流形上的Harnack不等式的研究
摘要
本文主要研究了完备Kahler流形上的Kahler-Ricci流的局部Harnack估计、完备流形上半线性抛物方程的Harnack不等式以及Ricci流下线性热方程的Harnack估计,并在此基础上对一些结论做了推广
随着几何流理论的成熟,几何分析在近20年里得到了充分的发展,成为当前几何研究中的一个重要方向。在这方面最重要的结果是Perelman给出了一个关于Poincare猜想的证明的概要[34],曹怀东和朱熹平则给出了一个完整的证明[9]。
几何流的Harnack不等式也称为Li-Yau-Harnack不等式,在几何分析中具有相当重要的意义。抛物方程的Harnack不等式起源于Moser在1964年的工作[32],他研究了线性散度型方程的情形。1986年,李伟光和丘成桐[30]用最大值原理得到流形上热方程的Harnack不等式,这是第一次将微分方程的Harnack不等式和微分几何结合起来。随后,Hamilton用同样的技巧得到了黎曼流形上一些非线性方程的Harnack不等式[18,19,21]。Chow在1991年计算了欧式空间中超曲面上的高斯曲率流的Harnack不等式[12]。1992年,曹怀东[6]得到了关于Kahler流形上的Kahler-Ricci流的Harnack不等式。Andrews用高斯映射的逆映射法得到一类欧式空间中超曲面上几何流的Harnack不等式[1]。近几年在该领域上也有一些很好的结果[8,10,28,29]。本文即在前人工作的基础上得到了一些结果及其推广：
在第一章中,我们研究了n维完备Kahler流形上的Kahler-Ricci流的局部Harnack不等式及其推论。曹怀东在[5]中最早证明了Kahler-Ricci流方程整体解的存在性。作为Mok在[31]中的一个结论,我们知道在Kahler-Ricci流方程下可以保持Kahler流形双截面曲率的正则性。Mori[32]和萧荫堂以及丘成桐[38]则证明了任意的具有正的爽截面曲率的Kahler流形X双全纯同胚于复射影空间。
最近,Hamilton证明了Ricci流的局部Harnack不等式,并由此得到Ricci流的Nonconic估计。在他的报告Curvature and Volume Bounds中使用这个估计证明了有限距离具有有限曲率。这在Poincare猜想证明中是很重要的一步。下面我们给出关于Ricci流的Nonconic估计。
定理A(Hamilton的Ricci流中的Nonconic估计)：设是Ricci流方程的解,t∈[0,T),Mn为黎曼流形。U是Mn上的一个连通开集,且在U×[0,t0],t00,且只依赖于n和C1.
在第一章中,我们对Kahler-Ricci流做了类似的研究,我们首先给出Kahler-Ricci流的方程在限定了一些曲率条件之后,我们得到了定理1.1.1和推论1.1.1及1.1.2。
定理1.1.1.(Kahler-Ricci流的局部Harnack估计)：设是方程(1.1)的解,在Br(O,t)×[0,r2]上满足曲率条件：则我们有某个只与n有关的常数B>0,使得如下估计成立：设则对任意的t>0,ω∈Tx,ω≠0,我们有这里
推论1.1.1.若是方程(1.1)的解,则数量曲率R满足估计：
推论1.1.2(Kahler-Ricci流的Nonconic估计)：若是方程(1.1)的解,我们有如下的估计：其中C仅与n有关。
在第二章中,我们给出了黎曼流形上半线性抛物方程的正解的Harnack估计,利用该估计我们可以得到一个积分形式的Harnack不等式。我们接下来研究完备黎曼流形上的半线性抛物方程这里函数V满足V=(x)+k(u).此类估计仍然有许多未解决的问题。我们希望通过利用一个梯度估计来得到方程正则解的Harnack估计。该方法最早在[11]和[40]中由Cheng.S.Y,丘成桐和Trudinger提出,用于解决椭圆方程的情况,此时解与时间无关。1986年,丘成桐和Peter Li在[30]中得到了线性抛物方程的正则解的Harnack估计。
定理2.1.1.设M是一个完备带边流形。设p是M上一点,Bp(2R)是以p为球心2R为半径的测地球,且与M的边界(?)M没有交点。我们用-K(2R)(K(2R)≥0)表示测地球Bp(2R)的Ricci曲率的下界。设V是定义在M×(0,∞)上的C2,1函数。假设及这里θ(2R),γ(2R),M(2R)是定义在Bp(2R)×[0,T]上的常数。若u(x,t)是方程在M×(0,T]上的正则解,则对任意的a>1和c>0,我们在Bp(R)上有以下估计其中Ci是仅和n有关的常数。
定理2.1.2.设M是一个完备带边流形。设p是M上一点,Bp(2R)是以p为球心2R为半径的测地球,且与aM没有交点。我们用-K(2R)(K(2R)≥0)表示测地球Bp(2R)的Ricci曲率的下界。设V是定义在M×(0,∞)上的C2,1函数,假设及这里θ(2R),γ(2R),M(2R)是定义在Bp(2R)×[0,T]上的常数。若u(x,t)是方程在M×(0,T]上的正则解,则对任意的a>1,0 在第三章中,我们主要研究Ricci流下关于线性热方程的Harnack估计。利用该估计我们得到了一些结论,包括一个整体结果和一个积分形式的Harnack不等式。
我们首先假设M是一个n维无边流形,是Ricci流方程的一个完备解。我们假定对所有的t∈[0,T]都有曲率一致有界,考虑定义在M×[0,T]上的函数u(x,t),我们假定u(x,t)满足方程这里记号△表示g(x,t)下的Laplace算子。这里需要强调的是△与t有关。我们有
定理3.1.1.设是Ricci流方程(1.1)的一个完备解。假定Ric(x,t)|≤Kg(x,t)对某个K>0和所有的(x,t)∈Bρ,T成立。设u：M×[0,T]→R是一个正则光滑函数满足热方程(1.4),q(x,t)是一个定义在M×(0,T)上的C2,1函数,|△q|≤θ,则存在一个仅依赖于流形M维数n的常数C'满足以下估计这里且t≠0.
