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 \(g\to\infty\), the shrinker converges to a multiplicity \(2\) plane.