The size of the singular set in mean curvature flow of mean-convex sets

Abstract

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.