Wave Motion: Theory, Modelling, and Computation
Volume 7 of the series Mathematical Sciences Research Institute Publications pp 1559
The Curve Shortening Flow
 C. L. Epstein
 , Michael Gage
Abstract
This is an expository paper describing the recent progress in the study of the curve shortening equation Here X is an immersed curve in ℝ^{2}, k the geodesic curvature and N the unit normal vector. We review the work of Gage on isoperimetric inequalities, the work of Gage and Hamilton on the associated heat equation and the work of Epstein and Weinstein on the stable manifold theorem for immersed curves. Finally we include a new proof of the Bonnesen inequality and a proof that highly symmetric immersed curves flow under (0.1) to points.
$${X_{{t\,}}} = \,kN $$
(0.1)
Keywords
curve shortening heat flow isoperimetric inequalities stable manifolds Title
 The Curve Shortening Flow
 Book Title
 Wave Motion: Theory, Modelling, and Computation
 Book Subtitle
 Proceedings of a Conference in Honor of the 60th Birthday of Peter D. Lax
 Pages
 pp 1559
 Copyright
 1987
 DOI
 10.1007/9781461395836_2
 Print ISBN
 9781461395850
 Online ISBN
 9781461395836
 Series Title
 Mathematical Sciences Research Institute Publications
 Series Volume
 7
 Series ISSN
 09404740
 Publisher
 Springer US
 Copyright Holder
 SpringerVerlag New York Inc.
 Additional Links
 Topics
 Keywords

 curve shortening
 heat flow
 isoperimetric inequalities
 stable manifolds
 Industry Sectors
 eBook Packages
 Editors

 Alexandre J. Chorin ^{(1)}
 Andrew J. Majda ^{(2)}
 Editor Affiliations

 1. Department of Mathematics, University of California
 2. Department of Mathematics, Princeton University
 Authors
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