Singularities of Mean Curvature Flow of Surfaces

TL;DR

This paper characterizes the local structure of singularities in mean curvature flow of surfaces, showing tangent flows are smooth shrinkers and providing a new Gauss-Bonnet based technique for analysis.

Contribution

It introduces a local Gauss-Bonnet formula to analyze singularities, establishing smoothness and multiplicity properties of tangent flows in mean curvature flow.

Findings

Tangent flows at singularities are smooth, possibly branched shrinkers.

Embedded initial surfaces in 3D have smooth, embedded shrinkers without branch points.

A new local Gauss-Bonnet formula is developed for analyzing singularities.

Abstract

This paper proves that, at the first singular time for a smoothly immersed surface moving by mean curvature flow in a n-manifold, each tangent flow is given by a smooth, branched shrinker, possibly with multiplicity. If n=3 and if the initial surface is embedded, then the shrinker is smoothly embedded without branch points, but possibly with multiplicity. A key ingredient of the proof is a new, local version of the Gauss-Bonnet formula.