Mean curvature flow with triple junctions in higher space dimensions

TL;DR

This paper studies the evolution of n-dimensional surface clusters with triple junctions under mean curvature flow, establishing local well-posedness using new parametrization and regularity methods in higher dimensions.

Contribution

It introduces a novel parametrization and a new existence and regularity approach for mean curvature flow with triple junctions in higher dimensions.

Findings

Proves local well-posedness of the flow

Develops a new parametrization of surface clusters

Establishes regularity results for the flow

Abstract

We consider mean curvature flow of n-dimensional surface clusters. At (n-1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120 degree angle condition. Using a novel parametrization of evolving surface clusters and a new existence and regularity approach for parabolic equations on surface clusters we show local well-posedness by a contraction argument in parabolic Hoelder spaces.