Length-preserving evolution of immersed closed curves and the isoperimetric inequality

Xiao-Liu Wang, Hui-Ling Li and Xiao-Li Chao

Vol. 290 (2017), No. 2, 467–479

DOI: 10.2140/pjm.2017.290.467

Abstract

It is shown that all immersed closed, locally convex curves with total curvature of $2m\pi$ and $n$-fold rotational symmetry ($m∕n\le 1$) finally evolve into $m$-fold circles under the length-preserving curvature flow. Sufficient conditions for the occurrence of the finite-time singularities in the flow are also established. As a byproduct, an isoperimetric inequality for rotationally symmetric, locally convex curves is proved via the flow method.

Keywords

curvature flow, nonlocal, blow-up, convergence, isoperimetric inequality

Mathematical Subject Classification 2010

Primary: 35B40, 35K59, 53C44

Milestones

Received: 21 January 2016

Revised: 8 February 2017

Accepted: 23 February 2017

Published: 25 July 2017