Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type

Abstract

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝn + 1 remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1, and convex. Our results generalize the well‐known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc.