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Evolution of closed curves by length-constrained curve diffusion


Authors: James McCoy, Glen Wheeler and Yuhan Wu
Journal: Proc. Amer. Math. Soc. 147 (2019), 3493-3506
MSC (2010): Primary 53C44, 58J35
DOI: https://doi.org/10.1090/proc/14473
Published electronically: March 15, 2019
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Abstract: We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow and that the only closed rotators are circles.


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Additional Information

James McCoy
Affiliation: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales, 2308 Australia
Email: james.mccoy@newcastle.edu.au

Glen Wheeler
Affiliation: Institute for Mathematics and its Applications, University of Wollongong, Wollongong, New South Wales, 2522 Australia
Email: glenw@uow.edu.au

Yuhan Wu
Affiliation: Institute for Mathematics and its Applications, University of Wollongong, Wollongong, New South Wales, 2522 Australia
Email: yw120@uowmail.edu.au

DOI: https://doi.org/10.1090/proc/14473
Keywords: Curvature flow, higher order quasilinear partial differential equation, curve diffusion flow, differential geometry of plane curves
Received by editor(s): June 28, 2018
Received by editor(s) in revised form: November 11, 2018
Published electronically: March 15, 2019
Additional Notes: Some of this research was conducted while the first author was visiting the University of Wollongong. The first author is the corresponding author. The research of the first and second authors was supported by Discovery Project grant DP150100375 of the Australian Research Council.
The research of the third author was supported by a University of Wollongong Faculty of Engineering and Information Sciences Postgraduate research scholarship.
Communicated by: Guofang Wei
Article copyright: © Copyright 2019 American Mathematical Society