定理3.1.2.假设M是Ricci流方程(1.1)的一个解。假定0≤Ric(x,t)≤kg(x,t),对k>0及所有的(x,t)∈M×[0,T]成立。设u：M×[0,T]→R是一个正则光滑函数满足热方程(1.4),q(x,t)是一个定义在M×(0,T)上的C2,1函数,且|△q|≤θ,我们对所有(x,t)∈M×[0,T]有估计
定理3.1.3.设是Ricci流方程(1.1)的一个完备解。假定|Ric(x,t)|≤Kg(x,t)对某个K>0和所有的(x,t)∈Bρ,T成立。设u：M×[0,T]→R是一个正则光滑函数满足热方程(1.4),q(x,t)是一个定义在M×(0,T)上的C2,1函数,|△q|≤θ.给定α>1,我们有估计对所有的(x1,t1)∈M×(0,T)和(x2,t2)∈M×(0,T)使得t1In this paper we mainly study the local Harnack estimate for Kahler-Ricci flow on complete Kahler manifold,Harnack estimate for semilinear parabolic equations on the complete manifolds and Harnack estimates for the linear heat equation under the Ricci flow.
By the maturity of the differential equation theory, geometric analysis has gotten full development over the past 20 years, and becomes an important field on geometric research at present. The most important result is that the sketch of the proof of the Poincare conjecture given by Perelman in [34], Huaidong Cao and Xiping Zhu finished it in the end [9].
The Harnack estimate of geometry flow is also called Li-Yau-Hamilton in-equality. It plays an important role in geometric analysis. Harnack inequalities of parabolic originated from the work of Moser[31] who treated the case of linear divergence-form equations. In 1986 Li and Yau got the Harnack inequality for the heat equation on manifold by the parabolic maximum principle in [29]. This is the first time to combine the Harnack inequality of differential equation with geometry. After that, Hamilton got some Harnack estimates of nonlinear differential equations on manifold by using the same method[17,18,20]. Chow obtained similar inequalities for Gauss curvature flow of hypersurface on Euclidean space in 1991 [11]. Moreover, in 1992 Cao got the Harnack estimate for Kahler-Ricci flow on Kahler manifold[6]. Andrews used the inverse of Gauss map to get the Harnack estimate of a class ge-ometric flow of hypersurface on Euclidean space[1]. Recently, There occurs many papers on the field,such as[8,10,28,29]. In this paper we get the following results on the basis of their works.
In chapter one we study the local Harnack estimate for Kahler-Ricci flow on complete Kahler manifold and give its corollary (the nonconic estimate of Kahler-Ricci flow). It was proved by H.D.Cao[5] that the solution of Kahler-Ricci flow equation exists for all time. By a result of Mok[31] one also knows that the positiv- ity of the bisectional curvature is preserved under the Kahler-Ricci flow equation. Mori and Siu and Yau proved that any compact Kahler manifold X of pos-itive holomorphic bisectional curvature is biholomorphic to a complex projective space.
Recently, Hamilton proved the local Harnack estimate of Ricci flow, and derived the nonconic estimate from it. On his report Curvature and Volume bounds he applied the nonconic estimate to prove that finite curvature within finite distance, which is an important step in the proof of Poincare conjecture. The following is nonconic estimate of Ricci flow.
Theorem A(Nonconic estimate of Hamilton's Ricci flow):Let Mn be a Riemannian manifold, (M,g(t)) is a solution to the equation t∈[0, T).U C Mn is an open connected set, and on U×[0, to), t0 where O∈U. If Mr2= C1 such that Br(O,t0) (?)U, then and V∈TpM, we have
where C> 0 depending on n and C1.
