Communications in Analysis and Geometry

Volume 18 (2010)

Number 1

On the saddle point property of Abresch–Langer curves under the curve shortening flow

Pages: 1 – 21



Thomas Kwok-Keung Au (Department of Mathematics, The Chinese University of Hong Kong)


In the study of the curve shortening flow on general closedcurves in the plane, Abresch and Langer posed a conjecturethat the homothetic curves can be regarded as saddle pointsbetween multi-folded circles and certain singular curves.In other words, these homothetic curves are the watershedbetween curves with a nonsingular future and those withsingular future along the flow. In this article, weprovide an affirmative proof to this conjecture.

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