A Study Of Harnack's Inequalities For Some Geometric Flows

In this paper we mainly study the local Harnack estimate for Yamabe flow on locally conformally flat manifold, Harnack estimates for curvature flows depend-ing on mean curvature on hypersurfaces of Euclidean spaces, Harnack estimates of heat equations with potentials on Kahler manifolds and their applications.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. There are two most important examples; one is the proof of the Riemannian Penrose inequality given by Huisken and Ilmanen[28] by using the inverse mean curvature flow, and the other is the proof of Poincare conjecture given by Huaidong Cao and Xiping Zhu[8]by using Ricci flow.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 equations originated from the work of Moser[33] 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 [31]. 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[19,20,22]. Chow obtained similar inequalities for Gauss curvature flow of hypersurface on Eu-clidean space and Yamabe flow respectively in 1991 and 1992[12,13]. Moreover, in 1992 Cao got the Harnack estimate for Kahler-Ricci flow on Kahler manifold[5]. 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[7,9,10,29,30]. In this paper we get the following results on the basis of their works.In Chapter one we study the local Harnack inequality of Yamabe flow on locally conformally flat manifold and give its corollary (the nonconic estimate of Yamabe flow). Let (Mn,go) be a smooth complete locally conformally flat n-dimensional manifold. The Yamabe flow is given by for x?Mn,t?0, and where R(x,t) is the scalar curvature of g(x, t).The Yamabe flow was originally conceived to attack the Yamabe problem. Yamabe problem states that given a compact Riemannian manifold (M, g0), there exists a metric g pointwise conformal to go with constant scalar curvature. In 1968 Trudinger [41] pointed out the proof given by Yamabe was wrong and corrected his proof in the case of non-positive scalar curvature. Aubin[2] proved it when M is not conformally flat, dimM?6 and the scalar curvature is positive in 1976. Schoen [35] gave the complete proof of the problem in 1984. There are several researches on Yamabe flow as follows. In 1988 Hamilton solved the convergence of the solution to Yamabe flow on two dimensional case[18]. In 1992 Chow showed that on a compact locally conformally flat manifold the solution to the normalized Yamabe flow exists for all time and converges in C?-norm to constant curvature metric, and obtained its Harnack estimate[13]. Yugang Ye gave the proof of the global existence of solution to Yamabe flow in [44].Recently, Hamilton proved the local Harnack estimate of Ricci flow, and derived the nonconic estimate from it. On his report?Curvature and Volume Bounds?[24] he applied the nonconic estimate to prove "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 solution to the equation (?)/(?)tgij= -2Rij,t?[0,T). U (?) Mn is an open connected set, and on U×[0, t0],t0< T the curvature satisfies the following condition:(?) Co and (?) M> 0 where O?U. If Mr2=C1 such that Br(O,t0) (?) U, then (?)(p,t)?Br/2(O,t0)×[t0-r2/4 and (?) V?TpMn, we have DR(V)2?CM2(Rc(V, V)+Cm|V|2). where C> 0 depending on n and C1.Jie Wang considered the mean curvature flow of hypersurface on Euclidean space and got its local Harnack estimate and nonconic estimate[43].Theorem B(Local Harnack inequality of mean curvature flow):If on Br(O, t)×[0, R2] we have the curvature condition where O?Mn, C0= M R, then for (?)(x,t)?BR/2(O,t)×[0,R2], (?) V?TpMn, we can find a positive constant B depending only on n and Co, s.t. the Local Harnack inequality holds,We consider a similar problem on Yamabe flow in Chapter one, and get Theorem 1.1.1 and Corollary 1.1.1. We denote the following curvature condition by (*):Theorem 1.1.1 (Local Harnack inequality of Yamabe flow):Suppose (Mn,g(x,t)) is a n-dimensional smooth complete locally conformally flat solu-tion to Yamabe flow for t?[0, r2], the curvature satisfies the condition (*) on Br(O,t) x [0,r2], then at (?) (x,t)?Br/2(O,t)×[0,r2], (?) V?TxMn, we can find some constant B> 0, depending only on n, such that the following local Harnack estimate holds,Corollary 1.1.1(Nonconic estimate of Yamabe flow):Under the same conditions as Theorem 1.1.1, at point (O,r2), we have(1)|(?)R(V)|2?C1M2(Rab+m(t)gab)VaVb, M?1(2)|(?)R(V)|2?C2(Rab+m(t)gab)VaVb,0< M< 1(3)(?)R(V)=0, if [(?)tR+nm(t)R+BM(1+1/r2+1/t)](O,r2)= 0 or (Rab+ m(t)gab)VaVb(O,r2)= 0 where C1 and C2 depend only on n.In Chapter two we give the Harnack estimates of curvature flows of complete hypersurface in Euclidean space, where the normal velocity is given by a smooth function depending only on the mean curvature. By use of the estimates, we get some corollaries including the integral Harnack inequality. Meanwhile, we give the conditions, with which the solutions to the flows are a translation soliton and an expanding soliton respectively. So we generalize the results in [22,40,42].Let Mn be a smooth manifold without boundary, and let Fo:Mn?Rn+1 be a smooth immersion which is convex. We consider a smooth evolving one-parameter family of hypersurface immersions described by a map F(·,t):Mn x [0, T)?Rn+1, where the evolution is given by the following equation: where v is the unit inward normal and f is a smooth function depending only on the mean curvature H. Definition A:Let F:Mn?Rn+1 be an immersion.f:??R is a smooth function,where?