We consider a similar problem on Kahler-Ricci flow in Chapter one, and get Theorem 1.1.1 and Corollary 1.1.1 with some curvature conditions. Firstly, we give the Kahler-Ricci flow equation:
Theorem 1.1.1 (Local Harnack estimate of Kahler-Ricci flow):we denote the curvature condition, Then let gij(x,t) be a solution of (1.1) on a complete Kahler manifold X for t∈[0,r2], and satisfies the curvature condition on Br(O,t)×[0,r2], then (?)(x,t)∈B2/r(O,t)×[0,r2], we can find some constant B> 0, depending only on n, s.t the following local Harnack estimate holds, let Then for any t> 0, and w∈TxX,w≠0, we have Here
Corollary 1.1.1 Under the same condition in Theorem 1.1.1, the scalar cur-vature R satisfies the estimate
Corollary 1.1.2 (Nonconic estimate of Kahler-Ricci flow)
Under the same conditions as Theorem 1.1.1, at point (0,r2), we have where C depends only on n.
In Chapter 2, we prove a Harnack inequality for positive solutions of the semi-linear parabolic equation on Riemannian manifold and get some results, including an integral Harnack inequality.
We will study the semilinear parabolic equations of the type on a complete Riemannian manifold. The function V satisfies V=h(x)+k(u). The geometric dependency of the estimates is complicated and sometimes unclear. Our goal is to prove a Harnack inequality for positive solutions of the equation by utilizing a gradient estimate derived in Section 2. The method of proof is originated [11] and [40], where they have studied the elliptic case, i.e. the solution is time independent. Later, professor Yau,S.T and Peter Li got some results in [30].
Theorem 2.1.1. Let M be a complete manifold with boundary, (?)M. Assume p∈M and let BP(2R) to be a geodesic ball of radius 2R around p which does not intersect (?)M. We denote-K(2R), with K(2R)≥0, to be a lower bound of the Ricci curvature on BP(2R). Let V be a function defined on M x (0,∞) which is C2 in the x-variable and C1 in the t-variable. Assume that and on BP(2R)×[0,T] for some constantsθ(2R),γ2R),M(2R). If u(x,t) is a positive solution of the equation on M x (0,T], then for any a> 1 and c> 0, u(x,t) satisfies the estimate on BP(R), where Ci are constants depending only on n.
Theorem 2.1.2. Let M be a complete manifold with boundary, (?)M. Assume p∈M and let Bp(2R) to be a geodesic ball of radius 2R around p which does not intersect (?)M. We denote-K(2R), with K(2R)≥0, to be a lower bound of the Ricci curvature on BP(2R). Let V be a function defined on M x (0,∞) which is C2 in the x-variable and C1 in the t-variable. If u(x, t)is a positive solution of the equation on M×(0, T], assume that and on Bp (2R)×[0,T] for some constantsθ(2R),γ(2R),M(2R). then for anyα>1, 0 In part 3, we study a Harnack inequality for positive solutions of the linear heat equation under the Ricci flow and get some results, including a global result and an integral Harnack inequality. Suppose M is a manifold without boundary. Let (M,g(x,t))t∈[0,T] be a complete solution to the Ricci flow We assume its curvature remains uniformly bounded for all t E [0, T]. Consider a positive function u(x,t) defined on M×[0,T]. we assume u(x,t) solves to the equation The symbolΔhere stands for the Laplacian given by g(x, t).
Theorem 3.1.1. Let (M, g(x,t))t∈[0,T] be a complete solution to the Ricci flow (1.3). Suppose|Ric(x,t)≤Kg(x,t) for some K>0 and all (x,t)∈BPp,T. Consider a smooth positive function u:M×[0, T]→R solving the heat equation (1.4) and q(x,t) is a C2,1 function defined on M x (0,T),|▽q|≤γ,|Δq|≤θThere exists a constant C1 that depends only on the dimension of M and satisfies the estimate for allα>1 and all with t≠0.
Theorem 3.1.2. Suppose the manifold M is a solution to the Ricci flow (1.3). Assume that 0≤Ric(x,t)≤kg(x,t) for some k<0 and all (x, t)∈M x [0, T]. Consider a smooth positive function u:Mx[0,T]→Rsatisfying the heat equation (1.4) and q(x,t) is a C2,1 function defined on M×(0, T) and|Δq|≤θThe estimate holds for all (x,t)∈M x [0,T].
Theorem 3.1.3 Let (M,g(x,t))t∈[0,T]be a complete solution to the Ricci flow (1.1). Assume that|Ric(x,t)|≤Kg(x,t) for some K>0 and all (x,t)∈Bρ,T. Suppose a smooth positive function u:M x [0,T]→R solving the heat equation (1.2) and q(x,t) is a C2,1 function defined on M x (0,T),|▽q|≤γ,|Δq|≤θ. Given a> 1, the estimate holds for all (x1,t1)∈M x (0,T) and (x2,t2) G M×(0,T) such that t1< t2-The constant C comes from Theorem 3.1.1.
