A NOTE ON PERELMAN S LYH TYPE INEQUALITY. Lei Ni. Abstract

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1 A NOTE ON PERELAN S LYH TYPE INEQUALITY Lei Ni Abstract We give a proof to the Li-Ya-Hamilton type ineqality claime by Perelman on the fnamental soltion to the conjgate heat eqation. The rest of the paper is evote to improving the known ifferential ineqalities of Li-Ya-Hamilton type via monotonicity formlae.. Introction In [P], Perelman prove a Li-Ya-Hamilton type also calle ifferential Harnack ineqality for the fnamental soltion of the conjgate heat eqation, on a manifol evolving by the Ricci flow. ore precisely, let, g ij t be a soltion to Ricci flow:. on [, T ] an let Hx, y, τ = t g ij = R ij e f 4πτ n where τ = T t be the fnamental soltion to the conjgate heat eqation τ +R =. ore precisely we shol write the fnamental soltion as Hy, t; x, T, which satisfies t + y + Ry, t Hy, t; x, T = for any y, t with t < T an lim t T Hy, t; x, T fy, t µ ty = fx, T. Define v H = [ τ f f + R + f n ] H. Here all the ifferentiations are taken with respect to y, an n = im R. Then v H on [, T ]. This reslt is a ifferential ineqality of Li-Ya type [LY], which has important applications in the later part of [P]. For example it is essential in proving the pseolocality theorem in Section of [P]. It is also crcial in localizing the entropy formla [N3]. athematics Sbject Classification. Primary 58J35. Key wors an phrases. Heat eqation, ifferential Harnack ineqality, entropy formla, local monotonicity formlae, mean vale theorem. The athor was spporte in part by NSF Grants an an Alfre P. Sloan Fellowship, USA. Receive September 5.

2 LEI NI. In Section 9 of [P], the following important ifferential eqation τ + R v = τ R ij + i j f τ g ij is state for any positive soltion to the conjgate heat eqation, whose integration on gives the celebrate entropy formla for the Ricci flow. One can conslt varios sorces e.g. [N] for the etaile comptations of this eqation, which can also be one throgh a straightforwar calclation, after knowing the reslt. [P] then procees with the proof of the claim v H in a clever way by checking that for any τ with T τ >, v Hyhy µ τ y, for any smooth fnction hy with compact spport. In orer to achieve this, in [P] the heat eqation t hy, t = with the initial ata hy, T τ = hy more precisely t = T τ, the given compactly spporte nonnegative fnction, is solve. Applying. to y, τ = Hx, y, τ, one can easily erive as in [P], via integration by parts, that.3 v H h µ τ = τ R ij + i j f τ τ g ij Hh µ τ. The Li-Ya type ineqality v H then follows from the above monotonicity, provie the claim that.4 lim v H h µ τ. τ The main prpose of this note is to prove.4, hence provie a complete proof of the claim v H. This will be one in Section 3 after some preparations in Section. It was written in [P] that it is easy to see that lim τ v Hh µ τ =. It trns ot that the proof fon here nees to se some graient estimates for positive soltions, qite precise estimate on the rece istance, a tool also introce by Perelman in [P], an the monotonicity formla.3. We shall focs on the proof of.4 for the case when is compact an leave the more technical etails of generalizing it to the noncompact setting to the later refinements. Inee the claim that lim τ v Hh µ τ = follows from a blow-p argment of [P], after we have establishe.4. Since or argment is a bit involve, this may not be the proof. In Section 4 we erive several monotonicity formlae, which improve varios Li-Ya-Hamilton ineqalities for linear heat eqation systems as well as for Ricci flow, incling the original Li-Ya s ineqality. In Section 5 we illstrate the localization of them by applying a general scheme of [EKNT].

