# 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… Expand

#### 63 Citations

Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

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Peer Reviewed Title: Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere Author:

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We classify closed, convex, embedded ancient solutions to the curve shortening flow on the sphere, showing that the only such solutions are the family of shrinking round circles, starting at an… Expand

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In this paper we classify convex compact ancient solutions to the affine curve shortening flow, namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking… Expand

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We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1)\times O(n)$ symmetry. We show they all have unique asymptotics as $t\to -\infty$ and… Expand

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We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not… Expand

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We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time $t \rightarrow 0^-$ the solutions collapse to a round point where $0$ is the… Expand

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We prove some estimates for convex ancient solutions (the existence time for the solution starts at1 ) to the power-of-mean curvature flow, when the power is strictly greater than 1 . As an… Expand

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