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The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1075–1079 | Cite as

Classifying Convex Compact Ancient Solutions to the Affine Curve Shortening Flow

  • Shibing ChenEmail author
Article

Abstract

In this paper we classify convex compact ancient solutions to the affine curve shortening flow, namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling argument inspired by Wang (Ann. Math., 173(1):1185–1239, 2011), affine invariance of the equation, and monotonicity of the affine isoperimetric ratio. It also provides a new simple proof for the corresponding classification result to the higher-dimensional affine normal flow.

Keywords

Affine curve shortening flow Ancient solutions 

Mathematics Subject Classification

53C44 35K55 

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Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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