Classification of compact ancient solutions to the curve shortening flow
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2008
- DOI:
- 10.48550/arXiv.0806.1757
- arXiv:
- arXiv:0806.1757
- Bibcode:
- 2008arXiv0806.1757D
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- 53C44