Embedded self-similar shrinkers of genus $0$

Abstract

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution of mean curvature flow in $\mathbb{R}^3$ with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.