Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generating Shrinkers by Mean Curvature Flow

David Hoffman, Francisco Martin, Brian White

TL;DR

This paper develops a flow-driven method to realize shrinkers as singularity blowups by flow-constructing initial surfaces and analyzing their mean curvature evolution. It connects with and extends existing desingularization and minimax approaches, producing new families with Platonic and polygonal symmetry, including one-ended and three-ended configurations in S^2 x R. By establishing topological and geometric controls on genus and ends, the authors prove that these shrinkers arise from smooth flows and analyze their limits as genus grows, obtaining entropy bounds and convergence to planar configurations. Overall, the work broadens the catalog of shrinkers, demonstrates flow-based blowup realizations, and deepens the link between mean curvature flow singularities and self-similar shrinkers in geometric analysis.

Abstract

We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.

Generating Shrinkers by Mean Curvature Flow

TL;DR

This paper develops a flow-driven method to realize shrinkers as singularity blowups by flow-constructing initial surfaces and analyzing their mean curvature evolution. It connects with and extends existing desingularization and minimax approaches, producing new families with Platonic and polygonal symmetry, including one-ended and three-ended configurations in S^2 x R. By establishing topological and geometric controls on genus and ends, the authors prove that these shrinkers arise from smooth flows and analyze their limits as genus grows, obtaining entropy bounds and convergence to planar configurations. Overall, the work broadens the catalog of shrinkers, demonstrates flow-based blowup realizations, and deepens the link between mean curvature flow singularities and self-similar shrinkers in geometric analysis.

Abstract

We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.

Paper Structure

This paper contains 11 sections, 52 theorems, 254 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Oriented Flows
  4. Platonic and Regular Polygon Shrinkers
  5. A Topological Lemma
  6. Surfaces in S2 x R
  7. Shrinkers with one end
  8. Shrinkers with three ends
  9. Limits of Shrinkers
  10. Connectivity
  11. Genus of Shrinkers

Key Result

Theorem 3

If $(p,t)$ is a singular point with $t\le \operatorname{T_\textnormal{fat}}$, and if there is an $\epsilon>0$ such that for every regular time $\tau \in (t-\epsilon,t)$, then $(p,t)$ has genus $\le g$.

Figures (9)

  • Figure 1: A numerical simulation of the shrinkers constructed in Section \ref{['desing-section']}, for genus $g=1, \ldots,6$. Figures courtesy of Francisco Torralbo (UGR).
  • Figure 2: A cubical shrinker, found numerically by David Chopp. Figure courtesy of David Chopp.
  • Figure 3: $\mathbf{S}^2\setminus \cup \mathcal{P}$ is a union of spherical triangles. In the case of the icosahedron, the figure shows the spherical triangles in the region corresponding to one face. The points $e_i$ are defined by $e_i:= E_i \cap \mathbf{S}^2$, $i=1,2,3$. Here, the $E_i$ are the rays defined at the beginning of the proof of Theorem \ref{['geometry-theorem']}.
  • Figure 4: The curve $\Gamma$ in the case $g=4$.
  • Figure 5: A numerical simulation of the shrinker $\Sigma_5$ in Theorem \ref{['desing-theorem']}. Figure courtesy of Mario Schulz.
  • ...and 4 more figures

Theorems & Definitions (134)

  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Corollary 4
  • Theorem 5
  • proof
  • Definition 6
  • Theorem 7
  • proof
  • ...and 124 more