Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

TL;DR

This paper proves that volume-preserving mean curvature flows in two and three dimensions become smooth after finite time and converge exponentially to a sphere, strengthening previous results on their regularity and convergence.

Contribution

It establishes finite-time smoothness and exponential convergence of volume-preserving mean curvature flows, using a new epsilon regularity theorem and PDE estimates at the discrete scheme level.

Findings

Flow becomes smooth after finite time.

Flow converges exponentially fast to a sphere.

Results hold in both planar and three-dimensional cases.

Abstract

The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.