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The Genus-Decreasing Property of Mean Curvature Flow, I

Brian White

TL;DR

This work addresses how topology, specifically genus, evolves under mean curvature flow for a compact surface in a complete 3-manifold with a Ricci lower bound, under the assumption that all singularities arise from multiplicity-one shrinking spheres or cylinders. It develops a framework based on Brakke flow, tangent-flow analysis, and a key geodesic-compactness lemma to prove that the genus of the regular set $g(t)$ is nonincreasing up to the first possible fattening time $T_{pos}$, and even across singular times via lower semicontinuity. The paper provides global and localized variants showing genus monotonicity in open regions and under refined configurations, and supplements the core arguments with a detailed appendix establishing unit-regularity and the behavior of shrinkers at critical times. Overall, it strengthens topological control along mean curvature flow, laying groundwork for genus-based approaches in subsequent works and connecting to prior foundational results on flow with surgery and level-set formulations.

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.

The Genus-Decreasing Property of Mean Curvature Flow, I

TL;DR

This work addresses how topology, specifically genus, evolves under mean curvature flow for a compact surface in a complete 3-manifold with a Ricci lower bound, under the assumption that all singularities arise from multiplicity-one shrinking spheres or cylinders. It develops a framework based on Brakke flow, tangent-flow analysis, and a key geodesic-compactness lemma to prove that the genus of the regular set is nonincreasing up to the first possible fattening time , and even across singular times via lower semicontinuity. The paper provides global and localized variants showing genus monotonicity in open regions and under refined configurations, and supplements the core arguments with a detailed appendix establishing unit-regularity and the behavior of shrinkers at critical times. Overall, it strengthens topological control along mean curvature flow, laying groundwork for genus-based approaches in subsequent works and connecting to prior foundational results on flow with surgery and level-set formulations.

Abstract

This paper proves that, in mean curvature flow of a compact surface in a complete -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.
Paper Structure (10 sections, 21 theorems, 72 equations)

This paper contains 10 sections, 21 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Terminology and Preliminaries
  3. Generalities
  4. Spherical Singularites
  5. Cylindrical Singularities
  6. Geodesics
  7. Genus
  8. A Localized Version
  9. Another Localized Version
  10. Appendix

Key Result

Theorem 1

Suppose $M$ is a compact, smoothly embedded surface in a complete, smooth Riemannian $3$-manifold with Ricci curvature bounded below. Let $M(t)$ be the result of flowing $M$ for time $t$ by level set flow. Let $T_\textnormal{pos}=T_\textnormal{pos}(M)$ be the largest time $\le \infty$ such that is a standard Brakke flow (see Definition standard-definition), and such that all shrinkers to the flow

Theorems & Definitions (42)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 32 more