Mean convex mean curvature flow with free boundary

TL;DR

This paper extends White's regularity theory for mean convex mean curvature flow to include free boundary conditions, introducing new bounds and theorems to handle the complexities of free boundary flows.

Contribution

It develops a priori bounds for the ratio of second fundamental form to mean curvature and introduces Bernstein-type and sheeting theorems for free boundary flows.

Findings

Established a bound for the ratio of second fundamental form to mean curvature.

Proved a Bernstein-type theorem for low entropy free boundary flows.

Ruling out multiplicity 2 planes as tangent flows in free boundary settings.

Abstract

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish such a bound via the maximum principle for a triple-approximation scheme, which combines ideas from Edelen, Haslhofer-Hershkovits, and Volkmann. Other important new ingredients are a Bernstein-type theorem and a sheeting theorem for low entropy free boundary flows in a halfslab, which allow us to rule out multiplicity 2 (half-)planes as possible tangent flows and, for mean convex domains, as possible limit flows.