A NEW ENTROPY FORMULA AND GRADIENT ESTIMATES FOR THE LINEAR HEAT EQUATION ON STATIC MANIFOLD

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1 International Jornal of Analysis an Applications ISSN Volme 6, Nmber 1 014, A NEW ENTROPY FORULA AND GRADIENT ESTIATES FOR THE LINEAR HEAT EQUATION ON STATIC ANIFOLD ABIBOLA ABOLARINWA Abstract. In this paper we prove a new monotonicity formla for the heat eqation via a generalize family of entropy fnctionals. This family of entropy formlas generalizes both Perelman s entropy for evolving metric an Ni s entropy on static manifol. We show that this entropy satisfies a pointwise ifferential ineqality for heat kernel. The conseqences of which are varios graient an Harnack estimates for all positive soltions to the heat eqation on compact manifol. 1. Introction an Preliminaries We sty the heat eqation efine on a compact Riemannian manifol with static metric g 1.1 g t x, t = 0, where g is the sal Laplace-Beltrami operator acting on fnctions in space with respect to metric g. Throghot, will be taken to be a close manifol i.e., compact withot bonary except when otherwise inicate. ost of or calclations are one in local coorinates, where {x i } is fixe in a neighborhoo of every point x. The Riemannian metric gx at any point x is a bilinear symmetric positive efinite matrix enote in local coorinates by g ij = s = g ij x i x j The Laplace-Beltrami operator acting on a smooth fnction f on is efine as the proct of ivergence an graient of f written as g f := iv gra f = 1 g g ij g x i x j f, where g = etg ij an the inverse metric g ij = g ij 1. By the above we note that gra f i = f i = g ij 1 f an ivf = xj g x i g F i. Also we have the metric norm f g = g ij i f j f 010 athematics Sbject Classification. 53C1, 58D17, 58J35. Key wors an phrases. Riemannian manifol, Heat eqation, Entropy, onotonicity formlas. c 014 Athors retain the copyrights of their papers, an all open access articles are istribte ner the terms of the Creative Commons Attribtion License. 1

2 ABIBOLA ABOLARINWA an application of Cachy-Schwarz ineqality on the expression yiels the following ineqality f = g ij i j f = trhessf Hessf 1 n f. The Riemann strctre allows s to efine Riemannian volme measre V x on V x = g ij x x i. By the ivergence theorem we have the following integration by parts formlas for fnctions f, h C f g h V = f, h g V = g f hv. For any smooth fnction f on, we have the Bochner ientity efine as f = f + f, f + Rc f, f, where Rc is the Ricci crvatre of whose tensor components will be written in local coorinates as R ij. We switch between coorinates to allow calclations to be explicit an we write in local coorinates f = f i, f = i j f = f ij an x i = i. Also we write time erivative t f = tf = f t. We aopt smmation convention with repeate inices smme p. Any fnction 0 < C [0, T ] which satisfies 1.1 is calle a positive soltion. If tens to a irac-elta δ-fnction as t goes to zero, will be calle the heat kernel, that is the niqe minimal positive soltion on. We are intereste in the behaviors of all positive soltions, in particlar, the heat kernel. We erive graient estimates an ifferential Harnack ineqalities via the monotone property of a new family of entropy fnctionals. It is well known that entropy monotonicity formlas are closely relate to the graient estimate for the heat eqation. The importance of graient estimates as well as those of Harnack ineqalities can not be overemphasise in the fiels of Differential geometry an Analysis among their nmeros applications. Differential Harnack ineqalities are se to sty the behaviors of soltions to the heat eqation in space-time. Li an Ya s paper [15] can be sai to mark the beginning of rigoros applications of these concepts. They erive graient estimates for positive soltions to the heat operator efine on a complete manifol with static metrics, from which they obtaine Harnack ineqalities. These ineqalities were in trn se to establish varios lower an pper bons on the heat kernel. Precisely, Li an Ya s reslts for static metrics are the following; Theorem A Li-Ya [15]. Let, g be an n-imensional complete Riemannian manifol. Sppose there exist some nonnegative constant k sch that the Ricci crvatre R ij g k. Let C,1 [0, T ] be any smooth positive soltion to the heat eqation 1.1 in the geoesic ball B ρ [0, T ]. Then, the following estimate hols { 1. sp x B ρ α } t nα + Cα α t ρ α 1 + kρ + nα k α 1.

3 LINEAR HEAT EQUATION ON STATIC ANIFOLD 3 for all x, t B ρ,t, t > 0 an some constants C epening only on n an α > 1. oreover, the following estimate { 1.3 sp x B ρ α } t nα + nα k t α 1 hols for complete noncompact manifol by letting ρ. The above reslts have been improve by Davies [7, Section 5.3] as follows { 1.4 sp x B ρ α } t nα + nα k t 4α 1. As α 1, the secon terms in both 1.3 an 1.4 blow p an we obtain a sharp estimate t n t. Note that α can be chosen as a constant fnction of time only sch in a way that it goes to 1 as t 0, see for instance Hamilton [11], Hang, Hang an Li [1] an Li an X [14]. Li an Ya erive their graient estimates sing the maximm principle, bt by now it is known how to se monotonicity formlas erive from classical entropies of Shannon from statistical thermoynamics an Fisher s information from information theory. Let > 0 be a positive soltion to 1.1 with the normalization conition V x = 1, then, the classical Shannon entropy is efine by 1.5 S 0 t = an the Fisher information efine by 1.6 F 0 t = x, t log x, tv x x, t V x. x, t A straightforwar comptation shows that t S 0t = log x, t x, tv x = F 0 t an t S 0t = t F 0t = Hess log + Rc log, log V x, where Rc is the Ricci crvatre of. We now efine normalise versions of S 0 an F 0 by St := S 0 t + n logt + n = log + n logt + n V x Ft := tf 0 t n = t log n V x.

4 4 ABIBOLA ABOLARINWA Here, the normalisation is one so that the entropies remain ientically zero for all time when is the heat kernel. It easily shown that S an F are ientically zero on = R n, the Ecliean space, for = Hx, y, t = t n exp x y. By the above calclation, Shannon entropy S 0 for a positive soltion to the heat eqation on static manifol is seen to be monotone ecreasing while its erivative is monotone nonecreasing on the conition that the Ricci crvatre of is nonnegative. Ths, the Shannon entropy is convex in this case. We can now efine another entropy W, t base on the above 1.7 W, t = F, t S, t = ts, t. t Obviosly, the entropy W, t reas W, t = t log n logt n V x. Let = t n e f be a positive soltion to the heat eqation, where f is a smooth fnction. Here we have f = log n logt, t n e f = 1, 1.8 Wf, t = an t f + f nt n e f V x t W = f 1 ts = t t t g e f + Rc f, f t. n This is exactly Ni s reslt in [17] which states that Wf, t is monotone nonincreasing on a close manifol with nonnegative Ricci crvatre. In the case the manifol is Ricci flat this is inee Perelman s entropy monotonicity formla [0] on a metric evolving by the Ricci flow. Notice that by application of integration by parts Ft can be written as 1.9 Ft = t log + n V x. This has a srprising connection to the Li-Ya graient estimate in Theorem A above. Clearly, the qantity ner the integral is eqivalent to the Harnack qantity of Li-Ya t log + n = + n. Li-Ya graient estimate [15] says F 0 when Rc 0, which implies This is in trn eqivalent to t + n t f n 0,

