Davies type estimate and the heat kernel bound under the Ricci flow
aa r X i v : . [ m a t h . DG ] F e b DAVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUNDUNDER THE RICCI FLOW
MENG ZHU
Abstract.
We prove a Davies type double integral estimate for the heat kernel H ( y, t ; x, l ) under the Ricci flow. As a result, we give an affirmative answer to aquestion proposed in [8]. Moreover, we apply the Davies type estimate to provide anew proof of the Gaussian upper and lower bounds of H ( y, t ; x, l ) which were firstshown in [6]. Introduction
On a complete Riemannian manifold ( M n , g ij ), the heat kernel H ( x, y, t ), is the small-est positive fundamental solution to the heat equation ∂u∂t = ∆ u. (1.1)Heat kernel estimates is of great importance and interest due to its relation to manyother properties of the manifolds, such as Harnack estimate, Sobolev inequality, LogSoblev inequality, Faber-Krahn type inequality, and Nash type inequality (see e.g. [21],[9], [12], [4], [18], [5]). Since the work of Nash [19] and Aronson [1], many methods havebeen discovered for deriving Gaussian upper and lower bounds of H ( x, y, t ), see e.g.,[7], [18], [10], [13], [11], [17]. One of the methods was developed by Li-Wang in [17].They obtained a Gaussian upper bound for H ( x, y, t ) based on the parabolic mean valueinequality and the following double integral upper estimate of the heat kernel provedby E.B. Davies [11]: Theorem 1.1.
Let ( M, g ) be a complete Riemannian manifold. For any two boundedsubsets U and U of M , one has: Z U Z U H ( x, y, t ) dµ ( x ) dµ ( y ) ≤ Vol ( U )Vol ( U ) e − d U ,U t , (1.2) where d ( U , U ) is the distance between U and U . In this paper, we consider the heat kernel of the time-evolving heat equation underthe Ricci flow on a complete manifold M n , i.e., ∂u∂t = ∆ t u, (1.3) Research is partially supported by China Postdoctral Science Foundation Grant No. 2013M531105. where ∆ t is the Laplacian with respect to a complete solution g ij ( t ), t ∈ [0 , T ) and T < ∞ , of the following Ricci flow equation ∂g ij ( t ) ∂t = − R ij g (0) = g (1.4)on M .The existence and uniqueness of the heat kernel H ( y, t ; x, l ) to (1.3) were proved in[14] and [8]. When M is compact, C. Guenther [14] obtained a Gaussian lower boundof H ( y, t ; x, l ). For general complete manifolds, L. Ni [20] first obtained Gaussian es-timates of H ( y, t ; x, l ) assuming uniformly bounded curvature and nonnegative Riccicurvature along the Ricci flow. Q. Zhang [23] proved both Gaussian upper and lowerbounds of H ( y, t ; x, l ) for type I ancient κ -solutions of the Ricci flow with nonnegativecurvature operator. In [22], G. Xu removed the assumption on the nonnegativity of thecurvature operator, and obtained similar Gaussian bounds of the heat kernel for generaltype I ancient κ -solutions of the Ricci flow. Assuming that M has nonnegative Riccicurvature and is not Ricci flat for all t , Gaussian estimates of H ( y, t ; x, l ) were givenby Cao-Zhang [3]. When the Ricci curvature is only assumed to be uniformly bounded,Chau-Tam-Yu [6] also showed Gaussian lower and upper bounds for H ( y, t ; x, l ), butthe constants in the estimates depend on more geometric quantities than the ones inCao-Zhang’s estimates (see [8] for more detail). While the authors in [6] and [8] mainlyused Grigor’yan’s method in [13] to prove the Gaussian upper bound of the heat kernel,they raised the following question: Question:
Is there a ”Ricci flow” version of Davies’ estimate as the one in Theorem 1.1?Our main goal is to give an affirmative answer to the question above. More specifically,we prove
Theorem 1.2.
Let ( M n , g ij ( t )) be a complete solution to (1.4) on [0 , T ) and T < ∞ .Suppose that H ( y, t ; x, l ) is the heat kernel of (1.3) , and Rc ≥ − K on [0 , T ) for somenonnegative constant K . Then for any open sets, U and U , with compact closure in M , and ≤ l < t < T , we have Z U Z U H ( y, t ; x, l ) dµ t ( y ) dµ l ( x ) ≤ e C K T e − d t ( U ,U e K T ( t − l ) Vol l ( U ) Vol t ( U ) , where C is a constant depends only on n . Using Theorem 1.2, we are able to provide a new proof of the Gaussian upper boundand lower bounds of H ( y, t ; x, l ) which was first shown by Chau-Tam-Yu [6]. For theGaussian upper bound, we have Theorem 1.3.
