Genus one singularities in mean curvature flow

Abstract

We show that for certain one-parameter families of initial conditions in ${\mathbb{R}}^3$, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut.

37 pages. Accepted by Geometry & Topology. v3: We added a remark to address the genus one example by Buzano-Nguyen-Schulz and their numerical simulation