随着几何流理论的成熟,几何分析在近20年里得到了充分的发展,成为当前几何研究中的一个重要方向。在这方面最重要的结果是Perelman给出了一个关于Poincare猜想的证明的概要[34],曹怀东和朱熹平则给出了一个完整的证明[9]。
几何流的Harnack不等式也称为Li-Yau-Harnack不等式,在几何分析中具有相当重要的意义。抛物方程的Harnack不等式起源于Moser在1964年的工作[32],他研究了线性散度型方程的情形。1986年,李伟光和丘成桐[30]用最大值原理得到流形上热方程的Harnack不等式,这是第一次将微分方程的Harnack不等式和微分几何结合起来。随后,Hamilton用同样的技巧得到了黎曼流形上一些非线性方程的Harnack不等式[18,19,21]。Chow在1991年计算了欧式空间中超曲面上的高斯曲率流的Harnack不等式[12]。1992年,曹怀东[6]得到了关于Kahler流形上的Kahler-Ricci流的Harnack不等式。Andrews用高斯映射的逆映射法得到一类欧式空间中超曲面上几何流的Harnack不等式[1]。近几年在该领域上也有一些很好的结果[8,10,28,29]。本文即在前人工作的基础上得到了一些结果及其推广：
在第一章中,我们研究了n维完备Kahler流形上的Kahler-Ricci流的局部Harnack不等式及其推论。曹怀东在[5]中最早证明了Kahler-Ricci流方程整体解的存在性。作为Mok在[31]中的一个结论,我们知道在Kahler-Ricci流方程下可以保持Kahler流形双截面曲率的正则性。Mori[32]和萧荫堂以及丘成桐[38]则证明了任意的具有正的爽截面曲率的Kahler流形X双全纯同胚于复射影空间。
最近,Hamilton证明了Ricci流的局部Harnack不等式,并由此得到Ricci流的Nonconic估计。在他的报告Curvature and Volume Bounds中使用这个估计证明了有限距离具有有限曲率。这在Poincare猜想证明中是很重要的一步。下面我们给出关于Ricci流的Nonconic估计。
定理A(Hamilton的Ricci流中的Nonconic估计)：设是Ricci流方程的解,t∈[0,T),Mn为黎曼流形。U是Mn上的一个连通开集,且在U×[0,t0],t0
在第一章中,我们对Kahler-Ricci流做了类似的研究,我们首先给出Kahler-Ricci流的方程在限定了一些曲率条件之后,我们得到了定理1.1.1和推论1.1.1及1.1.2。
定理1.1.1.(Kahler-Ricci流的局部Harnack估计)：设是方程(1.1)的解,在Br(O,t)×[0,r2]上满足曲率条件：则我们有某个只与n有关的常数B>0,使得如下估计成立：设则对任意的t>0,ω∈Tx,ω≠0,我们有这里
推论1.1.1.若是方程(1.1)的解,则数量曲率R满足估计：
推论1.1.2(Kahler-Ricci流的Nonconic估计)：若是方程(1.1)的解,我们有如下的估计：其中C仅与n有关。
在第二章中,我们给出了黎曼流形上半线性抛物方程的正解的Harnack估计,利用该估计我们可以得到一个积分形式的Harnack不等式。我们接下来研究完备黎曼流形上的半线性抛物方程这里函数V满足V=(x)+k(u).此类估计仍然有许多未解决的问题。我们希望通过利用一个梯度估计来得到方程正则解的Harnack估计。该方法最早在[11]和[40]中由Cheng.S.Y,丘成桐和Trudinger提出,用于解决椭圆方程的情况,此时解与时间无关。1986年,丘成桐和Peter Li在[30]中得到了线性抛物方程的正则解的Harnack估计。
定理2.1.1.设M是一个完备带边流形。设p是M上一点,Bp(2R)是以p为球心2R为半径的测地球,且与M的边界(?)M没有交点。我们用-K(2R)(K(2R)≥0)表示测地球Bp(2R)的Ricci曲率的下界。设V是定义在M×(0,∞)上的C2,1函数。假设及这里θ(2R),γ(2R),M(2R)是定义在Bp(2R)×[0,T]上的常数。若u(x,t)是方程在M×(0,T]上的正则解,则对任意的a>1和c>0,我们在Bp(R)上有以下估计其中Ci是仅和n有关的常数。
定理2.1.2.设M是一个完备带边流形。设p是M上一点,Bp(2R)是以p为球心2R为半径的测地球,且与aM没有交点。我们用-K(2R)(K(2R)≥0)表示测地球Bp(2R)的Ricci曲率的下界。设V是定义在M×(0,∞)上的C2,1函数,假设及这里θ(2R),γ(2R),M(2R)是定义在Bp(2R)×[0,T]上的常数。若u(x,t)是方程在M×(0,T]上的正则解,则对任意的a>1,0
我们首先假设M是一个n维无边流形,是Ricci流方程的一个完备解。我们假定对所有的t∈[0,T]都有曲率一致有界,考虑定义在M×[0,T]上的函数u(x,t),我们假定u(x,t)满足方程这里记号△表示g(x,t)下的Laplace算子。这里需要强调的是△与t有关。我们有
定理3.1.1.设是Ricci流方程(1.1)的一个完备解。假定Ric(x,t)|≤Kg(x,t)对某个K>0和所有的(x,t)∈Bρ,T成立。设u：M×[0,T]→R是一个正则光滑函数满足热方程(1.4),q(x,t)是一个定义在M×(0,T)上的C2,1函数,|△q|≤θ,则存在一个仅依赖于流形M维数n的常数C'满足以下估计这里且t≠0.
