Non-collapsing in Mean-convex Mean Curvature Flow

We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius inversely proportional to the mean curvature at that point, then this remains true for all positive times in the interval of existence. We follow [4] in defining a notion of ’non-collapsing’ for embedded hypersurfaces as follows: Recall that a hypersurface M is called mean-convex if the mean curvature H of M is positive everywhere. Definition 1. A mean convex hypersurface M bounding an open region Ω in R is δ-non-collapsed (on the scale of the mean curvature) if for every x ∈ M there there is an open ball B of radius δ/H(x) contained in Ω with x ∈ ∂Ω. It was proved in [4] that any compact mean-convex solution of the mean curvature flow is δ-non-collapsed for some δ > 0. Closely related statements are deduced by Brian White in [6]. In both of these works the result is derived only after a lengthy analysis of the properties of solutions of mean curvature flow. The purpose of this paper is to provide a self-contained proof of such a non-collapsing result using only the maximum principle. It is first necessary to reformulate the non-collapsing condition to allow the application of the maximum principle. Given a hypersurface M = X(M̄), define a function Z on M ×M by Z(x, y) = H(x) 2 ‖X(y)−X(x)‖ + δ 〈X(y)−X(x), ν(x)〉 . Then we have the following characterization: Proposition 2. M is δ-non-collapsed if and only if Z(x, y) ≥ 0 for all x, y ∈ M̄ . Proof. By convention we choose the unit normal ν to be outward-pointing, so that a ball in Ω of radius δ/H(x) with X(x) as a boundary point must have centre at the point p(x) = X(x)− δ H(x)ν(x). The statement that this ball is contained in Ω is equivalent to the statement that no points of M are of distance less than δ/H(x) from p: 0 ≤ ‖X(y)− p(x)‖ − ( δ H(x) 2 = 2H(x)Z(x, y) for all x and y in M̄ . Since H > 0 this is equivalent to the statement that Z is non-negative everywhere. The converse is clear. The main result of this paper is the following: 1