Contracting convex surfaces by mean curvature flow with free boundary on convex barriers

TL;DR

This paper studies the mean curvature flow of convex surfaces with free boundary on convex barriers, showing they contract to points and become asymptotically spherical, extending previous convergence results.

Contribution

It introduces a new perturbation method to establish convexity and pinching estimates for free boundary mean curvature flow on convex barriers.

Findings

Surfaces contract to a point in finite time.

Flow becomes asymptotic to a shrinking half-sphere.

Extension of Stahl's convergence result to general convex barriers.

Abstract

We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexity and pinching estimates for the flow. Our result can be compared to a celebrated convergence theorem of Huisken for mean curvature flow of convex hypersurfaces in Riemannian manifolds.