  We confirm a well-known conjecture that the round sphere is the only compact,
embedded self-similar shrinking solution to the mean curvature flow 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.