(?)R. F is called admissible, if H(x)??,(?)x?M.There are many results on curvature flow of hypersurface in Euclidean space. There are some works related with the chapter as follows:Hamilton got the Harnack inequality of mean curvature flow in 1995 in [22]; Smoczyk [40] proved the short-time existence of the solution to f-flow under the condition of f1> 0 and the Harnack estimate of f-flow on compact hypersurface of Rn+1; In [42] Jie Wang considered the Hk-flow on complete hypersurface of Euclidean space and got the same estimate. We generalize their results and obtain Theorem 2.1.1 in the chapter.Throughout this chapter, we denote Mt as the admissible solution to the flow, and t?[0, T) on which the solution exists. We always assume the solution satisfies the following condition (**) compact or complete with bounded|A|,|DA|,|D2A| at each time t, where A is the second fundamental form of Mt.Theorem 2.1.1:Assume that F0:Mn?Rn+1 is an admissible smooth and convex immersion, Mt is convex under the condition(**) and that f:[0,+?)?R is a smooth function such that for all x?[0,.+?) we have where a?R is a constant. Then we can find a small positive constant d such that holds for all tangent vectors V as long as t?[0, T) and d+(a+2)t> 0, where we have set c(t):=1/d+(1+2)t.For Ricci flow, Hamilton got three types of singularity model by blowing up in [23] and classified the type I singularity model in detail; He also proved the type II singularity model is a Ricci soliton in [21]; Huaidong Cao showed that the type III singularity model is an expanding Ricci soliton in [6]. For mean curvature flow, Huisken discussed the classification of singularties by blowing up and gave the detailed classification of the type I singularity model in [27]; Hamilton proved that the type II singularity model is a translating soliton in [22]; Binglong Chen got the conclusion that the type III singularity model is also an expanding soltion in [11]. For Hk-flow, Weiming Sheng and Chao Wu considered the compact manifolds in [39], obtained the gradient estimate of Hk flow, gave the monotonicity formula of the rescaling manifold in case the type I singularity model appears, and got the structure of the type I singularity model by the formula; Jie Wang proved that the type II and type III singularity model of Hk-flow are a Hk-translating soliton and a Hk-expending soliton respectively in [45]. We obtain the similar results on f-flow in the remaining parts of the chapter.Theorem 2.1.2:If f satisfies the assumptions of Theorem 2.1.1 and a+2> 0, then any strictly convex eternal solution under the condition (**) to the f-flow where the mean curvature attains its maximum value at some point in space-time, it must be a translation soliton.Theorem 2.1.3:If f satisfies the assumptions of Theorem 2.1.1 and a+2> 0; then any strictly convex solution under the condition (**) to the f-flow which exists for 0<t<+?, where (d+(a+2)t)Ha+2 attains its maximum value at some point in space-time, it must be an expanding soliton.In Chapter three we discuss the Harnack estimates of heat equations with potentials on Kahler manifolds and their applications.Let (M, go) be a n-dimensional compact Kahler manifold with positive holo-morphic bisectional curvature. Suppose gij(t) is one-parameter family of Kahler metric and satisfies the following Kahler-Ricci flow equation: where Rij is Ricci curvature of the metric gij.We consider the differential Harnack inequality for positive solution of parabolic equation on M where f:M×[0, T)?R+ is smooth.Perelman gave the proof of Harnack inequality for positive fundamental solution of conjugate heat equation under Ricci flow in [34]. Xiaodong Cao studied the Harnack inequalities for positive solutions of backward heat equations with potentials under Ricci flow in [9] and got the same results with Perelman. Recently, he and Hamilton proved the same inequalities for positive solutions of heat equations with potentials under Ricci flow [10]. In the chapter we derive the similar estimates on Kahler manifold under Kahler-Ricci flow and obtain the following theorems.Theorem 3.1.1:Let M be n-dimensional compact Kahler manifold, g(t) satisfies the above Kahler-Ricci flow with positive holomorphic bisectional curva-ture. Let f is a positive solution to heat equation (?)f/(?)t=?g(t)f+(R-n)f on M. Suppose then for (?) t?(0, T), we have P?0.Theorem3.1.2:Let M be n-dimensional compact Kahler manifold, g(t) satisfies the above Kahler-Ricci flow with positive holomorphic bisectional curva-ture. Let f is a positive solution to heat equation (?)f/(?)t=?g(t)f+(R - n)f on M. Suppose where d is a constant, then for (?) t?(0, T), we have Moreover, max((et - 1)Q) is non-increasing on t. Remark:The above two theorems hold on complete noncompact Kahler manifolds with nonnegative holomorphic bisectional curvature if we give some bounded assumptions.