On the mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow

TL;DR

This paper analyzes the local geometry near singularities in mean curvature flow, confirming that neighborhoods around blowup points are mean convex and isolated, supporting Ilmanen's conjecture.

Contribution

It provides a detailed description of neighborhoods around singularities in mean curvature flow, confirming mean convexity and isolation of singularities.

Findings

Neighborhoods around blowup points are mean convex.

Singularities are isolated from each other.

Supports Ilmanen's conjecture on singularity structure.

Abstract

We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isolated from the other ones.