Under the Ricci flow, assume that Rc ≥ − K on M × [0 , T ) for somenonnegative constant K and T < ∞ , and that Λ = R T sup M | Rc | ( t ) dt < ∞ . We have AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 3 the following upper bound: H ( y, t ; x, l ) ≤ C e C e C C K T min exp (cid:16) − d t ( x,y )8 e K T ( t − l ) (cid:17) Vol l ( B l ( y, q t − l )) , exp (cid:16) − d t ( x,y )8 e K T ( t − l ) (cid:17) Vol t ( B t ( x, q t − l )) , where constants C , C , , C and C depend only on n . Next, following a method of Li-Tam-Wang [16], we can use a gradient estimate of Q.Zhang in [24] (see also [2]) to show a Gaussian lower bound of H ( y, t ; x, l ). Theorem 1.4.
Let ( M n , g ij ( t )) , t ∈ [0 , T ) and T < ∞ , be a complete solution to theRicci flow (1.4) . Suppose that Rc ≥ − K on M × [0 , T ) and Λ = R T sup M | Rc | ( t ) dt < ∞ .Then we have the following lower bound: H ( y, t ; x, l ) ≥ C e − C e C C K T exp (cid:18) − d t ( x, y )( t − l ) (cid:19) Vol t ( B t ( x, q t − l )) , where C , C , C and C are positive constants only depending on n . This paper is organized as follows. In section 2, we review some known facts aboutthe fundamental solutions of heat-type equations with metric evolving under a group ofmore general equations than the Ricci flow, including the existence, uniqueness and themean value inequality. In section 3, we prove Theorem 1.2. The main idea is similar tothat in the fixed metric case. However, since the heat kernel is not self-symmetric in thiscase, we use the symmetry between the heat kernel and the adjoint heat kernel instead.In section 4, we first use the method in [17] to prove a slightly different version of theGaussian upper estimate from the one in Theorem 1.3. Then we show that Theorem1.3 can be derived from this upper estimate and an L bound of H . In section 5, wefinish the proof of Theorem 1.4. Acknowledgements:
The author would like to thank Professor Huai-Dong Cao formany valuable suggestions, and for his constant support and encouragement. I alsowant to thank Professors Qing Ding, Jixiang Fu, Jiaxing Hong, Hong-Quan Li, Jun Li,Jiaping Wang, Quanshui Wu, Guoyi Xu and Weiping Zhang for helpful conversationsand encouragement, and Professors Luen-Fai Tam and Qi S. Zhang for their interestsin this work. In addition, I want to express my appreciation to the Shanghai Center forMathematical Sciences, where this work was carried out, for its hospitality and support.2.
Preliminaries
In this section, we present some basic results regarding the heat kernel and adjointheat kernel of time evolving heat-type equations. The readers can refer to [8] for moredetail.
MENG ZHU
Let M n be a Riemannian manifold, and g ij ( t ), t ∈ [0 , T ) and T < ∞ , a completesolution to the following equation: ∂g ij ( t ) ∂t = − A ij g (0) = g , (2.1)where A ij ( t ) is a time-dependent symmetric 2-tensor. Consider the following heat op-erator with potential: L = ∂∂t − ∆ t + Q, where ∆ t is the Laplacian with respect to g ij ( t ) and Q : M × [0 , T ) → R is a C ∞ function. Definition 2.1.
Let R T = { ( t, l ) ∈ R | ≤ l < t < T } . A fundamental solution forthe operator L is a function H : M × M × R T → R that satisfies: (1) H is continuous, C in the first two space variables and C in the last two timevariables, (2) LH = ( ∂∂t − ∆ t,y + Q ) H ( y, t ; x, l ) = 0 , (3) lim t ց l H ( y, t ; x, l ) = δ x ,where H ( y, t ; x, l ) = H ( y, x, ( t, l )) .The heat kernel for L is defined to be the minimal positive fundamental solution. Definition 2.2.
The adjoint heat kernel is the minimal positive fundamental solution G : M × M × R T → R for the operator L ∗ = ∂∂l + ∆ l − Q − A , i.e., G satisfies (1) G is continuous, C in the first two space variables and C in the last two timevariables, (2) L ∗ G = ( ∂∂l + ∆ l,x − Q − A ) G ( x, l ; y, t ) = 0 , (3) lim l ր t G ( x, l ; y, t ) = δ y ,where G ( x, l ; y, t ) = G ( y, x, ( t, l )) , and A = tr g ( l ) ( A ij ) . First of all, we have the following existence and uniqueness of the heat kernel and theadjoint heat kernel according to [14] and [8].
Theorem 2.3.
Let M n be a complete manifold, and g ( t ) , t ∈ [0 , T ) for some T < ∞ ,a smooth family of Riemannian metrics on M . If R T − inf M Q ( t ) dt < ∞ , then thereexists a unique C ∞ minimal positive fundamental solution H ( y, t ; x, l ) for the operator L = ∂∂t − ∆ t,y + Q . As in the fixed metric case, the heat kernel H ( y, t ; x, l ) for L on M is the limit of theDirichlet heat kernels on a sequence of exhausting subsets in M . AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 5
Definition 2.4.
