Harnack inequalities for evolving hypersurfaces
Harnack inequalities for evolving hypersurfaces
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DOI：
10.1007/bf02571941
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发表时间：
1994-09
影响因子：
0.8
通讯作者：
B. Andrews
中科院分区：
文献类型：
--
作者：
B. Andrews
抛物线方程的 Harnack 不等式起源于 Moser [M] 的工作，他处理了线性散度形式方程的情况。在这种情况下，不等式根据抛物线域早期区域获得的值从下面估计解。这种类型的不等式最近出现在许多几何演化方程中，包括几个拟线性和完全非线性的例子。这些新进展始于 Li 和 Yau [LY]，他们展示了如何通过巧妙地使用抛物线极大值原理来获得热方程的 Harnack 不等式。汉密尔顿也采用了类似的技术，他证明了各种非线性演化方程的哈纳克不等式——二维里奇曲率的黎曼度量流[Hall，欧几里得空间中超曲面的平均曲率流，以及几个标量方程[Ha2]。 Chow 通过高斯曲率的幂 [Ch3] 处理了欧几里得空间中的超曲面流，还通过 Yamabe 泛函的梯度处理了黎曼度量的流 [-Ch4]。最近，Hamilton 证明了高维 Ricci 曲率流的 Harnack 不等式 [Ha3]。在本文中，M" 将是一个紧凑、光滑的 n 维黎曼流形。我们考虑由映射 ~ 0:[0, T) xM"--* IR"+ 1 描述的平滑演化的单参数沉浸族，其中演化由以下形式的方程控制：
Harnack inequalities for parabolic equations originate with the work of Moser [M] who treated the case of linear divergence-form equations. In this context, the inequality estimates a solution from below, in terms of the values it attains on an earlier region of the parabolic domain. Inequalities of this type have recently appeared for many geometric evolution equations, including several quasilinear and fully nonlinear examples. These new developments began with Li and Yau [LY], who showed how to obtain a Harnack inequality for the heat equation by clever use of the parabolic maximum principle. Similar techniques were employed by Hamilton, who proved Harnack inequalities for various nonlinear evolution equations-the flow of Riemannian metrics by their Ricci curvature in two dimensions [Hall, the mean curvature flow of hypersurfaces in Euclidean space, and several scalar equations [Ha2]. Chow has treated flows of hypersurfaces in Euclidean space by powers of the Gauss curvature [Ch3], and also the flow of Riemannian metrics by the gradient of the Yamabe functional [-Ch4]. Most recently, Hamilton has proved a Harnack inequality for the higher-dimensional Ricci curvature flow [Ha3].Throughout this paper, M" will be a compact, smooth n-dimensional Riemannian manifold. We consider a smoothly evolving one-parameter family of immersions described by a map~ 0:[0, T) xM"--* IR"+ 1, where the evolution is governed by an equation of the following form: