A collapsing ancient solution of mean curvature flow in ${\mathbb{R}}^3$

Abstract

We construct a compact, convex ancient solution of mean curvature flow in ${\mathbb{R}}^{n+1}$ with $O(1)\times O(n)$ symmetry that lies in a slab of width $\pi$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies in a slab of width $\pi$ and in no smaller slab.

In this new version, we have made significant revisions to accommodate the following additional material: we derive detailed asymptotics for the constructed solution, we show that this solution is the unique solution amongst compact, convex, rotationally symmetric ancient solutions in a slab, and we generalize the entire process to any dimension

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• Rotationally symmetric translating solutions to extrinsic geometric flows
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• Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow