Volume 24, issue 2 (2020)

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Closed ideal planar curves

Ben Andrews, James McCoy, Glen Wheeler and Valentina-Mira Wheeler

Geometry & Topology 24 (2020) 1019–1049

We use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the L2 sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply covered circle. Moreover, we show that curves in any homotopy class with initially small L3ks22 enjoy a uniform length bound under the flow, yielding the convergence result in these cases.

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constant mean curvature, curvature flow, geometric evolution equation
Mathematical Subject Classification 2010
Primary: 35K25, 53C44
Secondary: 58J35
Received: 14 October 2018
Revised: 17 July 2019
Accepted: 17 August 2019
Published: 23 September 2020
Proposed: Tobias H Colding
Seconded: Bruce Kleiner, John Lott
Ben Andrews
Applied and Nonlinear Analysis Group
Mathematical Sciences Institute
Australian National University
Canberra, ACT
James McCoy
Priority Research Centre for Computer-Assisted Research Mathematics and its Applications
School of Mathematical & Physical Sciences
University of Newcastle
Callaghan, NSW
Glen Wheeler
Institute for Mathematics and its Applications
School of Mathematics and Applied Statistics
University of Wollongong
Wollongong, NSW
Valentina-Mira Wheeler
Institute for Mathematics and its Applications
School of Mathematics and Applied Statistics
University of Wollongong
Wollongong, NSW