定理3.1.2.假设M是Ricci流方程(1.1)的一个解。假定0≤Ric(x,t)≤kg(x,t),对k>0及所有的(x,t)∈M×[0,T]成立。设u：M×[0,T]→R是一个正则光滑函数满足热方程(1.4),q(x,t)是一个定义在M×(0,T)上的C2,1函数,且|△q|≤θ,我们对所有(x,t)∈M×[0,T]有估计
定理3.1.3.设是Ricci流方程(1.1)的一个完备解。假定|Ric(x,t)|≤Kg(x,t)对某个K>0和所有的(x,t)∈Bρ,T成立。设u：M×[0,T]→R是一个正则光滑函数满足热方程(1.4),q(x,t)是一个定义在M×(0,T)上的C2,1函数,|△q|≤θ.给定α>1,我们有估计对所有的(x1,t1)∈M×(0,T)和(x2,t2)∈M×(0,T)使得t1
By the maturity of the differential equation theory, geometric analysis has gotten full development over the past 20 years, and becomes an important field on geometric research at present. The most important result is that the sketch of the proof of the Poincare conjecture given by Perelman in [34], Huaidong Cao and Xiping Zhu finished it in the end [9].
The Harnack estimate of geometry flow is also called Li-Yau-Hamilton in-equality. It plays an important role in geometric analysis. Harnack inequalities of parabolic originated from the work of Moser[31] who treated the case of linear divergence-form equations. In 1986 Li and Yau got the Harnack inequality for the heat equation on manifold by the parabolic maximum principle in [29]. This is the first time to combine the Harnack inequality of differential equation with geometry. After that, Hamilton got some Harnack estimates of nonlinear differential equations on manifold by using the same method[17,18,20]. Chow obtained similar inequalities for Gauss curvature flow of hypersurface on Euclidean space in 1991 [11]. Moreover, in 1992 Cao got the Harnack estimate for Kahler-Ricci flow on Kahler manifold[6]. Andrews used the inverse of Gauss map to get the Harnack estimate of a class ge-ometric flow of hypersurface on Euclidean space[1]. Recently, There occurs many papers on the field,such as[8,10,28,29]. In this paper we get the following results on the basis of their works.
In chapter one we study the local Harnack estimate for Kahler-Ricci flow on complete Kahler manifold and give its corollary (the nonconic estimate of Kahler-Ricci flow). It was proved by H.D.Cao[5] that the solution of Kahler-Ricci flow equation exists for all time. By a result of Mok[31] one also knows that the positiv- ity of the bisectional curvature is preserved under the Kahler-Ricci flow equation. Mori and Siu and Yau proved that any compact Kahler manifold X of pos-itive holomorphic bisectional curvature is biholomorphic to a complex projective space.
Recently, Hamilton proved the local Harnack estimate of Ricci flow, and derived the nonconic estimate from it. On his report Curvature and Volume bounds he applied the nonconic estimate to prove that finite curvature within finite distance, which is an important step in the proof of Poincare conjecture. The following is nonconic estimate of Ricci flow.
Theorem A(Nonconic estimate of Hamilton's Ricci flow):Let Mn be a Riemannian manifold, (M,g(t)) is a solution to the equation t∈[0, T).U C Mn is an open connected set, and on U×[0, to), t0
where C> 0 depending on n and C1.
We consider a similar problem on Kahler-Ricci flow in Chapter one, and get Theorem 1.1.1 and Corollary 1.1.1 with some curvature conditions. Firstly, we give the Kahler-Ricci flow equation:
Theorem 1.1.1 (Local Harnack estimate of Kahler-Ricci flow):we denote the curvature condition, Then let gij(x,t) be a solution of (1.1) on a complete Kahler manifold X for t∈[0,r2], and satisfies the curvature condition on Br(O,t)×[0,r2], then (?)(x,t)∈B2/r(O,t)×[0,r2], we can find some constant B> 0, depending only on n, s.t the following local Harnack estimate holds, let Then for any t> 0, and w∈TxX,w≠0, we have Here
Corollary 1.1.1 Under the same condition in Theorem 1.1.1, the scalar cur-vature R satisfies the estimate
Corollary 1.1.2 (Nonconic estimate of Kahler-Ricci flow)
Under the same conditions as Theorem 1.1.1, at point (0,r2), we have where C depends only on n.
In Chapter 2, we prove a Harnack inequality for positive solutions of the semi-linear parabolic equation on Riemannian manifold and get some results, including an integral Harnack inequality.
