorems for Ricci ow, see  by the thir author. 4.. Comparig global a local quatities. Corollaries 3 a 5 suggest a atural questio: how oes the purely local mootoe quatity P ;v (r)=r compare to Perelma s global mootoe quatity V ~ (t) = R v? A path to a partial M aswer begis with a observatio that geeralizes Example above. Cao, Hamilto, a Ilmae  prove that ay complete graiet shrikig solito (M ; g(t)) that exists up to a maximal time T < a satis es certai ocollapsig a curvature ecay hypotheses coverges as t % T to a icomplete (possibly empty) metric coe (C; ), which is smooth except at the parabolic vertex O. The covergece is smooth except o a compact set (possibly all of M ) that vaishes ito the vertex. 8 Furthermore, they prove that alog a sequece (x k ; k ) approachig O, a limit `O(x; ) := lim `(xk ; k )(x; ) exists for all x M a (t) > 0. They show that the cetral esity fuctio RF O (t) := V ~ O (t) = lim ~V (xk ; k )(t) k! of the parabolic vertex O is iepeet of time a satis es RF O (t) e, where is the costat etropy of the solito (M ; g()). O a compact solito, there is a poitwise versio of the Cao Hamilto Ilmae result, ue to Beett Chow a the thir author: Lemma 7. If (M ; g()) is a compact shrikig (ecessarily graiet) solito, the the limit `O(x; ) exists for all x M a (t) > 0. This limit agrees up to a costat with the solito potetial fuctio f(x; ). See  for a proof. Recall that the etropy of a compact Riemaia maifol (M ; g) is (M ; g) := if W (g; f; ) : f C0 ; > 0; (4) = e f = ; M where (4.) W (g; f; ) : = (jrfj + R) + f (4) = e f : M (Compare to Example 3.) Uer the couple system (4.a) (4.b) (4.c) t g = Rc ( t + )f = jrfj R + t = ; 8 See  for examples where (C; ) = lim&0 (M ; g()) is oempty.
19 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 9 the fuctioal W(g(t); f(t); (t)) is mootoe icreasig i time a is costat precisely o a compact shrikig graiet solito with potetial fuctio f, where (after possible ormalizatio) oe has (4.3) Rc +rrf g $ 0: Here a i the remaier of this sectio, the symbol $ eotes a ietity that hols o a shrikig graiet solito. We are ow reay to aswer the questio we pose above regarig the relatioship betwee P ;v (r)=r a V ~ (t). (Compare to Example 7.) Corollary 8. Let (M ; g(t)) be a compact shrikig Ricci solito that vaishes ito a parabolic vertex O at time T. The for all t < T a r > 0, oe has where P ;v (r) = R [jr`j + R( log RF O (t) := ~ V O (t) = P ;v(r) r ; r p 4 `)] t is compute with ` = `O. Proof. It will be easiest to regar everythig as a fuctio of (t) := T t > 0. Because (M ; g()) is a compact shrikig solito, there exist a time-iepeet metric g a fuctio f o M such that Rc(g) + r r f g = 0. The solutio of Ricci ow is the g() = (g), where f g >0 is a oe-parameter family of i eomorphisms such that = i a (x) = gra g f(x). The solito potetial fuctio satis es f(x; ) = f(x) a f = jrfj. (Notice that (4.3) implies that system (4.) hols.) Let = (4) = e `(x;), where ` is the reuce istace from the parabolic vertex O. By Lemma 7, ` = f + C. So Assumptios a are clearly satis e. Because j j = O[ = log( = )] a jr j = O( = ) M fg M fg as & 0, Assumptio 3 is satis e as well. Because = jrfj, Lemma 9 implies that P ;v (r) = ( + ` + jr`j ) t $ t: (Compare Remark.) Computig V ~ () = V ~ O (), oe s that ~V () = (4) = e `(x;) (g()) M = $ = = E $ P ;v (): Vol g() fx : (4) = e `(x;) zg z h Vol g() Vol g()fy : `(y; ) < log t i fx : (4) = e `(x;) g (z = = ) p 4 g
20 0 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING But o a shrikig graiet solito, P ;v (r)=r is iepeet of r > 0, while ~V () is iepeet of > 0. Sice they agree at r = a =, they agree everywhere. Sice the reuce istace a reuce volume are ivariat uer parabolic rescalig, similar cosieratios apply to solutios whose rescale limits are shrikig graiet solitos Localizig forwar reuce volume. I , Felma, Ilmae, a the thir author itrouce a forwar reuce istace t `+(x; t) := if p p s j t s j + R s: 0 Here the i mum is take over smooth paths from a origi (x; 0) to (x; t). De e u(x; t) = (4t) = e `+(x;t) a = log u. I , it is prove that ( t R)u 0 hols i the istributioal sese if (M ; g(t)) is a complete solutio of Ricci ow with boue oegative curvature operator for 0 t T. Followig the same argumets as i the proof of Corollary 3 the leas to the followig result for P ';u (r) = [jr`+j R( log r p 4t `+)]' t: Corollary 9. Let (M ; g(t)) be a complete solutio of Ricci ow with boue oegative curvature operator for 0 t T. Let ' be ay smooth oegative fuctio. The wheever 0 < r 0 < r < p 4T, oe has (4.4) P ';u (r ) r P ';u (r 0 ) r 0 r r 0 r + ( + log r)( ' t I irect aalogy with Corollary 5, oe also has the followig. + ') t r: Corollary 0. Let (M ; g(t)) be a complete solutio of Ricci ow with boue oegative curvature operator for 0 t T. If equality hols i (4.4) with ', the ( ; g(t)) is isometric to a expaig graiet solito for all r < p 4T. 5. Mea-value theorems for heat kerels I this sectio, we apply Theorem 7 to heat kerels of evolvig Riemaia maifols, especially those evolvig by Ricci ow, with statioary (i.e. time-iepeet) maifols appearig as a iterestig special case. Let (M ; g(t)) be a smooth family of Riemaia maifols evolvig by (.) for t [0; t]. We will agai abuse otatio by regarig certai evolvig quatities, where coveiet, as fuctios of x M a (t) := t t. A smooth fuctio : (M [0; t])(x; 0)! R + is calle a fuametal solutio of the cojugate heat equatio (5.) ( tr g h) = 0 with sigularity at (x; 0) if satis es (5.) at all (x; ) M (0; t], with lim &0 (; ) = x i the sese of istributios. We call a miimal fuametal solutio of (5.) a heat kerel.