Convexity estimates for mean curvature flow with free boundary

TL;DR

This paper extends convexity estimates for mean curvature flow with free boundary conditions, proving new results for singularities and providing an alternative proof for flow behavior in spherical barriers.

Contribution

It generalizes convexity estimates to free boundary flows in arbitrary embedded surfaces and offers a new proof for flow pinching in spherical barriers.

Findings

Convexity estimates hold for mean curvature flow with free boundary in bounded geometry surfaces.

Finite-time singularities exhibit convexity properties similar to classical flows.

Flow in spherical barriers pinches to umbilic, confirming expected geometric behavior.

Abstract

We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier $S$. Here $S$ can be any properly embedded, oriented surface in $R^{n+1}$ of bounded geometry. We also give an alternative proof that convex mean curvature flows with free boundary in $S^n$ pinch to umbilic.