Fattening in mean curvature flow

For each $g\ge 3$, we prove existence of a compact, connected, smoothly embedded, genus-$g$ surface $M_g$ 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 $M_g$ fattens at the first singular time. As $g\to \mathrm{\infty }$, the shrinker converges to a multiplicity $2$ plane.