5 LINEAR HEAT EQUATION ON STATIC ANIFOLD 5 which can be viewe as a generalize Laplacian comparison theorem. Inee, the Laplacian comparison theorem on is a conseqence of 1.10 by applying ineqality to the heat kernel an letting t tens to zero. One can also see that lim St = 0 for the heat kernel an hence St is monotone increasing on nonnegative Ricci crvatre manifol. Therefore, we have Wf, t 0 for the heat kernel for some t > 0 if an only if is isometric to R n. Note that on R n we have f = x /. Lei Ni also showe that these reslts hol for all complete manifols with Rc 0. Let be a complete Riemannian manifol with nonnegative Ricci crvatre, then at t = 1/, W 0 hols on if an only if is isometric to R n, See also Weissler []. This is inee eqivalent to Gross logarithmic Sobolev ineqalities [9] on R n. Ths, there is a strong relation between the log-sobolev ineqality an the geometry of the manifol which was originally iscovere by Bakry, Concoret an Leox [3] see also [4]. That is, 1 e f 1.11 f + f n t 0 n implies 1.1 R n R n log x n log 1 nπe Rn with eqality on any Gassian with R n µ. To get 1.1 from 1.11 one ses the monotone property of W on R n an asymptotic behavior of the positive soltion to the heat eqation, noting that soltion on R n converges after rescaling at infinity to constant mltiples of the sal Gassian. The remarkable papers [17] an [18] have shown a esirable interpolation between entropy formla of Ni on static manifols an that of Perelman [0] on evolving manifols. The new W ɛ f, t iscsse in this paper see section is an example of sch a family of entropies connecting both Ni s an Perelman s entropies. We emonstrate this in [, Chapter 3] an have applie it on manifol evolving by the Ricci-harmonic map flow in [1]. We remark that estimates an bons on parabolic eqations behave in similar way whether the metric is static or moving. This can be jstifie by the fact that heat iffsion on a bone geometry with either static or evolving metric behaves like heat iffsion in Ecliean space, many a times, their estimates even coincie. In this paper however, we prove the monotonicity formlas for a family of entropy fnctionals W ɛ f, t an iscss some of its analytic an geometric conseqences. The plan of the rest of the paper is as follows: In Section we introce a new family of entropy fnctionals an prove its monotonicity for a positive soltion to the heat eqation. The monotonicity erive here is se in Section 3 to erive pointwise ifferential Harnack ineqalities an graient estimates for the heat eqation. As a conseqence we obtain Harnack estimates for the fnamental soltion which also hols for all positive soltions in Section 4. We give Li-Ya-Hamilton type graient estimates for bone soltions in the last section.. A new entropy monotonicity formla We emphasize that the volme is kept fixe throghot the time of evoltion for the heat eqation on a close n-imensional manifol, g. We also impose the conition of nonnegativity on the Ricci crvatre of the nerlying manifol.

6 6 ABIBOLA ABOLARINWA Let = x, t be a positive soltion to the heat eqation.1 = t x, t = 0. Let f : 0, T ] R be smoothly efine as = t n e f with normalization conition x, tv x = 1. We introce a generalize family of entropy by [ ɛ t. W ɛ f, t = f + f + n ] ln ɛ nɛ e f V x, t n where 0 < ɛ. We remark that if ɛ =, we recover the Perelman s entropy as in the special case consiere by Ni in [17]. From this entropy formla we later erive the corresponing ifferential ineqality an graient estimate for the fnamental soltion, which in fact, hols for all positive soltions to the heat eqation. The same entropy is se by the athor in his Ph thesis [] to examine the srprising relation that exists between the entropy formla for heat eqation an the conjgate heat eqation ner the Ricci flow. We have also se its monotonicity properties combine with some Sobolev-type ineqalities to erive sharp pper bon for conjgate heat kernel along Ricci-harmonic map heat flow in [1]. Lemma.1. Let = t n e f be a positive soltion to the heat eqation = 0 on a close Riemannian manifol. Then.3 t f = f ij f, f R ij f i f j an.4 t f = f ij f, f f, t f R ij f i f j. oreover, if w = f f, then.5 t w = f ij R ij f i f j w, f. Proof. Since = t n e f, f = log n logt an t f = f f n t. 1.6 t f = t gij i f j f = g ij i f j f t f = f, t f. By Bochner ientity f = f ij + f, f + R ij f i f j 3 = fij + f, t f + f + R ij f i f j = fij + f, f + f, t f + R ij f i f j. Aing the last eqality to.6 proves.3. t f = f f f = f = fij f, t f + f R ij f i f j = fij f, f t f R ij f i f j. t w = t f t f = fij R ij f i f j f, f + t f.

