Self-shrinkers whose asymptotic cones fatten

TL;DR

This paper constructs a family of genus g self-shrinkers with specific symmetries and entropy properties, demonstrating non-uniqueness in mean curvature flow evolution after singularities.

Contribution

It introduces a new class of self-shrinkers with asymptotic cones that fatten, confirming a long-standing conjecture and providing examples of non-unique mean curvature flow evolutions.

Findings

Existence of genus g self-shrinkers with prescribed symmetry and entropy less than 2.

Sequence of self-shrinkers converges to a double plane as genus increases.

Examples of mean curvature flows with non-unique evolution after singularities.

Abstract

For each positive integer $g$ we use variational methods to construct a genus $g$ self-shrinker $\Sigma_g$ in $\mathbb{R}^3$ with entropy less than $2$ and prismatic symmetry group $\mathbb{D}_{g+1}\times\mathbb{Z}_2$. For $g$ sufficiently large, the self-shrinker $\Sigma_g$ has two graphical asymptotically conical ends and the sequence $\Sigma_g$ converges on compact subsets to a plane with multiplicity two as $g\to\infty$. Angenent-Chopp-Ilmanen conjectured the existence of such self-shrinkers in 1995 based on numerical experiments. Using these surfaces as initial conditions for large $g$, we obtain examples of mean curvature flows in $\mathbb{R}^3$ with smooth initial non-compact data that evolve non-uniquely after their first singular time.