Volume 24, issue 2 (2020)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 3, 1075–1614
Issue 2, 533–1073
Issue 1, 1–532

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Closed ideal planar curves

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

Geometry & Topology 24 (2020) 1019–1049
Abstract

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.

Keywords
constant mean curvature, curvature flow, geometric evolution equation
Mathematical Subject Classification 2010
Primary: 35K25, 53C44
Secondary: 58J35
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
Authors
Ben Andrews
Applied and Nonlinear Analysis Group
Mathematical Sciences Institute
Australian National University
Canberra, ACT
Australia
James McCoy
Priority Research Centre for Computer-Assisted Research Mathematics and its Applications
School of Mathematical & Physical Sciences
University of Newcastle
Callaghan, NSW
Australia
Glen Wheeler
Institute for Mathematics and its Applications
School of Mathematics and Applied Statistics
University of Wollongong
Wollongong, NSW
Australia
Valentina-Mira Wheeler
Institute for Mathematics and its Applications
School of Mathematics and Applied Statistics
University of Wollongong
Wollongong, NSW
Australia