Classification of compact ancient solutions to the curve shortening flow

Daskalopoulos, Panagiota; Hamilton, Richard; Sesum, Natasa

Classification of compact ancient solutions to the curve shortening flow

Authors

Daskalopoulos, Panagiota

Hamilton, Richard

Sesum, Natasa

Description

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.