Fattening in mean curvature flow

TL;DR

The paper constructs, for each $g\ge 3$, a compact genus-$g$ surface $M_g$ in $\mathbb{R}^3$ whose mean curvature flow has a single first singularity with tangent flow given by a genus $g-1$ shrinker with two ends, and shows that such $M_g$ fatten under the level-set flow for large $g$; as $g\to\infty$ the shrinkers converge to a multiplicity-$2$ plane. The approach blends Bamler–Kleiner theory with two geometric templates, pancakes and $g$-surfaces, organized into $g$-wheels, and analyzes a trichotomy of possible tangent flows to locate a critical wheel whose tangent flow yields the desired singular structure. Existence of critical $g$-wheels is established via a connectedness argument among thin and thick wheels, yielding the $M_g$ with the claimed singular behavior and fattening, while an auxiliary $g\to\infty$ analysis shows convergence to the doubled plane. Together, the results provide explicit fattening examples in mean curvature flow, clarifying how genus influences singularity type and connecting high-genus shrinkers to multiplicity-two planar limits.

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.

Figures (3)

• Figure 1: A shrinker $\Sigma_4$ with genus $3$ and $2$ ends. (Courtesy of David Chopp.)
• Figure 2: Thin and Thick $8$-Wheels
• Figure :

Theorems & Definitions (105)

• Theorem
• Theorem 1.1
• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Trichotomy Theorem, Abbreviated Version
• Definition 2.6
• Theorem 3.1
• Theorem 4.1