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


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 equator and collapsing to a point, or the curve is a fixed equator for all time. A Harnack inequality for the curve shortening flow on the sphere and an application of Gauss-Bonnet allow us to obtain curvature bounds for convex ancient solutions, which lead 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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