3 A NOTE ON PERELAN S LYH TYPE INEQUALITY 3. Estimates an reslts neee We shall collect some known reslts an erive some estimates neee for proving.4 in this section. We nee the asymptotic behavior of the fnamental soltion to the conjgate heat eqation for small τ. Let τ x, y be the istance fnction with respect to the metric gτ. Let B τ x, r V ol τ be the ball of rais r centere at x the volme with respect to the metric gτ. Theorem.. Let Hx, y, τ be the fnamental soltion to the backwar in t conjgate heat eqation. Then as τ we have that exp x,y. Hx, y, τ τ j 4πτ n j x, y, τ. By. we mean that exists T > an seqence j C [, T ] sch that exp x,y k Hx, y, τ τ j 4πτ n j x, y, τ = w k x, y, τ with j= j= w k x, y, τ = O τ k+ n as τ, niformly for all x, y. The fnction x, y, τ can be chosen so that x, x, =. This reslt was prove in etail, for example in [GL], when there is no zero orer term Ry, τy, τ in the eqation τ + R = an replacing x, y by τ x, y. However, one can check that the argment carries over to this case if one assmes that the metric gτ is C near τ =. One can conslt [SY, CLN] for intrinsic presentations. Let W h g, H, τ = v H h µ τ where h is the previosly escribe soltion to the heat eqation. It is clear that for any τ with T τ >, W h g, H, τ is a well-efine qantity. A priori it may blow p as τ. It trns ot that in or corse of proving that lim τ W h g, H, τ we nee to show first that exists C >, which may epens on the geometry of the Ricci flow soltion, gτ efine on [, T ], bt inepenent of τ as τ so that W h g, H, τ C for all T τ >. The following lemma see also [EKNT] for a localize version of it spplies the key estimates for this prpose. Lemma.. Let, gt be a smooth soltion to the Ricci flow on [, T ]. Assme that there exist k an k, sch that the

4 4 LEI NI Ricci crvatre R ij gτ k g ij τ an maxry, τ, R y, τ k, on [, t]. i If A is a positive soltion to the conjgate heat eqation on [, T ], then there exists C an C epening on k, k an n sch that for < τ min, T, k,. τ + C τ log A + C τ ii If is a positive soltion to the conjgate heat eqation on [, T ], then there exists B, epening on, gτ so that for τ mint, k,,.3 τ + C τ B log τ n µ τ + C τ. Remark.3. Here an thereafter we se the same C i B at ifferent lines if they iffer only by a constant epening on n. Notice that µ τ is inepenent of τ an eqal to if is the fnamental soltion. The proof of the lemma given below is a moification of some argments in [H]. Proof. Direct comptation, sing a nitary frame, gives τ = ij i j + R + 4R ij i j R, 4 + nk + R an A τ log [4 + nk + ] = + k A + R R log A nk k log Combining the above two eqations together we have that τ Φ where Φ = ϕ e k τ log A k + nk e k τ.

5 with ϕ = A NOTE ON PERELAN S LYH TYPE INEQUALITY 5 τ +[4+nk +]τ, which satisfies τ ϕ + [4 + nk + ] ϕ <. By the maximm principle we have that ϕ A e kτ log + k + nk e k τ. From this one can erive. easily. To prove the secon part, we claim that for, a positive soltion to the conjgate heat eqation, there exists a C epening on, gτ sch that.4 y, τ C τ n z, τ µ τ z. This is a mean-vale type ineqality, which can be prove via, for example the oser iteration. Here we follow [H]. We may assme that sp y, τ τ n y, τ is finite. Otherwise we may replacing τ by τ ɛ = τ ɛ an let ɛ after establishing the claim for τ ɛ. Now let x, τ [, ] be sch a space-time point that max τ n y, τ = τ n y, τ. Then we have that n n sp y, t τ [ τ,τ ] τ y, τ = n y, t. Noticing this pper bon, we apply. to on [ τ, τ ], an concle that τ n y, τ y, τ + C τ log + C τ. y, τ n y Let g = log,τ y,τ + C τ. The above can be written as which implies that g + C τ τ sp gy, τ gy, τ +. τ B τ y,q +C τ Rewriting the above in terms of we have that y, τ n y, τ e + n log +C = C 3 y, τ

6 6 LEI NI for all y B τ y, τ +C τ. Here we have also se τ. Noticing that V ol τ B τ y, τ + C τ C 4 τ n for some C 4 epening on the geometry of, gτ. have that C 5 y, τ µ τ y y, τ τ n Therefore we for some C 5 epening on C 3 an C 4. By the way we choose y, τ we have that τ n n y, τ τ y, τ C 5 y, τ µ τ y = C 5 y, τ µ τ y. This proves the claim.4. Now the estimate.3 follows from., applie to on [ τ, τ], an the jst prove.4, which ensres the neee pper bon for applying the estimate.. q.e.. If y, τ = e f 4πτ n is the fnamental soltion Hy, τ; x, expresse in terms of τ to the conjgate heat eqation we have that µ τ =. Therefore, by.3, we have that.5 τ f h µ τ + C τ log B + f + n log4π + C τ h µ τ. On the other han, integrating by parts we can rewrite W h g,, τ = τ f h µ τ τ f, h µ τ +τ Rh µ τ + f nh µ τ = I + II + III + IV. The I term can be estimate by.5, whose right han sie contains only one ba term fh µ τ in the sense that it col possibly blow p. The secon term II = τ, h µ = τ h µ τ is clearly bone as τ. In fact II as τ. The same conclsion obviosly hols for III. Smmarizing above, we rece the qestion of boning from above the qantity W h, g, τ to boning one single term V = fh µ τ

