Self-shrinkers whose asymptotic cones fatten

Abstract

For each positive integer $g$ we use variational methods to construct a genus $g$ self-shrinker $\hat{I}{\pounds }_g$ 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 $\hat{I}{\pounds }_g$ has two graphical asymptotically conical ends and the sequence $\hat{I}{\pounds }_g$ 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.

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