Subsequent Singularities in Mean-Convex Mean Curvature Flow

TL;DR

This paper proves that mean convex hypersurfaces in Euclidean space become nearly convex near singularities during mean curvature flow, extending previous results to higher dimensions using elliptic regularization.

Contribution

It extends the understanding of convexity behavior near singularities in mean curvature flow to all dimensions using Ilmanen's elliptic regularization.

Findings

Near singularities, surfaces are nearly convex in spacetime neighborhoods.

The result applies to all dimensions, not just n<7.

Uses elliptic regularization to analyze singularity neighborhoods.

Abstract

We use Ilmanen's elliptic regularization to prove that for an initially smooth mean convex hypersurface in Euclidean n-space moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. Previously this was known only (i) for n < 7, and (ii) for arbitrary n up to the first singular time.