Consistency of the flat flow solution to the volume preserving mean curvature flow

TL;DR

This paper proves that the flat flow solution for volume preserving mean curvature flow starting from a smooth set matches the classical solution until the first singularity, establishing uniqueness and regularity of the flat flow.

Contribution

It demonstrates the consistency and full regularity of the flat flow solution, providing an alternative proof without using the comparison principle.

Findings

Flat flow agrees with classical solution before singularity.

Flat flow is unique and smooth up to the first singular time.

Method applies to mean curvature flow, offering an alternative proof.

Abstract

We consider the flat flow solution, obtained via discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from C^{1,1}-regular set. We prove the consistency principle which states that (any) such flat flow agrees with the classical solution as long as the latter exists. In particular, the flat flow is unique and smooth up to the first singular time. We obtain the result by proving the full regularity for the discrete time approximation of the flat flow such that the regularity estimates are stable with respect to the time discretization. Our method can also be applied in the case of the mean curvature flow and thus it provides an alternative proof, not relying on comparison principle, for the consistency between the flat flow solution and the classical solution for C^{1,1}-regular initial sets.