Classification of compact ancient solutions to the curve shortening flow

@article{Daskalopoulos2008ClassificationOC,
  title={Classification of compact ancient solutions to the curve shortening flow},
  author={Panagiota Daskalopoulos and Richard S. Hamilton and Nata{\vs}a {\vS}e{\vs}um},
  journal={arXiv: Differential Geometry},
  year={2008}
}
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… 

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