We will study the semilinear parabolic equations of the type on a complete Riemannian manifold. The function V satisfies V=h(x)+k(u). The geometric dependency of the estimates is complicated and sometimes unclear. Our goal is to prove a Harnack inequality for positive solutions of the equation by utilizing a gradient estimate derived in Section 2. The method of proof is originated [11] and [40], where they have studied the elliptic case, i.e. the solution is time independent. Later, professor Yau,S.T and Peter Li got some results in [30].
Theorem 2.1.1. Let M be a complete manifold with boundary, (?)M. Assume p∈M and let BP(2R) to be a geodesic ball of radius 2R around p which does not intersect (?)M. We denote-K(2R), with K(2R)≥0, to be a lower bound of the Ricci curvature on BP(2R). Let V be a function defined on M x (0,∞) which is C2 in the x-variable and C1 in the t-variable. Assume that and on BP(2R)×[0,T] for some constantsθ(2R),γ2R),M(2R). If u(x,t) is a positive solution of the equation on M x (0,T], then for any a> 1 and c> 0, u(x,t) satisfies the estimate on BP(R), where Ci are constants depending only on n.
Theorem 2.1.2. Let M be a complete manifold with boundary, (?)M. Assume p∈M and let Bp(2R) to be a geodesic ball of radius 2R around p which does not intersect (?)M. We denote-K(2R), with K(2R)≥0, to be a lower bound of the Ricci curvature on BP(2R). Let V be a function defined on M x (0,∞) which is C2 in the x-variable and C1 in the t-variable. If u(x, t)is a positive solution of the equation on M×(0, T], assume that and on Bp (2R)×[0,T] for some constantsθ(2R),γ(2R),M(2R). then for anyα>1, 0
Theorem 3.1.1. Let (M, g(x,t))t∈[0,T] be a complete solution to the Ricci flow (1.3). Suppose|Ric(x,t)≤Kg(x,t) for some K>0 and all (x,t)∈BPp,T. Consider a smooth positive function u:M×[0, T]→R solving the heat equation (1.4) and q(x,t) is a C2,1 function defined on M x (0,T),|▽q|≤γ,|Δq|≤θThere exists a constant C1 that depends only on the dimension of M and satisfies the estimate for allα>1 and all with t≠0.
Theorem 3.1.2. Suppose the manifold M is a solution to the Ricci flow (1.3). Assume that 0≤Ric(x,t)≤kg(x,t) for some k<0 and all (x, t)∈M x [0, T]. Consider a smooth positive function u:Mx[0,T]→Rsatisfying the heat equation (1.4) and q(x,t) is a C2,1 function defined on M×(0, T) and|Δq|≤θThe estimate holds for all (x,t)∈M x [0,T].
Theorem 3.1.3 Let (M,g(x,t))t∈[0,T]be a complete solution to the Ricci flow (1.1). Assume that|Ric(x,t)|≤Kg(x,t) for some K>0 and all (x,t)∈Bρ,T. Suppose a smooth positive function u:M x [0,T]→R solving the heat equation (1.2) and q(x,t) is a C2,1 function defined on M x (0,T),|▽q|≤γ,|Δq|≤θ. Given a> 1, the estimate holds for all (x1,t1)∈M x (0,T) and (x2,t2) G M×(0,T) such that t1< t2-The constant C comes from Theorem 3.1.1.
引文
[1]B.Andrews, Harnack inequalities for evolving hypersurf aces, Math. Z. 217(1994),179-197.
[2]S.Brendle, A generalization of Hamilton's Harnack inequality for the Ricci flow, preprint
[3]T.Aubin, The scalar curvature,Differential Geometry and Relativity, edited by Cahen and Flato, Reider,1976.
[4]K.Brakke, The motion of a surface by its mean curvature, Mathematical Notes 20, Princeton University Press, Princeton, NJ,1978.
[5]H.D.Cao, Deformation of Kdhler metrics to Kdhler-Einstein metrics on com-pact Kdhler manifolds, Invent. Math,81(1985),359-372.
[6]H.D.Cao, On Harnack's inequalities for the Kdhler-Ricci flow, Invent.math. 109 (1992) 247-263.
[7]H.D.Cao, Limits of solutions to the Kahler-Ricci flow, J.Diff.Geom,45(1997),257-272.
[8]H.D.Cao and N.Li, Matrix Li-Yau-Hamilton estimates for the heat equation on Kahler manifolds, Math.Ann.,331 No.4(2005),795-807.
[9]H.D.Cao, X.P.Zhu, A complete proof of the Poincare and Geometrization conjectures—Application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. Vol.10, No.2, (2006) 165-492.
[10]X.D.Cao, Differential Harnack estimates for backward heat equations with po-tentials under the Ricci flow, J.Funct.Anal.255 No.4(2008),1024-1038.
[11]CHENG, S. Y. Yau, S. T., Differential equations on Riemanian manifolds and their geometric applications. Comm. Pure Appl. Math,33 (1980),199-211.
