Fattening in mean curvature flow

For each g 3, we prove existence of a compact, connected, smoothly embedded, genus-g surface Mg with the following property: under mean curvature flow, there is exactly one singular point at the first singular time, and the tangent flow at the singularity is given by a shrinker with genus (g-1) and with two ends. Furthermore, we show that if g is sufficiently large, then Mg fattens at the first singular time. As g, the shrinker converges to a multiplicity 2 plane.