Fattening in mean curvature flow

TL;DR

This paper constructs specific high-genus surfaces evolving under mean curvature flow that develop a unique singularity characterized by a genus-reducing shrinker, with fattening occurring for large genus.

Contribution

It proves the existence of high-genus surfaces with controlled singularities and describes their asymptotic behavior as genus increases.

Findings

Existence of genus-$g$ surfaces with a single singularity under mean curvature flow.

The tangent flow at singularity is a genus-$(g-1)$ shrinker with two ends.

Fattening occurs for sufficiently large genus, and the shrinkers converge to a multiplicity 2 plane as $g o \\infty$.

Abstract

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.