Asian Journal of Mathematics

Volume 27 (2023)

Number 2

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

Pages: 187 – 220



Sven Hirsch (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Martin Man-Chun Li (Department of Mathematics, Chinese University of Hong Kong, Shatin, N. T., Hong Kong)


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.


mean curvature flow, free boundary

Received 26 October 2021

Accepted 26 January 2023

Published 12 October 2023