7 A NOTE ON PERELAN S LYH TYPE INEQUALITY 7 from above as τ. We shall show later that lim τ V. To o this we nee to se the rece istance, introce by Perelman in [P] for the Ricci flow geometry. Let x be a fixe point in. Let ly, τ be the rece istance in [P], with respect to x, more precisely τ =. We collect the relevant properties of ly, τ in the following lemma Cf. [Ye, CLN]. Lemma.4. Let Ly, τ = ly, τ. i Assme that there exists a constant k sch that R ij gτ k g ij τ, Ly, τ is a local Lipschitz fnction on [, T ]; ii Assme that there exist constant k an k so that k g ij τ R ij gτ k g ij τ. Then.6 Ly, τ e k τ x, y + 4k n 3 τ an.7 x, y e k τ Ly, τ + 4k n 3 τ ; iii.8 τ + R exp Ly,τ. 4πτ n Proof. The first two claims follow from the efinition by straight forwar checking. For iii, it was prove in Section 7 of [P]. By now there are varios sorces where a etaile proof can be fon. See for example [Ye] an [CLN]. q.e.. As a conseqence of.6 an.7 exp Ly,τ lim τ 4πτ n = δ x y, which together with.8 implies that H, the fnamental soltion to the conjgate heat eqation, is bone from below as exp Ly,τ Hx, y, τ, 4πτ n by the heat kernel comparison principle cf. Proposition of [CLY], noticing the ality between the fnamental soltion of the heat eqation an the fnamental soltion of the conjgate heat eqation. Hence.9 fy, τ Ly, τ.

8 8 LEI NI This was prove in [P] making se of the ineqality v H. Since we are in the mile of proving v H, we provie the above alternative of obtaining Synthesis Now we assemble the reslts in the previos section to prove.4. As the first step we show that W h g, H, τ is bone thanks to the monotonicity.3, it is sfficient to bon it from above as τ, where Hx, y, τ is the fnamental soltion to the conjgate heat eqation with Hx, y, = δ x y. By the rection one in the previos section we only nee to show that V = fhh µ τ is bone from above as τ. By.9 we have that Ly, τ lim sp fhh µ τ lim sp Hx, y, τhy, τ µ τ y τ τ lim sp τ + lim τ x, y Hx, y, τhy, τ µ τ y e k τ x, y + k n 3 τ Hx, y, τhy, τ µ τ y. Here we have se.6 in the last ineqality. By Theorem., some elementary comptations give that x, y lim Hx, y, τhy, τ µ τ y = n hx,. τ Since ek τ x, y + k n 3 τ is a bone continos fnction even at τ =, we have that e k τ lim τ x, y + k n 3 τ Hx, y, τhy, τ µ τ y =. This completes or proof of the finiteness of lim sp τ fhh µ τ. In fact we have prove that 3. lim sp f n τ Hh µ τ. By the jst prove finiteness of W h g, H, τ as τ, an the entropy monotonicity.3, we know that the limit lim τ W h g, H, τ exists. Let lim W hg, H, τ = lim v H h µ τ = α τ τ

9 A NOTE ON PERELAN S LYH TYPE INEQUALITY 9 for some finite α. Hence lim τ Wh g, H, τ W h g, H, τ =. By.3 an the mean-vale theorem we can fin τ k sch that lim τ k R ij + i j f g ij Hh µ τk =. τ k τ k By the Cachy-Schwartz ineqality an the Höler ineqality we have that lim τ k R + f n Hh µ τk =. τ k τ k This implies that lim W hg, H, τ = lim τ k f f + f n Hh µ τk. τ τk Again integration by parts shows that τ k f f Hh µ τk = τ k H, h µ τk = τ k H h µ τk. Hence by 3. lim W hg, H, τ = lim τ τk f n Hh µ τ k. This proves α, namely.4. The claim that α = lim τ W h g, H, τ = can now be prove by the blow-p argment as in Section 4 of [P]. Assme that α <. One can easily check that this wol imply that lim τ µg, τ <. Here µg, τ is the invariant efine in Section 4 of [P]. In fact, noticing that hy, τ > for all τ τ where the τ is the one we fixe in the introction. Therefore by mltiple hx, more precisely hx, at τ = to the original hy, τ, we may assme that hx, = Hx, y, τhy, τ µ τ =. Let ũy, τ = Hx, y, τhy, τ an f = log ũ n log4π. Now irect comptation yiels that h W h g, H, τ = Wg, ũ, τ + τ h log h H µ τ. h Noticing that the secon integration goes to as τ, we can ece that Wg, ũ, τ < for sfficient small τ if α <. This, together with the fact ũ µ =, implies that µg, τ < for sfficiently small τ. Now Perelman s blow-p argment in the Section 4 of [P] gives a contraiction with the sharp logarithmic Sobolev ineqality on the Ecliean space [G]. One can conslt, for example [N, STW], for more etails of this part.

10 LEI NI Remark 3.. The metho of proof here follows a similar iea se in [N], where the asymptotic limit of the entropy as τ was compte. Note that we have to se properties of the rece istance, introce in Section 7 of [P], in or proof, while the similar, bt slightly easier, claim that lim τ Wg, H, τ = appears mch earlier in Section 4 of [P]. Remark 3.. R. Hamilton aske whether or not the LYH-type estimate v still hols for more general positive soltion to the conjgate heat eqation, other than the fnamental soltion. The proof presente here can be aapte to show that it still hols for finite sm of fnamental soltions. Namely let y, τ = k i= Hy, t; x i, T. Then the estimate.4, hence v, still hols for sch. The proof can be easily moifie to give the asymptotic behavior of the entropy efine in [N] for the fnamental soltion to the linear heat eqation, with respect to a fixe Riemannian metric. Inee if we restrict to the class of complete Riemannian manifols with nonnegative Ricci crvatre we have the following estimates. Proposition 3.3. For any δ >, there exists Cδ sch that H y, τ 3. x, y, τ Hx, Cδ + x, y H τ 4 δτ an 3.3 Hx, y, τ + H y, τ x, y, τ Hx, H τ Cδ + 4 x, y. 4 δτ The previos argment for the Ricci flow case can be transplante to show that τ f f + f n where Hy, τ; x, = e f is the fnamental soltion to the heat 4πτ n/ operator τ. This gives a rigoros argment for the ineqality.5 Theorem. of [N], for both the compact manifols an complete manifols with non-negative Ricci or Ricci crvatre bone from below. For the fll etaile accont please see [CLN]. 4. Improving Li-Ya-Hamilton estimates via monotonicity formlae The proof of.4 inicates a close relation between the monotonicity formlae an the ifferential ineqalities of Li-Ya type. The hinge is simply Green s secon ientity. This was iscsse very generally in [EKNT]. oreover if we chose h in the introction to be the fnamental soltion to the time epenent heat eqation t centere at x, t we can have a better pper bon on v H x, t in

11 A NOTE ON PERELAN S LYH TYPE INEQUALITY terms of the a weighte integral which is non-positive. In fact, this follows from the representation formla for the soltions to the nonhomogenos conjgate heat eqation. ore precisely, since hy, t; x, t is the fnamental soltion to the heat eqation to make it very clear, v H is efine with respect to H = Hy, t; x, T, the fnamental soltion to the conjgate heat eqation centere at x, T with T > t, we have that lim hy, t; x, t v H y, t µ t y = v H x, t. t t On the other han from. we have that by Green s secon ientity t hv H µ t = τ Therefore hv H µ t v H x, t = lim t T T t R ij + f ij τ Hh µ t. τ Using the fact that lim t T v H = we have that T v H x, t = T t R ij + f ij t T t R ij + f ij τ g ij Hh µ t t. Hh µ t t, which sharpens the estimate v H by proviing a non-positive pper bon. Noticing also the ality hy, t; x, t = Hx, t ; y, t for any t > t cf. [F] we can express everything in terms of the fnamental soltion to the backwar conjgate heat eqation. Below we show a few new monotonicity formlae, which expan the list of examples shown in [EKNT], an more importantly improve the earlier establishe Li-Ya-Hamilton estimates in a similar way as the above. For the simplicity let s jst consier the Kähler-Ricci flow case even thogh often the iscssions are also vali for the Riemannian Ricci flow case, after replacing the assmption on the nonnegativity of the bisectional crvatre by the nonnegativity of the crvatre operator whenever necessary. We first let, g α βx, t m = im C be a soltion to the Kähler- Ricci flow: t g α β = R α β. Let Υ α βx, t be a Hermitian symmetric tensor efine on [, T ], which is eforme by the complex Lichnerowicz-Laplacian heat eqation or L-heat eqation in short: t Υ γ δ = R βᾱγ δυ α β Rγ p k p δ + R p δυ γ p.

