Non-fattening 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 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. This generalizes the well known fact that the level set flow of a mean convex initial hypersurface M does not fatten. This also provides the first instance where non-fattening is concluded from local information around the singular set.

arXiv comment19 pages. Some new material (see definition 10 and theorem 11) clarifying the nature of the hypotheses has been added