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Closed ideal planar
curves
Ben Andrews, James McCoy, Glen Wheeler and Valentina-Mira Wheeler |
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Geometry & Topology 24 (2020) 1019–1049
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Abstract |
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We use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the 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 enjoy a uniform length bound under the flow, yielding the convergence result in these cases. |
Keywords
constant mean curvature, curvature flow, geometric
evolution equation
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Mathematical Subject Classification 2010
Primary: 35K25, 53C44
Secondary: 58J35
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References |
Publication
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
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Authors |
| Ben Andrews | |
| Applied and Nonlinear Analysis
Group Mathematical Sciences Institute Australian National University Canberra, ACT Australia |
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| James McCoy | |
| Priority Research Centre for
Computer-Assisted Research Mathematics and its
Applications School of Mathematical & Physical Sciences University of Newcastle Callaghan, NSW Australia |
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| Glen Wheeler | |
| Institute for Mathematics and its
Applications School of Mathematics and Applied Statistics University of Wollongong Wollongong, NSW Australia |
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| Valentina-Mira Wheeler | |
| Institute for Mathematics and its
Applications School of Mathematics and Applied Statistics University of Wollongong Wollongong, NSW Australia |
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