Deforming a hypersurface by its Gauss-Kronecker curvature
Deforming a hypersurface by its Gauss-Kronecker curvature
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DOI：
10.1002/cpa.3160380615
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发表时间：
1985-11
影响因子：
3
通讯作者：
K. Tso
中科院分区：
文献类型：
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作者：
K. Tso
最近，人们对研究曲线缩短问题很感兴趣：考虑平面上一条简单的闭合曲线，并以与其曲率相等的速度沿内法线变形。我们希望确定这种变形的存在及其极限行为。M.Gage[6]，[7]证明了当曲线是严格凸的时，变形减小了等周比，而且，如果它收缩到点p，则通过将以p为中心的曲线“爆破”以使其封闭面积为T而得到的归一化曲线一定在某种意义上趋向于单位圆。R.哈密尔顿[9]证明了变形的存在，即曲线保持光滑和凸且收缩到时间A/27r处的一点，其中A是初始曲线的封闭面积。这个问题可以推广到高维情形：给定一个闭超曲面X(S)，S E S“，我们研究了初值问题AX-(S，t)=-k(S，t)N(S，t)的解，其中k是Gauss-Kronecker曲率，N是X(S，t)处的外单位法线。变量S E S“仍然是闭超曲面X(S，t)的一个参数化子。本文证明了定理0.1。设X(S)，S E S“，是W*+I中的光滑、闭、严格凸超曲面，则(0.1)有唯一的光滑解X(S，t)，它是[0，T*)中每个t的严格凸超曲面，其中T*=V/An，V是X(S)所包围的体积，An是S的表面积.此外，对于t‘>；t，X(.，t’)被X(.，t)严格封闭.当t趋于T*时，超曲面收缩到顶点.
Recently there has been some interest in studying a curve-shortening problem: Consider a simple closed curve in the plane and deform it along the inner normal at a rate equal to its curvature. We wish to establish the existence of this deformation and its limiting behavior. M. Gage [6],[7] has shown that when the curve is strictly convex the deformation decreases the isoperimetric ratio and, furthermore, if it shrinks to a point p, the normalized curves, obtained by “blowing up” the curves centered at p so that its enclosed area is T, must tend to the unit circle in a certain sense. R. Hamilton [9] proved that the deformation exists, that is, the curves stay smooth and convex and shrink to a point at time A/27r, where A is the enclosed area of the initial curve. This problem can be generalized to higher dimensions as follows: Given a closed hypersurface X (s), s E S”, we study the solution of the initial value problem: ax-(s, t)=-k (s, t) N (s, t), at where k is the Gauss-Kronecker curvature and N is the outer unit normal at X (s, t). The variable s E S “should remain as a parametrization of the closed hypersurfaces X (s, t) for all t. In this paper we shall proveTHEOREM 0.1. Let X (s), s E S “, be a smooth, closed, strictly-convex hypersurface in W*+ I. Then (0.1) has a unique, smooth solution X (s, t) which is a strictly convex hypersurface for each t in [0, T*) where T*= V/an, V is the volume enclosed by X (s), and an is the surface area of S”. Furthermore, X (., t’) is strictly enclosed by X (., t) for t ‘> t. As t tends to T*, the hypersurfaces shrink to apoint.