12 LEI NI Let ivυ α = g γ δ γ Υ α δ an ivυ β = g γ δ δυ γ β. Consier the qantity [ α Z = g βg γ δ β γ + γ β Υα δ + R α δυ γ β + ] K γ Υ α δv β + βυ α δv γ + Υα δv βv γ + t = [gα β βivυ α + g γ δ γ ivυ δ] +g α βg γ δ[r α δυ γ β + γ Υ α δv β + βυ α δv γ + Υ α δv βv γ ] + K t where K is the trace of Υ α β with respect to g α βx, t. In [NT] the following reslt, which is the Kähler analoge of an earlier reslt in [CH], was showe by the maximm principle. Theorem 4.. Let Υ α β be a Hermitian symmetric tensor satisfying the L-heat eqation on [, T ]. Sppose Υ α βx, an satisfies some growth assmptions in the case is noncompact. Then Z on, T ] for any smooth vector fiel V of type,. The se of the maximm principle in the proof can be replace by the integration argment as in the proof of.4. For any T t >, in orer to prove that Z at t it sffices to show that when t = t, t Zh µ t for any compact-spporte nonnegative fnction h. Now we solve the conjgate heat eqation τ + R hy, τ = with τ = t t an hy, τ = = hy, the given compact-spporte fnction at t. By the pertrbation argment we may as well as assme that Υ >. Let Z m y, t = inf V Zy, t. It was shown in [NT] that t Z m = Y + Y Z m t where an Y = Υ pq Rp q + R p qα βrᾱβ + α R p q Vᾱ + ᾱr p q V α + R p qα βvᾱv β + R p q t Y = Υ γᾱ [ p V γ R p γ t g p γ ] [ p V α R α p t g pα ]. +Υ γᾱ p V γ p V α Notice that in the above expressions, at every point y, t the vector V y, t is the vector minimizing Z m. This implies the monotonicity t Z m h µ t = t Y + Y h µ t. t

13 A NOTE ON PERELAN S LYH TYPE INEQUALITY 3 Since lim t t Z m =, which is certainly the case if Υ is smooth at t = an can be assme so in general by shifting t with a ɛ >, we have that t Z m h µ t=t. This proof via the integration by parts implies the following monotonicity formla. Proposition 4.. Let, gt, Υ an Z be as in Theorem 4.. For any space-time point x, t with < t T, let ly, τ be the rece istance fnction with respect to x, t. Then 4. t In particlar, exp l t Z m πτ m µ t t Y + Y 4. t Zx, t t t Y + Y exp l πτ m µ t. exp l πτ m µ t t. Notice that 4. sharpens the original Li-Ya-Hamilton estimate of [NT], by encoing the rigiity sch as Hamilton-Cao s characterization on the singlarity moels, erive ot of the eqality case in the Li- Ya-Hamilton estimate Z, into the integral of the right han sie. The reslt hols for the Riemannian case if one ses comptation from [CH]. In [N3], the athor iscovere a new matrix Li-Ya-Hamilton ineqality for the Kähler-Ricci flow. We also showe a family of eqations which connects this matrix ineqality to Perelman s entropy formla. ore precisely we showe that for any positive soltion to the forwar conjgate heat eqation t R =, we have that 4.3 Υ α β := α β log + R α β + t g α β ner the assmption that, gt has bone nonnegative bisectional crvatre. Using the above argment we can also obtain a new monotonicity relate to 4.3. Inee, tracing. of [N3] gives that t Q = RQ R α βυ βᾱ t Q + Υ α β + α β log + Y 3 where Q = g α βυ α β an Y 3 = R + R α β + α R ᾱ log + α log ᾱr + R α β ᾱ log β log + t R. Hence we have the following monotonicity formla, noticing that Y 4 := RQ R α βυ βᾱ.

