Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications

TL;DR

This paper develops nonlocal estimates for the Volume Preserving Mean Curvature Flow (VPMCF), showing that singularities resemble ancient solutions and establishing the applicability of monotonicity methods at finite times, with insights into flow asymptotics.

Contribution

It introduces new nonlocal estimates for VPMCF, links singularity blowups to ancient solutions, and demonstrates the use of monotonicity methods at finite times.

Findings

Blowups of finite time singularities are ancient solutions to MCF.

Monotonicity methods can be applied at finite times.

Provides asymptotic behavior insights of the flow.

Abstract

We obtain estimates on nonlocal quantities appearing in the Volume Preserving Mean Curvature Flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time singularities of VPMCF are ancient solutions to Mean Curvature Flow (MCF), prove that monotonicity methods may always be applied at finite times and obtain information on the asymptotics of the flow.