Self-shrinkers whose asymptotic cones fatten

Abstract

For each positive integer $g$ we use variational methods to construct a genus $g$ self-shrinker $Σ_g$ 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 $Σ_g$ has two graphical asymptotically conical ends and the sequence $Σ_g$ 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.