A note on the Abresch—Langer conjecture

Abstract

A saddle-point property of the self-similar solutions in the curve shortening flow was conjectured by Abresch and Langer and confirmed by Au. An improvement on Au's solution is presented.

On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

“…We finally mention that, after Abresch and Langer's celebrated study [1], the behavior of multiply winding rotationally symmetric curves is investigated by several authors for the curve shortening flow [3,20,24,25,44,45] and other second order geometric flows [13,[46][47][48]; in particular, a kind of stability result for multiply covered circles is obtained by Wang [45], in the same spirit of our result. We remark that all these studies focus on locally convex curves, thus using the property of second order.…”

On the isoperimetric inequality and surface diffusion flow for multiply winding curves

“…We finally mention that, after Abresch and Langer's celebrated study [1], the behavior of multiply winding rotationally symmetric curves is investigated by several authors for the curve shortening flow [25,24,3,44,45,20] and other second order geometric flows [47,48,13,46]; in particular, a kind of stability result for multiply covered circles is obtained by Wang [45], in the same spirit of our result. We remark that all these studies focus on locally convex curves, thus using the property of second order.…”

Evolution of highly symmetric curves under the shrinking curvature flow

“…Recently, the asymptotic stability of multi-circles has been proved by Wang [13]. And the saddle point property of Abresch-Langer curves (first conjectured by Abresch-Langer [11]) has been confirmed by Au [14] and improved by Chou-Wang [15]. We must also notice that in 1987, Epstein and Gage [16] showed that a highly symmetric, locally convex curve must evolve into a round point.…”

“…An example is the pentagram. In fact, the works of [13][14][15] and of Epstein-Gage [16] supply a picture for what type of curves can evolve into round points.…”