14 4 LEI NI Proposition 4.3. Let, gt an x, t be as in Proposition 4.. Then exp l 4.4 t Q t πτ m µ t t Υ α β + α β log exp l + Y 3 + Y 4 πτ m µ t. In particlar, 4.5 t Qx, t t t Υ α β + α β log exp l + Y 3 + Y 4 πτ m. Again the avantage of the above monotonicity formla is that it encoes the conseqence on eqality case which is that, gt is an graient expaning soliton into the the right han sie integral. Withot Ricci flow, we can apply the similar argment to prove Li- Ya s ineqality an obtain a monotonicity formla. ore precisely, let, g n = im R be a complete Riemannian manifol with nonnegative Ricci crvatre. Let x, t be a positive soltion to the heat eqation on [, T ]. Li an Ya prove that log + n t. Another way of proving the above Li-Ya s ineqality is throgh the above integration by parts argment an the ifferential eqation t Q = Υ ij t Q + R ij i j where Υ ij = i j + t g ij i j an Q = g ij Υ ij = log + n t. This together with Cheeger-Ya s theorem [CY] on lower bon of the heat kernel, gives the following monotonicity formla, which also give characterization on the manifol if the eqality hols somewhere for some positive. Proposition 4.4. Let, g be a complete Riemannian manifol with non-negative Ricci crvatre. Let x, t be a space-time point with t >. Let τ = t t. Then 4.6 t Qy, t tĥx, y, τ µx t i j log + t g ij + R ij i log j log Ĥ µ,

15 A NOTE ON PERELAN S LYH TYPE INEQUALITY 5 where Ĥx, y, τ = 4πτ n exp x,y with x, y being the istance fnction between x an y. In particlar, we have that log + n 4.7 t x, t t t t i j log + t g ij + R ij i log j log Ĥ. 4πt n It is clear that 4.7 improves the estimate of Li-Ya slightly by proviing the lower estimate, from which one can see easily that the eqality for Li-Ya s estimate holing somewhere implies that = R n this was first observe in [N], with the help of an entropy formla. The expression in the right han sie of 4.6 also appears in the linear entropy formla of [N]. One can write own similar improving reslts for the Li-Ya type estimate prove in [N], which is a linear analoge of Perelman s estimate v H, an the one in [N], which is a linear version of Theorem 4 above. For example, when is a complete Riemannian manifol with the nonnegative Ricci crvatre, if = Hx, y, t = e f, the fnamental soltion to the heat eqation centere at x at t =, letting W = t f f + f n, we have that W. If Ĥ is the pseo backwar heat kernel efine as in Proposition 4.4 we have that W t Ĥx, y, τ µy = t i j f t g ij + R ij i f j f Ĥx, y, τ µy an t W x, t t i j f t g ij + R ij i f j f Ĥ. If we assme frther that is a complete Kähler manifol with nonnegative bisectional crvatre an y, t is a strictly plrisbharmonic soltion to the heat eqation with w = t, then where t Z w my, t = t Z w mĥx, y, t µy = t Y 5 Ĥx, y, t µy, inf w t + α wvᾱ + ᾱwv α + α βvᾱv β + w V T, t

16 6 LEI NI an Y 5 = γᾱ [ p V γ ] [ t g p γ p V α ] t g pα + γᾱ p V γ p V α + R α βs t stv β Vᾱ with V being the minimizing vector in the efinition of Zm. w In particlar, t y, t x log t, t t Y 5 Ĥx, y, t µy t. This sharpens the logarithmic-convexity of y, t prove in [N]. Finally we shol remark that in all the iscssions above one can replace the pseo backwar heat kernel Ĥy, t; x, t = exp x,y 4t t or exp ly,τ 4πτ n 4πt t n, centere at x, t in the case of Ricci flow, which we wrote before as Ĥy, x, τ by absing the notation, by the fnamental soltion to the backwar heat eqation even by constant in the case of compact manifols. Also it still remains interesting on how to make effective ses of these improve estimates, besies the rigiity reslts ot of the ineqality being eqality somewhere. There is also a small point that shol not be glosse over. When the manifol is complete noncompact, one has to jstify the valiity of the Green s secon ientity for example in Proposition 4.4 we nee to jstify that Ĥ Q Q Ĥ µ =. This can be one when t is sfficiently small together with integral estimates on the Li-Ya-Hamilton qantity cf. [CLN]. The local monotonicity formla that shall be iscsse in the next section provies another way to avoi possible technical complications case by the non-compactness. 5. Local monotonicity formlae In [EKNT], a very general scheme on localizing the monotonicity formlae is evelope. It is for any family of metrics evolve by the eqation t g ij = κ ij. The localization is throgh the so-calle heat ball. ore precisely for a smooth positive space-time fnction v, which often is the fnamental soltion to the backwar conjgate heat eqation or the pseo backwar heat kernel Ĥx, y, τ = e r x,y 4πτ n or e ly,τ 4πτ n the case of Ricci flow, with τ = t t, one efines the heat ball by E r = {y, t v r n ; t < t }. For all interesting cases we can check that E r is compact for small r cf. [EKNT]. Let ψ r = log v + n log r. in

