Flow Study Of The Shape On The Harnack Inequality

In 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 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 HereCorollary 1.1.1 Under the same condition in Theorem 1.1.1, the scalar cur-vature R satisfies the estimateCorollary 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, 00 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.