Embedded self-similar shrinkers of genus 0

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