17 A NOTE ON PERELAN S LYH TYPE INEQUALITY 7 For any Li-Ya-Hamilton qantity Q we efine the local qantity: P r := ψr + ψ r tr g κ Q µ t t. E r The finiteness of the integral can be verifie via the localization of Lemma., a local graient estimate. The general form of the theorem, which is prove in Theorem of [EKNT], reas as the following. Theorem 5.. Let Ir = P r r. Then n r [ n Q 5. Ir Ir = r r n+ E r t + tr gκ v v ] +ψ r t Q µ t t r. It gives the monotonicity of Ir in the cases that Q, which is ensre by the Li-Ya-Hamilton estimates in the case we shall consier, an both t + tr gκ v an t Q are nonnegative. The nonnegativity of t + tr gκ v comes for free if we chose v to be the pseo backwar heat kernel. The nonnegativity of t Q follows from the comptation, which we may call as in [N3] the pre-li- Ya-Hamilton eqation, ring the proof of the corresponing Li-Ya- Hamilton estimate. Below we illstrate examples corresponing to the monotonicity formlae erive in the previos section. These new ones expan the list of examples given in [EKNT]. For the case of Ricci/Kähler-Ricci flow, for a fixe x, t, let v = e ly,τ 4πτ n, the pseo backwar heat kernel, where l is the rece istance centere at x, t. Example 5.. Let Z m, Y an Y be as in Proposition 4.. Q = t Z m. Then r Ir n [ t r n+ ψ r Y + Y ] µ t t E r an r n [ Qx, t I r + t r n+ ψ r Y + Y ] µ t t r. E r Example 5.3. Let, Q = t Q, Υ α β, Y 3 an Y 4 be as in Proposition 4.3. Then r Ir an n r n+ Qx, t I r+ E r t ψ r r Let Υ α β + α β log + Y 3 + Y 4 n r n+ t ψ r E r Υ α β + α β log + Y 3 + Y 4.

18 8 LEI NI For the fixe metric case, we may choose either v = Hx, y, τ, the backwar heat kernel or v = Ĥx, y, τ = e backwar heat kernel. x,y 4πτ n, the pseo Example 5.4. Let an Q be as in Proposition 4.4. Let Q = t Q an f = log. Then n Ir r r n+ t ψ r i j f + E r t g ij + R ij i f j f µ t an Qx, t I r+ r Example 5.5. Let = n r n+ t ψ r i j f + E r t g ij e f 4πt n + R ij i f j f be the fnamental soltion to the reglar heat eqation. Let W = t f f +f n an Q = W. Then n Ir r r n+ tψ r i j f E r t g ij + R ij i f j f µ t an Qx, t I r+ r n r n+ tψ r i j f E r t g ij + R ij i f j f Note that this provies another localization of entropy other than the one in [N3] see also [CLN]. Example 5.6. Let be a complete Kähler manifol with nonnegative bisectional crvatre. Let, Zm w an Y 5 be as in the last case consiere in Section 4. Let Q = t Zm. w Then r Ir n r n+ t Y 5 ψ r µ t E r an x, t log t r n x, t I r + r n+ t Y 5 ψ r µ t r. E r Acknowlegement. We wol like to thank Ben Chow an Peng L for continosly pressing s on a nerstanable proof of.4. We starte to seriosly work on it after the visit to Klas Ecker in Agst an a stimlating iscssion with him. We wol like to thank him, Dan Knopf an Peter Topping for iscssions on a relate isse. We are also gratefl to Ben Chow for correcting an error in eriving Lemma. in the earlier version...

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