Let Ω ⊂ M be a bounded subset, the Dirichlet Heat Kernel on Ω for L ,denoted by H Ω ( y, t ; x, l ) is the fundamental solution to ∂u∂t = ∆ t u − Qu in Ω and satisfiesi) lim t ց l H Ω = δ x for x ∈ Int (Ω) ;ii) H Ω ( y, t ; x, l ) = 0 , for y ∈ ∂ Ω and x ∈ Int (Ω) . Proposition 2.5 (see e.g. [8]) . Let Ω i ⊂ M be a sequence of exhausting bounded sets,and H Ω i ( y, t ; z, s ) the Dirichlet Heat Kernel on Ω i . Then lim i →∞ H Ω i ( y, t ; z, s ) = H ( y, t ; z, s ) uniformly on any compact subset of M × M × R T . Moreover, the heat kernel and the adjoint heat kernel satisfy the following importantproperties:
Proposition 2.6 (see e.g. [8]) . With the assumptions above, we have (1) H Ω ( y, t ; x, l ) = G Ω ( x, l ; y, t ) , for x, y ∈ Ω ; (2) H ( y, t ; x, l ) = G ( x, l ; y, t ) ; (3) H Ω ( y, t ; x, l ) = R Ω H Ω ( y, t ; z, s ) H Ω ( z, s ; x, l ) dµ g ( s ) ( z ) ; (4) H ( y, t ; x, l ) = R M H ( y, t ; z, s ) H ( z, s ; x, l ) dµ g ( s ) ( z ) ;where Ω is an open subset in M with compact closure, and G Ω ( x, l ; y, t ) is the Dirichletheat kernel for the adjoint operator ∂∂l + ∆ l − Q − A . Let Λ = Z T sup M | A ij | g ( t ) ( t ) dt. Assume that there exists a metric g ′ and a positive constant ˆ C ≥ C − g ′ ≤ g (0) ≤ ˆ Cg ′ , then we have ˆ C − e − g ′ ≤ g ( t ) ≤ ˆ Ce g ′ for all time t ∈ [0 , T ).Suppose that u : M × [0 , T ) → R is a positive subsolution to ∂u∂t ≤ ∆ t u − Qu.
Define the parabolic cylinder: P g ′ ( x, τ, r, − r ) = B g ′ ( x, r ) × [ τ − r , τ ] , where B g ′ ( x, r ) represents the geodesic ball of radius r centered at x in M with respectto the metric g ′ .By Moser iteration, Chau-Tam-Yu [6] got the following mean value inequality (seealso [8]): MENG ZHU
Theorem 2.7.
In the above setting, assume that Rc ( g ′ ) ≥ − K on M with some K ≥ . Then sup P g ′ ( x ,t ,r , − ( r ) ) u ≤ C ˆ C n ( n +3)2 e C Λ+ C √ K r + ˇ Ct r Vol g ′ ( B g ′ ( x , r )) Z P g ′ ( x ,t , r , − (2 r ) ) u ( x, s ) dµ g ′ ( x ) ds, where C , C and C are constants only depending on n , and ˇ C = − inf M × [0 ,T ) { Q + A } . Davies type estimate for the heat kernel under the Ricci flow
Assume that ( M n , g ij ( t )) is a complete solution to the Ricci flow (1.4) for t ∈ [0 , T )and T < ∞ .Denote by H ( y, t ; x, l ) >
0, 0 ≤ l < t < T , the Heat Kernel of (1.3) under the Ricciflow, i.e., ∂H∂t = ∆ t,y H lim t ց l H = δ x . (3.1)Then G ( x, l ; y, t ) = H ( y, t ; x, l ) is the adjoint Heat Kernel to the following conjugateheat equation: ∂G∂l = − ∆ l,x G + RG lim l ր t G = δ y , (3.2)where R is the scalar curvature of M .In the remaining of this paper, we will use the following notations: B t ( x, r ) := B g ( t ) ( x, r ), Vol s ( U ) := Vol g ( s ) ( U ), dµ s := dµ g ( s ) , where U is a subset of M and dµ g ( s ) denotes the volume element of g ( s ). Proof of Theorem 1.2. : Let Ω i ⊂ M be a sequence of exhausting bounded sets, and H Ω i ( y, t ; z, s ) the Dirichlet Heat Kernel on Ω i . Since U and U are bounded, we mayassume that ( U S U ) ⊂ Ω i for each i .By Propositions 2.5 and 2.6, we have Z U Z U H ( y, t ; x, l ) dµ t ( y ) dµ l ( x )= lim i →∞ Z U Z U Z Ω i H Ω i ( y, t ; z, t + l H Ω i ( z, t + l x, l ) dµ t + l ( z ) dµ t ( y ) dµ l ( x )= lim i →∞ Z Ω i Z U Z U H Ω i ( y, t ; z, t + l H Ω i ( z, t + l x, l ) dµ t ( y ) dµ l ( x ) dµ t + l ( z )= lim i →∞ Z Ω i u i ( z, t + l v i ( z, t + l dµ t + l ( z ) , where u i ( z, s ) = Z U H Ω i ( z, s ; x, l ) dµ l ( x ) , AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 7 and v i ( z, s ) = Z U H Ω i ( y, t ; z, s ) dµ t ( y ) = Z U G Ω i ( z, s ; y, t ) dµ t ( y ) . Here G Ω i ( z, s ; y, t ) denotes the adjoint Dirichlet Heat Kernel on Ω i .From Rc ≥ − K , we have for any s ∈ [0 , T ), g ij ( s ) ≤ e K T g ij (0) , and d s ( x, y ) ≤ e K T d ( x, y ) , where d s ( x, y ) is the distance function at time s .Let ξ ( z, s ) = d ( z,U )2 C K T ( s − l ) for C K T = e K T and s > l . Since |∇ d ( z, U ) | g ( s ) ≤ e K T |∇ d ( z, U ) | g (0) ≤ e K T , we have ∂ξ∂s + 12 |∇ ξ | g ( s ) ≤ − d ( z, U )2 C K T ( s − l ) + d ( z, U ) · e K T C K T ( s − l ) = 0 . We compute dds Z Ω i u i ( z, s ) e ξ ( z,s ) dµ s ( z )= Z Ω i (2 u i ∆ s,z u i − Ru i + u i · ∂ξ ( z, s ) ∂s ) e ξ ( z,s ) dµ s ( z ) . Notice that2 Z Ω i u i ∆ s,z u i e ξ dµ s ( z ) = − Z Ω i ( |∇ u i | + u i ∇ u i · ∇ ξ ) e ξ dµ s ( z ) + 2 Z ∂ Ω i u i ∂ ν u i e ξ dS ≤ Z Ω i u i |∇ ξ | e ξ dµ s ( z ) , where we have used the Cauchy-Schwartz inequality for − u i ∇ u i · ∇ ξ and the fact that u i | ∂ Ω i = 0.Thus, dds Z Ω i u i ( z, s ) e ξ ( z,s ) dµ s ( z ) ≤ C K Z Ω i u i ( z, s ) e ξ ( z,s ) dµ s ( z ) , where C is a constant only depending on n . Hence, we have Z Ω i u i ( z, s ) e ξ ( z,s ) dµ s ( z ) ≤ e C K ( s − l ) lim h → l Z Ω i u i ( z, h ) e ξ ( z,h ) dµ h ( z ) ≤ e C K ( t − l ) Vol l ( U ) . Let η ( z, s ) = d ( z,U )2 C K T ( t − s ) for C K T = e K T , then ∂η∂s − |∇ η | g ( s ) ≥ d ( z, U )2 C K T ( t − s ) − d ( z, U ) · e K T C K T ( t − s ) = 0 . MENG ZHU
Moreover, dds Z Ω i v i ( z, s ) e η ( z,s ) dµ s ( z )= Z Ω i ( − u i ∆ s,z v i + v i · ∂η ( z, s ) ∂s ) e η ( z,s ) dµ s ( z ) . Since, − Z Ω i v i ∆ sz v i e η dµ s ( z ) = 2 Z Ω i ( |∇ v i | + v i ∇ v i · ∇ η ) e η dµ s ( z ) − Z ∂ Ω v i ∂ ν v i e η dS ≥ − Z Ω i v i |∇ η | e η dµ s ( z ) , we have dds Z Ω i v i ( z, s ) e η ( z,s ) dµ s ( z ) ≥ . It implies that, for large i , Z Ω i v i ( z, s ) e η ( z,s ) dµ s ( z ) ≤ lim h → t Z Ω i v i ( z, h ) e η ( z,h ) dµ h ( z )= Vol t ( U ) . Now since 12 ξ ( z, t + l η ( z, t + l ≥ d ( U , U )4 C K T ( t − l ) , we have e d U ,U CK T ( t − l ) Z U Z U H ( y, t ; x, l ) dµ t ( y ) dµ l ( x )= lim i →∞ e d U ,U CK T ( t − l ) Z Ω i u i ( z, t + l v i ( z, t + l dµ t + l ( z ) ≤ lim i →∞ Z Ω i u i ( z, t + l e ξ ( z, t + l ) · v i ( z, t + l e η ( z, t + l ) dµ t + l ( z ) ≤ e C K t − l )2 Vol l ( U ) Vol t ( U ) , i.e., Z U Z U H ( y, t ; x, l ) dµ t ( y ) dµ l ( x ) ≤ e − d U ,U CK T ( t − l ) e C K t − l )2 Vol l ( U ) Vol t ( U ) . The Theorem follows from the fact that d t ( U , U ) ≤ e K T d ( U , U ). (cid:3) AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 9 A Gaussian upper bound of H ( y, t ; x, l )Recall that by the Volume Comparison Theorem, we have the following Lemma (seee.g. [15]): Lemma 4.1.
