Vol. 290, No. 2, 2017

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ISSN: 0030-8730
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
Abstract

It is shown that all immersed closed, locally convex curves with total curvature of 2mπ and n-fold rotational symmetry (mn 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
Authors
Xiao-Liu Wang
School of Mathematics
Southeast University
Jiulonghu Campus, Jiangning
211189 Nanjing
China
Hui-Ling Li
School of Mathematics
Southeast University
Jiulonghu Campus, Jiangning
211189 Nanjing
China
Xiao-Li Chao
School of Mathematics
Southeast University
Jiulonghu Campus, Jiangning
211189 Nanjing
China