[12]B.chow, On Harnack's inequality and entropy fot the Gaussian curvature flow, Comm.Pure Appl.Math,44(1991),469-483.
[13]K.Ecker and G.Huisken, Mean curvature evolution of entire graphs, Ann.Math, 130(1989),453-471.
[14]K.Ecker and G.Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent.Math,105(1991),547-569.
[15]M.Gage and R.Hamilton, The heat equation shrinking convex plane curves, J.Diff.Geom,23(1996),69-96.
[16]Mihai Bailesteanu, X.D. Cao, Artem Pulemotov, Gradient estimates for the heat equation under the Ricci flow, Arxiv:0910.1053vl[math. DG].
[17]R.Hamilton, The Ricci flow on surfaces, Contemp Math 71,Amer.Math.Soc, Providence, RI,(1988),237-262.
[18]R.Hamilton, A matrix Harnack estimate for the heat equation, Comm.Anal.Geom,1(1993),113-126.
[19]R.Hamilton, The Harnack estimate for the Ricci flow, J.Diff.Geom,37(1993),113-126.
[20]R.Hamilton, Eternal solutions to the Ricci flow, J.Diff.Geom,38(1993),1-11.
[21]R.Hamilton, The Harnack estimate for the mean curvature flow, J.Diff.Geom,41(1995),215-226.
[22]R.Hamilton, The formation of singularities in the Ricci flow, Surveys in Dif-ferential Geometry,2,International Press,(1995),7-136.
[23]R.Hamilton, Curvature and Volume Bounds Ⅱ, preprint.
[24]G.Huisken, Flow by mean curvature of convex surface into sphere, J.Diff.Geom,20(1984),237-266.
[25]G.Huisken, Contracting convex hypersurfaces in the Riemannian manifolds by their mean curvature, Invent.Math,84(1986),463-480.
[26]G.Huisken, Asymptotic behavior for singularities of the mean curbature flow,J.Diff. Geom,31(1990),285-299.
[27]G.Huisken and T.Ilmanen, The inverse mean curvature flow and the Rieman-nian Penrose inequality,59 No.3(2001),353-437.
[28]N.Li, A note Perelman's LYE inequality, arXiv:math.DG/0602337 v1 feb 15,2006.
[29]N.Li, A matrix Li-Yau-Hamilton estimate for Kdhler-Ricci flow, J.Diff.Geom, 75 No.2(2007),303-358.
[30]Peter Li, S.-T. Yau, On the parabolic kernel of the Kdhler Schrodinger oper-ator, Acta Math.156(1986) 153-201.
[31]N.Mok, The uniformization theorem for compact Kdhler manifolds of non-negative holomorphic bisectional curvature, J.Diff.Geom,27(1988),179-214.
[32]Mori,S, Projective manifolds with ample tangent bundles, Ann.Math.110,593-606(1979)
[33]J.Moser, A Harnack inequality for parabolic differential equations, Com-mun.Pure Appl.Math,17(1964),101-134.
[34]G.Perelman, The entropy formula for the Ricci flow and its geometric appli-cations, arXiv:math.DG/0211159 vl November 11,2002.
[35]R.Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J.Diff.Geom.,20(1984),479-495.
[36]R.Schoen and S.T.Yau, Lectures on Differntial Geometry, in Conference Pro-ceedings and Lecture Notes in Geometry and Topology,Vol.l, International Press Publications,1994.
[37]F.Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math.Z,251(2005),721-733.
[38]Siu,Y.-T,Yau,S.-T, Compact Kahler manifolds of positive bisectional curva-ture, Invent.Math.59,189-204(1980).
[39]K.Smoczyk, Harnack inequalities for curvature flows depending on mean cur-vature, New York J.Math,3(1997),103-118.
[40]N.Trudinger, Remarks concerning the conformal deformation of Rieman-nian structures on compact manifolds, Ann. Scuola Norm.Sup. Pisz Cl.Sci.(4),3(1968),265-274.
[41]Yau, S. T, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math,28 (1975),201-228.
[42]Yau, S. T, On the Harnack inequalities of partial differential equations, Comm in analysis and geometry, (1994) 431-450.
[43]J.Wang, Local Harnack estimate for mean curvature flow in Euclidean space, preprint.
[2]S.Brendle, A generalization of Hamilton's Harnack inequality for the Ricci flow, preprint
[3]T.Aubin, The scalar curvature,Differential Geometry and Relativity, edited by Cahen and Flato, Reider,1976.
[4]K.Brakke, The motion of a surface by its mean curvature, Mathematical Notes 20, Princeton University Press, Princeton, NJ,1978.
[5]H.D.Cao, Deformation of Kdhler metrics to Kdhler-Einstein metrics on com-pact Kdhler manifolds, Invent. Math,81(1985),359-372.
[6]H.D.Cao, On Harnack's inequalities for the Kdhler-Ricci flow, Invent.math. 109 (1992) 247-263.
[7]H.D.Cao, Limits of solutions to the Kahler-Ricci flow, J.Diff.Geom,45(1997),257-272.
