Self-shrinkers whose asymptotic cones fatten

For each positive integer g we use variational methods to construct a genus g self-shrinker Σg in R³ with entropy less than $2$ and prismatic symmetry group Dg+1₂. For g sufficiently large, the self-shrinker Σg has two graphical asymptotically conical ends and the sequence Σg converges on compact subsets to a plane with multiplicity two as g→∞. 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 R³ with smooth initial non-compact data that evolve non-uniquely after their first singular time.