7 LINEAR HEAT EQUATION ON STATIC ANIFOLD 7 This ens the proof of the lemma. We are now set to establish the monotone property of the W ɛ f, t-entropy. By the monotonicity formla for this entropy fnctional, we will erive graient estimates an the corresponing ifferential Harnack ineqalities for the fnamental soltion to the heat eqation on a static manifol. Proposition.. Let be any close manifol an = t n e f be any positive soltion to the heat eqation = t = 0 on 0.T ]. Denoting.7 P ɛ = ɛ t f f + f + n ln ɛ nɛ, where 0 < ɛ. Then.8 t P ɛ ɛ t fij π π ɛt g ij +R ij f i f j P ɛ, f 1 ɛ f. Proof. Here we write P ɛ = ɛ t w + f + n 1 ln ɛ nɛ t. Since f = ln n lnt, taking = e f implies f = f n lnt. we notice also that f = f, f = f an f ij = f ij, then t f = f n t. Now by irect ifferentiation an application of Lemma.1, we have the following comptation t P ɛ = ɛ t t w + ɛ w + t f + n 1 t ln ɛ nɛ t = ɛ t fij R ij f i f j w, f + ɛ = ɛ t Notice that an f f f n t fij π n ɛ t R ijf i f j + ɛ f f ɛ t w, f f. ɛ t w, f = P ɛ f, f = P ɛ, f f f f = t f + f Then we have t P ɛ ɛ t fij + π n ɛ t π ɛt f + R ijf i f j P ɛ, f + ɛ f f = ɛ t fij π ɛt g ij + R ij f i f j P ɛ, f 1 ɛ f. Theorem.3. Let be a close Riemannian manifol. Assme that = t n e f is a positive soltion to the heat eqation t = 0, then, we have the following monotonicity formla for W ɛ f, t efine in..9 [ t W ɛf, t = ɛ t π fij π ɛt g ij +R ij f i f j + 1 ɛ f ] e f V x t n

8 8 ABIBOLA ABOLARINWA with f, t satisfying.10 an 0 < ɛ. e f V x = 1 t n Proof. Combining Proposition. with the fact that = 0 an f =, we have t P ɛ = t P ɛ + P ɛ t P ɛ, = ɛ t fij π π ɛt g ij + R ij f i f j P ɛ, f 1 ɛ f P ɛ,. Integrating over, we have [ ɛ t P ɛ V x = f f + f + n ] ln ɛ nɛ V x [ ɛ t = f + f + n ] ln ɛ nɛ V x + ɛ t f f V x = W ɛ f, t, in the sense that the secon integral in the RHS vanishes on a close manifol since f f =. Therefore t W ɛf, t = P ɛ V x t = t P ɛ + P ɛ t V x [ ] = t P ɛ + P ɛ t V x = t P ɛ V x, where we have se integration by parts an = 0. Using the evoltion t P ɛ from Proposition., we get the esire reslt. oreover, if the manifol has nonnegative Ricci crvatre, i.e, R ij 0, it becomes obvios from.9 that W ɛ /t 0. We remark that Kang an Zhang [13] have a reslt in this irection, it is state as follows; Let be a close Riemannian manifol with nonnegative Ricci crvatre. Let be the fnamental soltion to the heat eqation with f = ln n lnt, we have.11 tα f f + f α n 0 for any constant α 1. Inee, if α =, this is exactly the ifferential ineqality t f f + f n 0

9 LINEAR HEAT EQUATION ON STATIC ANIFOLD 9 prove in [17]. Diviing throgh by α t, with α 1 an t 0, we obtain f f α + f αt n t 0 as t, which is precisely the Li-Ya graient estimate. For α >, the graient estimate is an interpolation of Perelman s estimate an Li-Ya estimate. For 1 α, it is consiere in [13]. In Ecliean space R n, if is the fnamental soltion to the heat eqation then.11 becomes an eqality. 3. Graient Estimates for Heat Eqation on Static anifol The monotonicity formla in the last section may be viewe as a local version of the Perelman s W-entropy formla in [0]. In what follows, we want to show that the local entropy satisfies a pointwise ifferential ineqality for the heat kernel. We have the following fashione after [17, Theorem 1.] with the proof follows from the argment of [16, Proposition 3.6]. Theorem 3.1. Let be a close manifol with nonnegative Ricci crvatre an Hx, y, t = H = t n e f be the heat kernel, where H tens to a δ-fnction as t 0 an satisfies HV x = 1. Then for all t > 0, we have 3.1 P ɛ = ɛ t f f + f + n ln ɛ nɛ 0. Proof. Let h be any compactly spporte smooth fnction for all t 0 > 0. Sppose h, t is a positive soltion to the backwar heat eqation t + h = 0, This is Perelman s argment in [0, Corollary 9.3], then, it is clear that t HhV = 0 an we have by irect calclation that [ ] hp ɛ HV x = t hp ɛ H + h t P ɛ H V x t [ ] = t + hp ɛ H + h t P ɛ H V x = h t P ɛ HV x 0. The ineqality is e to Theorem.3 since R ij 0. We are left to showing that for everywhere positive fnction h, t, the limit of hp ɛhv x is nonpositive as t 0. We assme the claim apriori i.e, lim hp ɛhv = 0 an concle the reslt. For completeness, we evote the next effort to jstifying the claim 3. lim hp ɛ HV 0. Or argment follows from [16], for etail see [17, 19, 0], the calclation in [13] is also similar. If H tens to a irac δ-fnction, say at a point p, for t 0, then f satisfies fx, t p, x. This is in relation to l- length of Perelman. This yiels 3.3 lim fhhv lim sp p, x hhv = n hp, 0.