Let ( M n , g ij ) be a complete Riemannian manifold. If Rc ≥ − K , forsome constant K ≥ , then for any point x ∈ M and any < r ≤ R , we have Vol( B ( x, R )) ≤ ( Rr ) n e √ ( n − K R Vol( B ( x, r )) , where B ( x, R ) is the geodesic ball of radius R in M centered at x .In particular, letting r → , we have V ol ( B ( x, R )) ≤ CR n e √ ( n − K R for some constant C only depending on n . Since we have obtained a Davies type estimate in Theorem 1.2, similarly to themethod in [17], we can show a Gaussian upper bound of the heat kernel (see [6]) byusing the mean value inequality in Theorem 2.7. In the following, unless otherwisestated, C , C , C , C , · · · , and ˜ C , ˜ C , ˜ C , ˜ C , · · · all represent positive constants onlydepending on n . Theorem 4.2.
Under the Ricci flow, assume that Rc ≥ − K on M × [0 , T ) and Λ = R T sup M | Rc | ( t ) dt < ∞ . We have the following upper bound H ( y, t ; x, l ) ≤ C e C Λ+ C K T + C √ K T exp (cid:18) − d t ( x, y )8 e K T ( t − l ) (cid:19)r V ol l ( B l ( y, q t − l )) r V ol t ( B t ( x, q t − l )) . Proof.
We have by Theorem 2.7 that, for 0 ≤ l < t < T and r with t − l ≥ r , H ( y , t ; x , l ) ≤ C e C Λ+ C K T + C √ K r r V ol l ( B l ( y , r )) Z t t − r Z B l ( y , r ) H ( y, s ; x , l ) dµ l ( y ) ds. (4.1)Let ˜ l = t − l , ˜ s = t − s , and e G ( x, ˜ l ; y, ˜ s ) = G ( x, t − ˜ l ; y, t − ˜ s ) = G ( x, l ; y, s ) = H ( y, s ; x, l ) . Then, e G ( x, ˜ l ; y, ˜ s ) satisfies ∂ e G ( x, ˜ l ; y, ˜ t ) ∂ ˜ l = e ∆ ˜ l,x e G − ˜ R e G ( x, ˜ l ; y, ˜ t ) , where e ∆ ˜ l is the laplacian with respect to the solution ˜ g (˜ l ) := g ( t − ˜ l ) of the backwardRicci flow: ∂ ˜ g (˜ l ) ∂ ˜ l = 2 e R ij (˜ l ) . Thus, according to Theorem 2.7, we have the following mean value inequality: e G ( x , ˜ l ; y, ˜ s ) ≤ ˜ C e ˜ C Λ+ ˜ C K T + ˜ C √ K r r V ol ˜ g (0) ( B ˜ g (0) ( x , r )) Z ˜ l ˜ l − r Z B ˜ g (0) ( x , r ) e G ( x, ˜ l ; y, ˜ s ) dµ ˜ g (0) ( x ) d ˜ l, where 4 r ≤ t − l − r ≤ s − l .Hence Formula (4.1) becomes H ( y , t ; x , l ) ≤ C e C Λ+ C K T + C √ K r r V ol l ( B l ( y , r )) Z r Z B l ( y , r ) e G ( x , ˜ l ; y, ˜ s ) dµ l ( y ) d ˜ s ≤ C e C Λ+ C K T + C √ K r r V ol l ( B l ( y , r )) ˜ C e ˜ C Λ+ ˜ C K T + ˜ C √ K r r V ol ˜ g (0) ( B ˜ g (0) ( x , r )) · Z r Z B l ( y , r ) Z ˜ l ˜ l − r Z B ˜ g (0) ( x , r ) e G ( x, ˜ l ; y, ˜ s ) dµ ˜ g (0) ( x ) d ˜ l dµ l ( y ) d ˜ s = C e C Λ+ C K T + C √ K r r V ol l ( B l ( y , r )) ˜ C e ˜ C Λ+ ˜ C K T + ˜ C √ K r r V ol t ( B t ( x , r )) · Z t t − r Z B l ( y , r ) Z l +4 r l Z B t ( x , r ) H ( y, s ; x, l ) dµ t ( x ) dl dµ l ( y ) ds. Now let r = r = q t − l , we have by Theorem 1.2 that H ( y , t ; x , l ) ≤ C e C Λ+ C K T + C √ K T ( t − l ) V ol l ( B l ( y , q t − l )) V ol t ( B t ( x , q t − l )) · Z t t l Z t l l Z B l ( y , q t − l ) Z B t ( x , q t − l ) H ( y, s ; x, l ) dµ t ( x ) dµ l ( y ) dl ds ≤ C e C Λ+ C K T + C √ K T r V ol t ( B t ( x , q t − l )) r V ol l ( B l ( y , q t − l )) V ol l ( B l ( y , q t − l )) V ol t ( B t ( x , q t − l )) · exp − d t ( B l ( y , q t − l ) , B t ( x , q t − l ))4 e K T ( t − l ) AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 11 ≤ C e C Λ+ C K T + C √ K T r V ol l ( B l ( y , q t − l )) r V ol t ( B t ( x , q t − l )) · exp − d t ( B t ( y , e K T q t − l ) , B t ( x , e K T q t − l ))4 e K T ( t − l ) . In the last step above, we have used Lemma 4.1 to get
