Classification of compact ancient solutions to the curve shortening flow

Panagiota Daskalopoulos; Richard Hamilton; Natasa Sesum

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arXiv · 0806.1757

Classification of compact ancient solutions to the curve shortening flow

Panagiota Daskalopoulos·Richard Hamilton·Natasa Sesum

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Abstract

We consider an embedded convex ancient solution $Γ_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $Γ_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex.

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Panagiota Daskalopoulos, Richard Hamilton, Natasa Sesum. 2008-06-10. Classification of compact ancient solutions to the curve shortening flow. https://arxiv.org/abs/0806.1757

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