Abstract
We prove that the only closed, embedded ancient solutions to the curve shortening flow on \(\mathbb {S}^2\) are equators or shrinking circles, starting at an equator at time \(t=-\infty \) and collapsing to the north pole at time \(t=0\). To obtain the result, we first prove a Harnack inequality for the curve shortening flow on the sphere. Then an application of the Gauss–Bonnet, easily allows us to obtain curvature bounds for ancient solutions leading to backwards smooth convergence to an equator. To complete the proof, we use an Aleksandrov reflection argument to show that maximal symmetry is preserved under the flow.
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Acknowledgments
Both authors would like to thank Professor Bennett Chow for suggesting this problem and providing much useful guidance on laying out the program. The second author is especially thankful, this paper arising from her Ph.D. thesis under Professor Chow’s supervision. This paper was completed while the first author was a SEW Visiting Assistant Professor at UCSD, acting as an informal Ph.D. advisor to the second author’s Ph.D. research at UCSD.
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Bryan, P., Louie, J. Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere. J Geom Anal 26, 858–872 (2016). https://doi.org/10.1007/s12220-015-9574-x
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DOI: https://doi.org/10.1007/s12220-015-9574-x