V ol t ( B t ( z, r t − l ≤ C e C √ K T V ol t ( B t ( z, r t − l t ∈ [0 , T ).Since d t ( B t ( y , e K T r t − l , B t ( x , e K T r t − l ( , if d t ( x , y ) ≤ e K T p t − l ) d t ( x , y ) − e K T p t − l ) , if d t ( x , y ) > e K T p t − l ) , it follows that when d t ( x , y ) ≤ e K T p t − l ), exp − d t ( B t ( y , e K T q t − l ) , B t ( x , e K T q t − l ))4 e K T ( t − l ) =1 ≤ e e K T exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19) . When d t ( x , y ) > e K T p t − l ), we have exp − d t ( B t ( y , e K T q t − l ) , B t ( x , e K T q t − l ))4 e K T ( t − l ) = exp − ( d t ( x , y ) − e K T p t − l )) e K T ( t − l ) ! ≤ exp (cid:18) − / · d t ( x , y ) + 2 e K T ( t − l )4 e K T ( t − l ) (cid:19) = e e K T · exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19) . Therefore, we get H ( y , t ; x , l ) ≤ C exp (cid:16) C Λ + C KT + C p K T (cid:17) exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19)r V ol l ( B l ( y , q t − l )) r V ol t ( B t ( x , q t − l )) . (cid:3) The rest of the proof of Theorem 1.3 follows [6]. We include them here for the purposeof completeness. The following Lemma shows an L bound of the Dirichlet heat kernel. Lemma 4.3.
Let Ω be a compact manifold with nonempty boundary ∂ Ω , and g ( t ) , t ∈ [0 , T ) , a solution to the Ricci flow (1.4) on Ω . Denote by H Ω ( y, t ; x, l ) and G Ω ( x, l ; y, t ) the Dirichlet heat kernel for ∂∂t − ∆ t,y and ∂∂l + ∆ l,x − R ( x, l ) on Ω , respectively. Thenwe have e − R tl sup M R ( t ) dt ≤ Z Ω H Ω ( y, t ; x, l ) dµ t ( y ) ≤ e − R tl inf M R ( t ) dt , (4.2) and Z Ω G Ω ( x, l ; y, t ) dµ l ( x ) ≡ for any x, y ∈ int (Ω) .Proof. Since ddt Z Ω H Ω ( y, t ; x, l ) dµ t ( y ) = Z Ω (∆ t,y H Ω ( y, t ; x, l ) − R · H Ω ( y, t ; x, l )) dµ t ( y )= Z ∂ Ω ν z ( H Ω ( z, t ; x, l )) dS − Z Ω R · H Ω ( y, t ; x, l ) dµ t ( y )= − Z Ω R · H Ω ( y, t ; x, l ) dµ t ( y )and ddl Z Ω G Ω ( x, l ; y, t ) dµ l ( x ) = Z Ω − ∆ l,x G Ω ( x, l ; y, t ) dµ l ( x )= − Z ∂ Ω ν z ( G Ω ( z, l ; y, t )) dS = 0 , the Lemma follows immediately. (cid:3) Since the heat kernel on a complete manifold is the limit of the Dirichlet heat kernelson a family of exhausting open subsets of the manifold, Lemma 4.3 implies that
Corollary 4.4.
Let ( M n , g t ) , t ∈ [0 , T ) , be a complete solution to the Ricci flow (1.4) .Then we have the following estimates for the heat kernel H ( y, t ; x, l ) and the adjointheat kernel G ( x, l ; y, t ) : e − R tl sup M R ( t ) dt ≤ Z M H ( y, t ; x, l ) dµ t ( y ) ≤ e − R tl inf M R ( t ) dt , (4.4) AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 13 and Z M G ( x, l ; y, t ) dµ l ( x ) ≡ . (4.5)From Corollary 4.4 and the mean value inequality, we can also show the followingrough C bound of H . Lemma 4.5.
