Communications in Analysis and Geometry
Volume 18 (2010)
On the saddle point property of Abresch–Langer curves under the curve shortening flow
Pages: 1 – 21
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.