On the saddle point property of Abresch—Langercurves under the curve shortening flow

摘要

In the study of the curve shortening flow on general closed curves in the plane, Abresch and Langer posed a conjecture that the homo-thetic curves can be regarded as saddle points between multi-folded circles and certain singular curves. In other words, these homoth-etic curves are the watershed between curves with a nonsingular future and those with singular future along the flow. In this article, we provide an affirmative proof to this conjecture.