A local regularity theorem for mean curvature flow with triple edges

TL;DR

This paper establishes a regularity theorem for mean curvature flow with triple edges, showing that weak closeness to a static union of half-planes implies smoothness, and proves short-time existence from initial clusters with triple edges.

Contribution

It extends classical mean curvature flow regularity results to flows with triple edges and proves smooth short-time existence from initial configurations with such edges.

Findings

Weak closeness implies smooth regularity for flows with triple edges

Short-time existence of mean curvature flow starting from initial clusters with triple edges

Extension of regularity results to integral Brakke flows

Abstract

Mean curvature flow of clusters of n-dimensional surfaces in R^{n+k} that meet in triples at equal angles along smooth edges and higher order junctions on lower dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.