Flat flow solution to the mean curvature flow with volume constraint

TL;DR

This paper revisits the construction of a global weak solution to volume-preserving mean curvature flow, proposing a direct implementation of the volume constraint in the discrete scheme, with a new density estimate proof.

Contribution

It introduces a novel approach to incorporate volume constraints directly into the discrete minimizing movement scheme for mean curvature flow.

Findings

Successful implementation of volume constraint in the discrete scheme

Proof of a new density estimate based on second variation

Enhanced understanding of volume-preserving mean curvature flow

Abstract

In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai-Seis-Spadaro (2016). This method is based on the gradient flow approach due to Almgren-Taylor-Wang (1993) and Luckhaus-Strurzenhecker (1995) and my aim is to replace the volume penalization by implementing the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on the second variation condition of the energy.