Let ( M n , g ( t )) , t ∈ [0 , T ) and T < ∞ , be a complete solution to the Ricciflow. Assume that Rc ≥ − K for all time t , and that Λ = R T sup M | Rc | ( t ) dt < ∞ .Then there exist constants ˜ C , ˜ C , , ˜ C and ˜ C such that H ( y, t ; x, l ) ≤ min ˜ C e ˜ C Λ+ ˜ C K T + ˜ C √ K T Vol l ( B l ( y, q t − l )) , ˜ C e ˜ C Λ+ ˜ C K T + ˜ C √ K T Vol t ( B t ( x, q t − l )) . Proof.
In Theorem 2.7, by choosing the parabolic cylinder P g ( l ) ( y, t, r , − ( r ) ) with r = q t − l , we have H ( y, t ; x, l ) ≤ sup P g ( l ) ( y,t,r , − ( r ) ) H ( · , · ; x, l ) ≤ C e C Λ+ C K T + C √ K T ( t − l ) V ol l ( B l ( y, r )) Z t t + l Z B l ( y, √ t − l ) H ( z, s ; x, l ) dµ l ( z ) ds. By (4.4), we can see that Z t t + l Z B l ( y, √ t − l ) H ( z, s ; x, l ) dµ l ( z ) ds ≤ e n Λ Z t t + l Z M H ( z, s ; x, l ) dµ s ( z ) ds ≤ e n Λ Z t t + l e C K ( s − l ) ds = e n Λ · e C K ( t − l ) − e C K t − l )2 C K . Hence, H ( y, t ; x, l ) ≤ C e C Λ+ C K T + C √ K T Vol l ( B l ( y, q t − l )) · e C K ( t − l ) − e C K t − l )2 C K ( t − l ) ≤ C e C Λ+ C K T + C √ K T Vol l ( B l ( y, q t − l )) . Similarly, using (4.5), we can get H ( y, t ; x, l ) ≤ ˜ C e ˜ C Λ+ ˜ C K T + ˜ C √ K T Vol t ( B t ( x, q t − l )) . (cid:3) Now we are ready to prove Theorem 1.3.
Proof of Theorem 1.3:
Let σ = q t − l and r = d t ( x , y ). By Theorem 4.2, we have H ( y , t ; x , l ) ≤ C e C Λ+ C K T + C √ K T exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19)r V ol l ( B l ( y , q t − l )) r V ol t ( B t ( x , q t − l )) . Since B t ( x , σ ) ⊂ B t ( y , σ + r ) ⊂ B l ( y , e Λ ( σ + r )),Vol − l ( B l ( y , σ )) ≤ Vol − t ( B t ( x , σ )) · Vol t ( B l ( y , e Λ ( σ + r )))Vol l ( B l ( y , σ )) ≤ e n Λ Vol − t ( B t ( x , σ )) · Vol l ( B l ( y , e Λ ( σ + r )))Vol l ( B l ( y , σ )) ≤ e n Λ Vol − t ( B t ( x , σ )) · e n Λ ( σ + r ) n e √ ( n − K e Λ ( σ + r ) σ n ≤ e n Λ Vol − t ( B t ( x , σ )) · e n Λ (1 + r σ ) n e C √ K T e Λ (1+ r σ ) . Let ˆ C = e K T , ˆ C = C √ K T e Λ and η = r σ , we have exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19) · (1 + r σ ) n e C √ K T e Λ r σ = exp (cid:16) − ˆ C η + ˆ C η (cid:17) (1 + η ) n ≤ C e C e C C K T . Thus, H ( y , t ; x , l ) ≤ C e C e C C K T exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19) V ol t ( B t ( x , q t − l )) . Similarly, one can show that H ( y, t ; x, l ) ≤ C e C e C C K T exp (cid:18) − d t ( x, y )8 e K T ( t − l ) (cid:19) V ol l ( B l ( y, q t − l )) . (cid:3) Remark 4.6.
