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Evolution of non-simple closed curves in the area-preserving curvature flow

Part of: Global differential geometry Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  28 December 2017

Xiao-Liu Wang ,
Wei-Feng Wo  and
Ming Yang
Xiao-Liu Wang
Affiliation:
School of Mathematics, Southeast University, Nanjing 210096, People's Republic of China (xlwang@seu.edu.cn)
Wei-Feng Wo
Affiliation:
Department of Mathematics, Ningbo University, Ningbo 315211, People's Republic of China
Ming Yang
Affiliation:
School of Mathematics, Southeast University, Nanjing 210096, People's Republic of China
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Abstract

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The convergence and blow-up results are established for the evolution of non-simple closed curves in an area-preserving curvature flow. It is shown that the global solution starting from a locally convex curve converges to an m-fold circle if the enclosed algebraic area A0 is positive, and evolves into a point if A0 = 0.

Keywords

curvature flownon-localblow-upconvergence

MSC classification

Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 35B40: Asymptotic behavior of solutions 35K59: Quasilinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 3 , June 2018 , pp. 659 - 668
Copyright
Copyright © Royal Society of Edinburgh 2018