Classification of compact ancient solutions to the curve shortening flow

2008/06/10 by Panagiota Daskalopoulos, Daskalopoulos, Panagiota, Richard Hamilton +3 · 1 citation
Mathematics · #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:53C44

paper · pdf · doi:10.48550/arxiv.0806.1757

arxiv created 2008/06/10 · arxiv updated 2009/12/01

Abstract

We consider an embedded convex ancient solution Γ_t to the curve shortening flow in ℝ². We prove that there are only two possibilities: the family Γ_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.

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