10 10 ABIBOLA ABOLARINWA eanwhile, by the strong aximm principle we have hx, 0 > 0 an lim hhv = hx, 0, hence by scaling argment, we assme that hx, 0 = 1. All these will soon become clearer. Rewriting P ɛ an sing integrating by parts methos we have ɛ t P ɛ hhv = f n t hhv ɛ t f, h HV π + fhhv + n [ ln ɛ ɛ ] Hh V. Thogh, the H appearing in the last eqation is actally the heat kernel on an evolving manifol in Ni s reslt [19] while h satisfies the forwar heat eqation, his argment still hols in or case, we only nee the asymptotic behavior of heat kernel on a fixe metric. We shol also note that since h, t 0 is compactly spporte an by strong maximm principle we have h, t 0, h, t 0 an h, t 0 bone on. This implies that there exists a bone soltion h, t 0. It trns ot that we nee to show that there exists a constant B 0 which may epen on the geometry of the nerlying manifol an inepenent of t as t 0, sch that P ɛhhv Bn. Now we claim that the first two terms on the right han sie of the last eqation vanish as t 0, we can see this in the following argment. By integration by parts an the fact that H = H f, we have t f, h HV = t H, h V = t H hv is bone since h is bone as state earlier. Ths, the secon term is bone an goes to zero as t 0. We nee a bon of Li-Ya type to obtain a bon for the first term f. See Lemma 3. below for the statement of the reslt [5] see also [6, Corollary 16.3] an Soplet an Zhang [1]. By this we have for the heat kernel in the present case that 3.4 t f Bn, δ + x, y 4 δt which is also clearly seen to be bone from above as t 0 by the jstification of asymptotic behavior of the heat kernel. We have now rece the analysis to 3.5 lim P ɛ hhv lim sp f + nq hhv, where q = ln ɛ ɛ. For simplicity, we can choose ɛ sch that ɛ as t 0 so that the whole problem is rece to fining 3.6 lim f n hhv. Using the asymptotic behavior of the heat kernel, i.e, f as t 0. Recall Cf. [8, 19] as t 0 Hx, y, t t n x, y exp j x, y, tt j := w k x, y, t where x, y is the istance fnction an w k x, y, t satisfies niformly for all x, y w k x, y, t = O t k+1 n δ x, y exp j=o,

11 LINEAR HEAT EQUATION ON STATIC ANIFOLD 11 an δ is jst a nmber epening only on the geometry of, g. The fnction can be chosen sch that 0 x, y, 0 = 1. Thogh, the above asymptotic reslt oes not reqire any crvatre assmption, a reslt e to Cheeger an Ya [5] states that on manifol with nonnegative Ricci crvatre which is or case, the heat kernel satisfies Hx, y, t t n x, y exp which implies Therefore fx, t x, y. lim f n hhv lim x, y n hy, thx, y, tv y = lim x, y n e x,y/ Hy, tv y. t n It is easy to see that for any δ > 0, the integration of the above integran in the omain x, y δ converges to zero exponentially fast. Therefore 3.7 lim f n hhv lim x,y δ x, y n e x,y t n hy, tv y. Whenever δ is chosen sfficiently small, x, y is asymptotically sfficiently close to the Ecliean istance. By stanar approximation, we have 3.8 lim f n x y hhv lim x,y δ n xy e t n h p yv y, where h p is the pllback of h, 0 to the Ecliean space from the region x, y δ. Splitting the last integran as in [13] we are left with lim f n x y hhv h p x lim R n xy e V y n t n y = h p lim R n e y t n V y n h p. The last eqality is e to convoltion properties of the heat kernel. Lastly we show that the RHS evalates to 0. Recall, sing stanar Gass integral, that y e α y n1 y = n y e αy y e αy y R n = n π α 3 π n1 n π n, = α α α so that we have y R n e y t n V y = = n, 1 t n n n1 y e 1 y y e 1 y y by taking α = 1/ in the above. We can then concle the claim.

