The size of the singular set in mean curvature flow of mean-convex sets
The size of the singular set in mean curvature flow of mean-convex sets
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DOI：
10.1090/s0894-0347-00-00338-6
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发表时间：
2000-04
影响因子：
3.9
通讯作者：
B. White
中科院分区：
文献类型：
--
作者：
B. White
在本文中，我们研究了当正平均曲率的超曲面以每个点都等于该点表面的平均曲率的速度移动时形成的奇点。最方便的是用 Chen-Giga-Goto [CGG] 和 Evans-Spruck [ES] 的水平集流（也称为“最大流”[I2]）来描述结果。在水平集流下，R 中的任何闭集 K 都会生成闭集 Ft(K) (t ≥ 0) 的单参数族，其中 F0(K) = K。如果 K 的边界是光滑紧致超曲面，则在某个区间 [0, ) 中 t 的 Ft(K) 边界也是如此，对于这样的 t，演化与偏微分方程和微分几何经典定义的平均曲率运动一致（如[H1]）。然而，如果 K 是紧的，那么 Ft(K) 的边界必然在某个有限时间变得奇异。我们的目标是证明奇异集必然非常小。假设初始集合 K 是紧致的且均凸的，即对于所有 t > 0，Ft(K) ⊂ Interior(K)。如果 M = ∂K 是平滑超曲面，则我们有以下等价的均凸特征：当且仅当 M 相对于向内单位法线的平均曲率处处非负时，K 才是均凸。给定一个紧集 K，令 K 为 Ft(K) 扫过的时空区域：K = {(x, t) ∈ R ×R : t ≥ 0, x ∈ Ft(K)}。 (*) 如果 (1) X 的邻域中 K 是有边界的光滑流形，且 (2) X 处 ∂K 的切平面不是水平的（即不是 R × [0]），则 K 边界中的点 X = (x, t) 称为正则点。请注意，如果 X = (x, t) 是正则点，则在 x 的邻域中，Ft(K) 是 R 中的有边界光滑流形。∂K 中 t > 0 的点 X = (x, t) 不是正则点，称为奇异点。
In this paper, we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point. It is most convenient to describe the results in terms of the level set flow (also called “biggest flow” [I2]) of Chen-Giga-Goto [CGG] and Evans-Spruck [ES]. Under the level set flow, any closed set K in R generates a one-parameter family of closed sets Ft(K) (t ≥ 0) with F0(K) = K. If the boundary of K is a smooth compact hypersurface, then so is the boundary of Ft(K) for t in some interval [0, ), and for such t’s the evolution coincides with motion by mean-curvature as defined classically by partial differential equations and differential geometry (as in [H1]). However, if K is compact, then the boundary of Ft(K) will necessarily become singular at some finite time. Our goal is to show that the singular sets are necessarily quite small. This we do provided the initial set K is compact and mean-convex in the sense that Ft(K) ⊂ interior(K) for all t > 0. In case M = ∂K is a smooth hypersurface, we have the following equivalent characterization of mean-convexity: K is mean-convex if and only if the mean-curvature of M with respect to the inward unit normal is everywhere non-negative. Given a compact set K, let K be the region in spacetime swept out by the Ft(K): K = {(x, t) ∈ R ×R : t ≥ 0, x ∈ Ft(K)}. (∗) A point X = (x, t) in the boundary of K is called a regular point if (1) X has a neighborhood in which K is a smooth manifold-with-boundary, and (2) the tangent plane to ∂K at X is not horizontal (i.e., is not R × [0]). Note that if X = (x, t) is a regular point, then in a neighborhood of x, Ft(K) is a smooth manifold-with-boundary in R. A point X = (x, t) in ∂K with t > 0 that is not a regular point is called a singular point.