[8]H.D.Cao and N.Li, Matrix Li-Yau-Hamilton estimates for the heat equation on Kahler manifolds, Math.Ann.,331 No.4(2005),795-807.
[9]H.D.Cao, X.P.Zhu, A complete proof of the Poincare and Geometrization conjectures—Application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. Vol.10, No.2, (2006) 165-492.
[10]X.D.Cao, Differential Harnack estimates for backward heat equations with po-tentials under the Ricci flow, J.Funct.Anal.255 No.4(2008),1024-1038.
[11]CHENG, S. Y. Yau, S. T., Differential equations on Riemanian manifolds and their geometric applications. Comm. Pure Appl. Math,33 (1980),199-211.
[12]B.chow, On Harnack's inequality and entropy fot the Gaussian curvature flow, Comm.Pure Appl.Math,44(1991),469-483.
[13]K.Ecker and G.Huisken, Mean curvature evolution of entire graphs, Ann.Math, 130(1989),453-471.
[14]K.Ecker and G.Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent.Math,105(1991),547-569.
[15]M.Gage and R.Hamilton, The heat equation shrinking convex plane curves, J.Diff.Geom,23(1996),69-96.
[16]Mihai Bailesteanu, X.D. Cao, Artem Pulemotov, Gradient estimates for the heat equation under the Ricci flow, Arxiv:0910.1053vl[math. DG].
[17]R.Hamilton, The Ricci flow on surfaces, Contemp Math 71,Amer.Math.Soc, Providence, RI,(1988),237-262.
[18]R.Hamilton, A matrix Harnack estimate for the heat equation, Comm.Anal.Geom,1(1993),113-126.
[19]R.Hamilton, The Harnack estimate for the Ricci flow, J.Diff.Geom,37(1993),113-126.
[20]R.Hamilton, Eternal solutions to the Ricci flow, J.Diff.Geom,38(1993),1-11.
[21]R.Hamilton, The Harnack estimate for the mean curvature flow, J.Diff.Geom,41(1995),215-226.
[22]R.Hamilton, The formation of singularities in the Ricci flow, Surveys in Dif-ferential Geometry,2,International Press,(1995),7-136.
[23]R.Hamilton, Curvature and Volume Bounds Ⅱ, preprint.
[24]G.Huisken, Flow by mean curvature of convex surface into sphere, J.Diff.Geom,20(1984),237-266.
[25]G.Huisken, Contracting convex hypersurfaces in the Riemannian manifolds by their mean curvature, Invent.Math,84(1986),463-480.
[26]G.Huisken, Asymptotic behavior for singularities of the mean curbature flow,J.Diff. Geom,31(1990),285-299.
[27]G.Huisken and T.Ilmanen, The inverse mean curvature flow and the Rieman-nian Penrose inequality,59 No.3(2001),353-437.
[28]N.Li, A note Perelman's LYE inequality, arXiv:math.DG/0602337 v1 feb 15,2006.
[29]N.Li, A matrix Li-Yau-Hamilton estimate for Kdhler-Ricci flow, J.Diff.Geom, 75 No.2(2007),303-358.
[30]Peter Li, S.-T. Yau, On the parabolic kernel of the Kdhler Schrodinger oper-ator, Acta Math.156(1986) 153-201.
[31]N.Mok, The uniformization theorem for compact Kdhler manifolds of non-negative holomorphic bisectional curvature, J.Diff.Geom,27(1988),179-214.
[32]Mori,S, Projective manifolds with ample tangent bundles, Ann.Math.110,593-606(1979)
[33]J.Moser, A Harnack inequality for parabolic differential equations, Com-mun.Pure Appl.Math,17(1964),101-134.
[34]G.Perelman, The entropy formula for the Ricci flow and its geometric appli-cations, arXiv:math.DG/0211159 vl November 11,2002.
[35]R.Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J.Diff.Geom.,20(1984),479-495.
[36]R.Schoen and S.T.Yau, Lectures on Differntial Geometry, in Conference Pro-ceedings and Lecture Notes in Geometry and Topology,Vol.l, International Press Publications,1994.
[37]F.Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math.Z,251(2005),721-733.
[38]Siu,Y.-T,Yau,S.-T, Compact Kahler manifolds of positive bisectional curva-ture, Invent.Math.59,189-204(1980).
[39]K.Smoczyk, Harnack inequalities for curvature flows depending on mean cur-vature, New York J.Math,3(1997),103-118.
[40]N.Trudinger, Remarks concerning the conformal deformation of Rieman-nian structures on compact manifolds, Ann. Scuola Norm.Sup. Pisz Cl.Sci.(4),3(1968),265-274.
[41]Yau, S. T, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math,28 (1975),201-228.
[42]Yau, S. T, On the Harnack inequalities of partial differential equations, Comm in analysis and geometry, (1994) 431-450.
[43]J.Wang, Local Harnack estimate for mean curvature flow in Euclidean space, preprint.