12 1 ABIBOLA ABOLARINWA Lemma 3.. On a complete Riemannian anifol, g with nonnegative Ricci crvatre, the following estimate hols for the graient of the heat kernel Hx, y, t an all δ > 0, 3.9 for all x, y in an t > 0. H H H t Bn, δ + x, y 4 δt 4. Harnack Estimates for the Heat Kernel The following ifferential Harnack qantity for linear heat eqation on static manifol follows immeiately as an application of the reslts in the last sbsection. Corollary 4.1. Let be a close manifol with crvatre bone from below by Rc 0. Then we have ɛ t 4.1 f f + f + n ln ɛ ɛ 0, π where f = lnt n H an H is the positive minimal soltion to the heat eqation x t Hx, y, t = 0. Remark 4.. Note that the qantity f f can be expresse as in terms of, which is similar to Li-Ya graient estimate [15] on a manifol t with nonnegative Ricci crvatre, t + n t 0. This is eqivalent to the ifferential Harnack ineqality t f n, where f = lnt n, which can be regare as a generalize Laplacian comparison theorem in space for Heat kernel on. However, we have from 4.1 that f n [ ɛ π n [ ɛ π Define ln ɛ ] ɛ t ln ɛ ] ɛ n 8π = n f f [ ɛ ln ɛ ]. 4. Qx, t = ɛ tfx, t π [ 4.3 t Qx, t = ɛ π fx, t + ɛ π t t f nɛ ɛ ] ln π ɛ. Still as ɛ = π we recover Ni s generalize Laplacian. From Corollary 4.1, we have the ifferential Harnack ineqality as follows ɛ t f f + f + n ln ɛ ɛ 0. π ltiplying throgh by π ɛ t, we have f + 1 f π ɛ t f nπ ɛ ln t ɛ ɛ 0 π f + 1 f π ɛ t f + n t nπ ɛ t ln ɛ 0.

13 LINEAR HEAT EQUATION ON STATIC ANIFOLD 13 Recall that t H = 0 implies f = t f + f + n t, then we have t f 1 f π ɛ t f nπ ɛ t ln ɛ t f + 1 f π ɛ t f nπ ɛ t ln ɛ = π ɛ f + n t ln ɛ. By the Yong s ineqality we have on the path γt, γt : [t 1, t ] is a minimizing geoesic connecting points x 1 an x sch that γt 1 = x 1 an γt = x. t fγt, t = tf + f, γ t t f + 1 f + 1 γ t = 1 γ t π ɛ f + n t ln ɛ since we have from 4.1 that f n ɛ ln ɛ, inserting this qantity in the above ineqality gives the following Harnack Estimates 4.4 t fγt, t 1 γ t n. After the sal integration of 4.4 an exponentiation we have the following Corollary 4.3. With the notation an assmption of Corollary 4.1, we have the following ifferential Harnack estimates x, t 4.5 x 1, t 1 t1 n [ 4 1 t ] exp γ t t. t Remark 4.4. If is a close manifol with nonnegative Ricci crvatre an = t n e f is the heat kernel on. Then W ɛ f, t 0 0 for some t 0 > 0, if an only if is isometric to Ecliean space R n. Recall that we have obtaine that t W ɛf, t 0 an W ɛ f, t 0 which in trn imply that we mst have W ɛ f, t 0 for 0 t t 0. For instance, in the case ɛ = π, we have Taking the trace of the above yiels f ij 1 t g ij = 0 an f ij 1 t g ij = t f n = 0. Becase fx, t fx, t = p,x for t small, we have lim f = p, x. Hence 4.6 implies that 4.7 p, x = n so that we can get for the area A p r of B p r an the volme V p r of the ball B p r, the following qotient A p r V p r = n r. t 1