If one assumes Rc ≥ on [0 , T ) , and Λ = R T sup M | Rc | ( t ) dt < ∞ , thenthe upper bound above can be improved to H ( y, t ; x, l ) ≤ C e C Λ exp (cid:18) − d t ( x, y )8( t − l ) (cid:19) min l ( B l ( y, q t − l )) , t ( B t ( x, q t − l )) . AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 15 A Gaussian lower bound of H ( y, t ; x, l )In this section, we prove Theorem 1.4 following [3]. Proof of Theorem 1.4:
Let W be a large constant to be determined later. By Theorem1.3 and (4.4), we have Z B t ( x , √ W ( t − l )) H ( y, t ; x , l ) dµ t ( y ) ≥ t (cid:16) B t ( x , p W ( t − l )) (cid:17) Z B t ( x , √ W ( t − l )) H ( y, t ; x , l ) dµ t ( y ) ! = 1Vol t (cid:16) B t ( x , p W ( t − l )) (cid:17) (cid:18)Z M H ( y, t ; x , l ) dµ t ( y ) − Z M − B t ( x , √ W ( t − l )) H ( y, t ; x , l ) dµ t ( y ) ! ≥ t (cid:16) B t ( x , p W ( t − l )) (cid:17) · e − C Λ − Z M − B t ( x , √ W ( t − l )) ˜ C ˆ C exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19) Vol t ( B t ( x , q t − l )) dµ t ( y ) , where ˆ C = e ˜ C e ˜ C C K T .Since Z M − B t ( x , √ W ( t − l )) exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19) Vol t ( B t ( x , q t − l )) dµ t ( y ) ≤ e − W e K T Z M − B t ( x , √ W ( t − l )) exp (cid:18) − d t ( x , y )16 e K T ( t − l ) (cid:19) Vol t ( B t ( x , q t − l )) dµ t ( y ) ≤ e − W e K T Vol t ( B t ( x , q t − l )) Z ∞ √ W ( t − l ) e − ρ e K T ( t − l d Vol t ( B t ( x , ρ )) dρ dρ = e − W e K T Vol t ( B t ( x , q t − l )) − Z ∞ √ W ( t − l ) Vol t ( B t ( x , ρ )) · d ( e − ρ e K T ( t − l ) dρ dρ + Vol t ( B t ( x , ρ )) e − ρ e K T ( t − l (cid:12)(cid:12)(cid:12)(cid:12) ∞ √ W ( t − l ) ! ≤ e − W e K T Z ∞ √ W ( t − l ) Vol t ( B t ( x , ρ ))Vol t ( B t ( x , q t − l )) e − ρ e K T ( t − l ρ e K T ( t − l ) dρ ≤ e − W e K T Z ∞ √ W ( t − l ) ( ρ √ t − l ) n e ( n − √ K ρ e − ρ e K T ( t − l ρ e K T ( t − l ) dρ. Set η = ρ √ t − l . If we choose W = C e C Λ+ C K T big enough so that Z ∞√ W η n e ( n − √ K T η e − η e K T η e K T dη ≤
12 ˜ C ˆ C e W e K T , and e − W e K T ≤ e − C Λ , then Z M − B t ( x , √ W ( t − l )) ˜ C ˆ C exp (cid:18) − d t ( x , y )8 e K T ( t − l ) (cid:19) Vol t ( B t ( x , q t − l )) dµ t ( y ) ≤ ˜ C ˆ C e − W e K T Z ∞√ W η n e ( n − √ K T η e − η e K T η e K T dη ≤ e − W e K T . Thus, we can get Z B t ( x , √ W ( t − l )) H ( y, t ; x , l ) dµ t ( y ) ≥ t (cid:16) B t ( x , p W ( t − l )) (cid:17) (cid:18) e − C Λ − e − W e K T (cid:19) ≥ t (cid:16) B t ( x , p W ( t − l )) (cid:17) e − C Λ . By Corollary 4.4, there exists a point y ∈ B t ( x , p W ( t − l )) such that H ( y , t ; x , l ) ≥ t (cid:16) B t ( x , p W ( t − l )) (cid:17) e − C Λ . AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 17
From Theorem 3.3 in [24], we know H ( y , t ; x , l ) ≤ C H δ ( y , t ; x , l ) K δ δ e d t y ,y δ ( t − l , (5.1)where δ is any positive number, and K = max M × [ t l ,t ] { H ( y, t ; x , l ) } .Since for t ∈ [ t + l , t ], we haveVol t ( B t ( x , r t − l ≥ Vol t ( B t ( x , r t − l
16 )) ≥ e − n Λ Vol t ( B t ( x , e − Λ r t − l
16 )) , according to Lemma 4.5, we have for any ( y, t ) ∈ M × [ t + l , t ], H ( y, t ; x , l ) ≤ ˜ C e ˜ C Λ+ ˜ C K T + ˜ C √ K T Vol t ( B t ( x , q t − l )) ≤ ˜ C e C Λ+ ˜ C K T + ˜ C √ K T Vol t ( B t ( x , e − Λ q t − l )) , i.e., K ≤ ˜ C e C Λ+ ˜ C K T + ˜ C √ K T Vol t ( B t ( x , e − Λ q t − l )) . In (5.1), letting δ = 1 and noticing that d t ( y , y ) ≤ d t ( y , x ) + d t ( x , y )) ≤ W ( t − l ) + 2 d t ( x , y ) , we get H ( y , t ; x , l ) ≥ C e − C Λ − C K T − C √ K T e − W Vol t ( B t ( x , e − Λ q t − l ))Vol t ( B t ( x , p W ( t − l ))) e − d t x ,y t − l . By Lemma 4.1, we haveVol t ( B t ( x , p W ( t − l )) ≤ C e n Λ W n e C √ W K T Vol t ( B t ( x , e − Λ r t − l
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AVIES TYPE ESTIMATE AND THE HEAT KERNEL BOUND UNDER THE RICCI FLOW 19
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