Extension of two-dimensional mean curvature flow with free boundary

TL;DR

This paper proves that a 2D mean curvature flow with free boundary conditions can be extended indefinitely if its mean curvature and perimeter remain bounded, under certain geometric conditions.

Contribution

It establishes a new extension criterion for free boundary mean curvature flows based on curvature and perimeter bounds.

Findings

Flow can be extended as long as mean curvature remains bounded.

Flow can be extended as long as perimeter remains bounded.

Extension depends on support surface being mean convex and smooth.

Abstract

Given a mean curvature flow of compact, embedded $C^2$ surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.