# Non-fattening of mean curvature flow at singularities of mean convex type

## TL;DR

This paper proves that mean curvature flow remains unique and non-fattening past singularities if all singularities are of mean convex type, extending known results to more general singularities using local information.

## Contribution

It establishes non-fattening of level set flow at singularities of mean convex type, generalizing previous results and providing new local criteria for non-fattening.

## Key findings

- Level set flow does not fatten at mean convex singularities.
- Uniqueness of flow is maintained past singularities of mean convex type.
- First result linking local singularity structure to non-fattening.

## 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.