Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Genus one singularities in mean curvature flow

Adrian Chun-Pong Chu, Ao Sun

TL;DR

The paper develops a novel topology-aware framework for mean curvature flow by introducing homology descent, termination, and breakage to track how loops in the complement of a flowing surface disappear at cylindrical and spherical singularities. Using this framework, it proves that for certain one-parameter torus families, a genus one singularity must appear and is robust to perturbations; it further constructs an embedded genus one self-shrinker with entropy below the shrinking doughnut and derives consequences for entropy-minimization and ancient/eternal flows. The results bridge topological changes with singularity models, showing that genus-one phenomena can be forced by bifurcation-like arguments in MCF and offering new insights into the landscape of genus-one self-shrinkers and their entropies. Overall, the work advances understanding of how topology and entropy constrain singularities in three-dimensional mean curvature flow and opens avenues for Genus-one minimization and generic singularity theory in geometric flows.

Abstract

We show that for certain one-parameter families of initial conditions in $\mathbb R^3$, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut.

Genus one singularities in mean curvature flow

TL;DR

The paper develops a novel topology-aware framework for mean curvature flow by introducing homology descent, termination, and breakage to track how loops in the complement of a flowing surface disappear at cylindrical and spherical singularities. Using this framework, it proves that for certain one-parameter torus families, a genus one singularity must appear and is robust to perturbations; it further constructs an embedded genus one self-shrinker with entropy below the shrinking doughnut and derives consequences for entropy-minimization and ancient/eternal flows. The results bridge topological changes with singularity models, showing that genus-one phenomena can be forced by bifurcation-like arguments in MCF and offering new insights into the landscape of genus-one self-shrinkers and their entropies. Overall, the work advances understanding of how topology and entropy constrain singularities in three-dimensional mean curvature flow and opens avenues for Genus-one minimization and generic singularity theory in geometric flows.

Abstract

We show that for certain one-parameter families of initial conditions in , when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut.
Paper Structure (29 sections, 38 theorems, 50 equations, 13 figures)

This paper contains 29 sections, 38 theorems, 50 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main ideas: Change in homology under MCF
  3. Heuristic observation
  4. Three new concepts
  5. Structure of cylindrical singularities
  6. Outline of proofs
  7. Theorem \ref{['thm:main1']}
  8. Theorem \ref{['thm:main2']}
  9. Theorem \ref{['thm_genus1_least_entropy']}
  10. Theorem \ref{['thm_eternal']}
  11. Open questions
  12. Organizations.
  13. Preliminaries
  14. Weak solutions of MCF
  15. Setting and notations
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\{M^s\}_{s\in [0,1]}$ be a smooth family of tori in $\mathbb R^3$ such that for the MCF starting from $M^0$ (resp. $M^1$), the inward (resp. outward) torus neck will pinch. Then there exists $s_0\in [0,1]$ such that the MCF starting from $M^{s_0}$ would develop a singularity that is not multipl

Figures (13)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4: The picture at time $t$, for all $t<T$ sufficiently close to $T$.
  • Figure 5:
  • ...and 8 more figures

Theorems & Definitions (84)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Definition 1.9
  • Conjecture 1.10
  • ...and 74 more