Classification of compact ancient solutions to the curve shortening flow

Details Classification of compact ancient solutions to the curve shortening flow Classification of compact ancient solutions to the curve shortening flow

2008-06-10

arxiv.org/abs/0806.1757v1

Mathematics

Differential Geometry

Scientific paper

We consider an embedded convex ancient solution $\Gamma_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $\Gamma_t$ 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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