14 14 ABIBOLA ABOLARINWA This shows that V p r is exactly the same as the volme fnction of Ecliean balls. This argment shows that the Li-Ya Harnack ineqality, which is eqivalent to t f n 0 for = t n e f becomes an eqality if an only if the manifol with Rc 0 is isometric to R n an is precisely the heat kernel. If t = 1 an = R n, the ineqality W ɛ f, t 0 0 for ɛ =, is eqivalent to R f + f nπ n e f V 0 n for all f with the conition π n e f V = 1. The above implies a sharp Gross logarithmic Sobolev ineqality on R n. For etails abot logarithmic-sobolev ineqalities see for instance [9, 10, ]. In the same vein or al entropy also yiels a version of logarithmic Sobolev ineqality. This will not be iscsse here. Remark 4.5. Note that f ij π ɛt g ij 0 = f n π ɛt which in trns = t n π ɛt. It trns ot that W ɛ f, t being finite with being the heat kernel, also has strong topological an geometric conseqences. For instance, in the case has nonnegative crvatre, it implies that has finite fnamental grop. In fact one can show that is of maximm volme growth if an only if the entropy W ɛ f, t is niformly bone for all t 0, where is the heat kernel. This analogy was originally iscovere in [0] for ancient soltion to the Ricci flow with bone nonnegative crvatre, where Perelman claims that ancient soltion to the Ricci flow with nonnegative crvatre operator is κ-noncollapse if an only if the entropy is niformly bone for any fnamental soltion to the conjgate heat eqation. Lastly, in this sbsection we make some comment to show how sharp the al entropy for the heat eqation. Recall 4.9 W ɛ f, t = [ ɛ t f + f + n ln ɛ with f = lnt n H an HV = 1 an 0 < ɛ. Rewrite W ɛ f, t as 4.10 W ɛ f, t = ɛ Hence, we have the following ] nɛ HV t f + f nhv + 1 ɛ fhv + n ln ɛ HV. Proposition 4.6. For 0 < ɛ, f = lnt n H with HV = 1, we have the following monotonicity formla on a manifol with nonnegative Ricci crvatre; 4.11 t W ɛf, t f ɛ π t ij 1 t g ij + R ij f i f j HV. Proof. The proof follows from a straight forwar comptation on W ɛ sing the iea of [17, Theorem 1.1]. 4.1 t W ɛf, t = ɛ t f + f n HV + 1 ɛ t fhv. t

15 LINEAR HEAT EQUATION ON STATIC ANIFOLD 15 We are only left to jstify the non-positivity of t fhv. Then we have by integration by parts t fhv = = = = t f H + f t H V t f H + f H + f t H V t + fhv f f n t HV, where we have se the facts that t H = 0 an t f = f f n t. Taking f = lnt n H, then the integran in the RHS of the last eqality becomes 4.13 f f n t = H H H H n t 0, which is precisely the Li-Ya Harnack ineqality since we are on nonnegative Ricci crvatre manifol. Hence or claim. 5. LYH graient estimates for positive soltions In the next we give sefl estimates fon by Hamilton [11]. He was inspire by the reslts of Li an Ya [15], hence the estimates are poplarly referre to as Li-Ya-Hamilton LYH estimates. We state an prove the reslt for bone soltions on a close manifol. As an application of this LYH-type estimates we can obtain a sharp pper bon on the heat kernel. Theorem 5.1. Let, g be a close Riemannian manifol with R ij kg ij, where k 0. Sppose is a positive soltion to the heat eqation with <. Then 5.1 t 1 + kt log Proof. Let f = log so that f = log = Define a heat type operator L := t f,.. an t f = f. The iea to this proof is to apply the heat-type operator L on the qantity t 1 + kt log an then se weak maximm principle. Recall from the calclation in Lemma.1 an the Bochner ientity that Hence t f = f i f ti & f = f ij + f j f jji + R ij f i f j. t f = f ij R ij f i f j + f, f

16 16 ABIBOLA ABOLARINWA an t t f = f + t t f = f t f ij tr ij f i f j + t f, f. Using the conition that R ij kg ij, we have 5. t t f = 1 + kt f + f, t f. On the other han L 1 + kt log = k log = k log ktl t + log 1 + kt f, log log. Compting t log = t log t log = f = f, log + f = f, log + f. Then L 1 + kt log = k log + f, log 1 + kt f, log = k log + f, 1 + kt log kt f. + f Combining the expressions in 5. an 5.3 we arrive at A 5.5 L t f 1 + kt log k log since k 0 an 0 log <. Note that at t = 0, A A log 0 an t f 1 + kt log 0. Hence, by the weak maximm principle we have A t f 1 + kt log 0 for all t 0. This completes the proof.

17 LINEAR HEAT EQUATION ON STATIC ANIFOLD 17 The above reslt can be extene to the case of complete noncompact manifol, althogh, a little more effort will be reqire. The iea here is to se ɛ-reglarization metho by spposing that the soltion ɛ, replacing by ɛ = + ɛ for a sfficiently small ɛ > 0 an letting ɛ go to zero after the analysis for ɛ is complete. An application of this reslt shows we can bon the maximm of a positive soltion by its integral see [11]. Frthermore, the estimate yiels sharp lower an pper bons for the fnamental soltion, See [6]. References [1] A. Abolarinwa, Differential Harnack an logarithmic Sobolev ineqalities along Ricci- Harmonic map flow, To appear [] A. Abolarinwa, Analysis of eigenvales an conjgate heat kernel ner the Ricci flow, PhD Thesis, University of Sssex, 014. [3] D. Bakry, D. Concoret an. Leox, Optimal heat kernel bons ner logarithmic Sobolev ineqalities, ESAI Probab. Statist., 1,1995, [4] D. Bakry an. Leox, A logarithmic Sobolev form of the Li-Ya parabolic ineqality, Revist. at. Iberoamericana, 006, [5] J. Cheeger an S-T. Ya, A lower bon for the heat kernel, Comm. Pre Appl. ath , [6] B. Chow, S. Ch, D. Glickenstein, C. Genther, J. Ienber. T. Ivey, D. Knopf, P. L, F. Lo an L. Ni, The Ricci Flow: Techniqes an Applications. Part II, Analytic Aspect, AS, Provience, RI, 008. [7] E. B. Davies, Heat Kernel an Spectral theory. Cambrige University Press [8] N. Garofalo an E. Lanconelli, Asymptotic behavior of fnamental soltions an potential theory of parabolic operators with variable coefficients, ath. Ann , [9] L. Gross Logarithmic Sobolev ineqalities, America J. ath , [10] L. Gross Logarithmic Sobolev ineqalities an contractivity properties of semigrops, Dirichlet Form, Lectre Notes in athematics Volme 1563, 1993, [11] R. Hamilton, A matrix Harnack estimate for the heat eqation, Commn. Anal. Geom., 1, 1993, [1] G. Hang, Z. Hang, H. Li, Graient estimates an ifferential Harnack ineqalities for a nonlinear parabolic eqation on Riemannian manifols, Ann. Glob. Anal. Geom., , [13] S. Kang, Qi S. Zhang, A graient estimate for all positive soltions of the conjgate heat eqation ner Ricci flow, J. Fnct. Anal., , [14] J. Li, X. X, Differential Harnack ineqalities on Riemannian manifols I: Linear heat eqation, Avances in ath., 6 011, [15] P. Li, S-T. Ya, On the parabolic kernel of the Schröinger operator, Acta ath , [16] R. üller, Differential Harnack Ineqalities an the Ricci Flow. Eropean athematics Society, 006. [17] L. Ni, The Entropy Formla for Linear Heat Eqation, Jornal of Geom. Analysis 14004, [18] L. Ni, Aena to The Entropy Formla for Linear Heat Eqation, Jornal of Geom. Analysis 14004, [19] L. Ni, A note on Perelman s Li-Ya-Hamilton ineqality, Comm. Anal. Geom 14006, [0] G. Perelman, The entropy formla for the Ricci flow an its geometric application, arxiv:math.dg/011159v1 00. [1] P. Soplet, Qi S. Zhang, Sharp graient estimate an Yas Lioville theorem for the heat eqation on noncompact manifols, Bll. Lonon ath. Soc , [] F. B. Weissler, Logarithmic Sobolev Ineqalities for the Heat-Diffsion Semigrop, Trans Am ath. Soc., , Department of athematics, University of Sssex, Brighton, BN1 9QH, UK