# 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…

## 64 Citations

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

- Mathematics
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We prove that the only closed, embedded ancient solutions to the curve shortening flow on $$\mathbb {S}^2$$S2 are equators or shrinking circles, starting at an equator at time $$t=-\infty $$t=-∞ and…

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

- Mathematics
- 2014

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…

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- Mathematics
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- Physics, Mathematics
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We construct a compact, convex ancient solution of mean curvature flow in $\mathbb R^{n+1}$ with $O(1)\times O(n)$ symmetry that lies in a slab of width $\pi$. We provide detailed asymptotics for…

Convergence of curve shortening flow to translating soliton

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- 2018

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\mathbb{R}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha >\frac12$. We show…

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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…

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- 2016

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…

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- 2018

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…

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- Mathematics, Physics
- 2012

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…

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