Existence of weak solution to volume preserving mean curvature flow in higher dimensions

TL;DR

This paper establishes the existence of a global weak solution to volume preserving mean curvature flow in higher dimensions using a varifold approach and the Allen--Cahn equation, demonstrating convergence under natural initial conditions.

Contribution

It introduces a novel construction of weak solutions via integral varifolds for volume preserving mean curvature flow in higher dimensions, utilizing a non-local Allen--Cahn equation approach.

Findings

Constructed a family of integral varifolds as weak solutions.

Proved convergence of Allen--Cahn solutions to varifolds.

Established solutions under initial data close to a sphere.

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 ball with the same volume. To construct the flow, we use the Allen--Cahn equation with 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.