Contracting convex surfaces by mean curvature flow with free boundary on 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 halfsphere 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.

Classification of convex ancient free boundary mean curvature flows in the ball

“…In particular, Stahl proved that convex hypersurfaces with free boundary on a totally umbilic barrier remain convex and shrink to a "round" point on the barrier hypersurface. Hirsch and Li [14] proved that the same conclusion holds for "sufficiently convex" barriers in R 3 .…”