Classification of convex ancient solutions to curve shortening flow on the sphere

Paul Bryan*, Janelle Louie

*Corresponding author for this work

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We prove that the only closed, embedded ancient solutions to the curve shortening flow on S2 are equators or shrinking circles, starting at an equator at time t = -∞ 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.

Original languageEnglish
Pages (from-to)858-872
Number of pages15
JournalJournal of Geometric Analysis
Volume26
Issue number2
DOIs
Publication statusPublished - Apr 2016
Externally publishedYes

Keywords

  • Curve shortening flow
  • Ancient solutions
  • Aleksandrov reflection
  • Harnack

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