The Existence of a Weak Solution to Volume Preserving Mean Curvature Flow in Higher Dimensions

Abstract

In this paper, we construct a family of integral varifolds, which is a global weak solution to the volume preserving mean curvature flow in the sense of $$L^2$$-flow. This flow is also a distributional BV-solution for a short time, when the perimeter of the initial data is sufficiently close to that of a ball with the same volume. To construct the flow, we use the Allen–Cahn equation with a non-local term motivated by studies of Mugnai, Seis, and Spadaro, and Kim and Kwon. We prove the convergence of the solution for the Allen–Cahn equation to the family of integral varifolds with only natural assumptions for the initial data.