Genus one singularities in mean curvature flow

TL;DR

This paper demonstrates that in specific families of initial conditions in three-dimensional space, mean curvature flow inevitably develops genus one singularities, which are stable under perturbations, and constructs a genus one self-shrinker with low entropy.

Contribution

It establishes the inevitability and robustness of genus one singularities in certain mean curvature flows and constructs a novel embedded genus one self-shrinker with low entropy.

Findings

Genus one singularities must appear in certain families of mean curvature flows.

Such singularities are stable under perturbations of initial conditions.

Constructed a genus one self-shrinker with entropy below that of a shrinking doughnut.

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.