The Genus-Decreasing Property of Mean Curvature Flow, I

TL;DR

This paper demonstrates that for certain mean curvature flows of compact surfaces, the genus of the regular set decreases over time under specific singularity conditions, with additional local results supporting this behavior.

Contribution

It establishes the genus-decreasing property of mean curvature flow for surfaces with particular singularities, extending understanding of topological changes during the flow.

Findings

Genus decreases over time under specified singularity conditions

Proves local versions of the genus-decreasing property

Applicable to mean curvature flow in 3-manifolds with Ricci curvature bounds

Abstract

This paper proves that, in mean curvature flow of a compact surface in a complete $3$-manifold with Ricci curvature bounded below, the genus of the regular set is a decreasing function of time as long as the only singularities are given by shrinking sphere and shrinking cylinder tangent flows. The paper also proves some local versions of that fact.