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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

Online
ISSN
1435-5345
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Volume 2011, Issue 653

Issues

Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem

Ben Andrews
  • Mathematical Sciences Institute, John Dedman Building, The Australian National University, Canberra, ACT 0200, Australia
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/ Paul Bryan
  • Mathematical Sciences Institute, John Dedman Building, The Australian National University, Canberra, ACT 0200, Australia
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Published Online: 2011-01-07 | DOI: https://doi.org/10.1515/crelle.2011.026

Abstract

A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length 2π. The estimate bounds the length of any chord from below in terms of the arc length between its endpoints and elapsed time. Applying the estimate to short segments we deduce directly that the maximum curvature decays exponentially to 1. This gives a self-contained proof of Grayson's theorem which does not require the monotonicity formula or the classification of singularities.

About the article

Received: 2009-10-17

Revised: 2009-11-01

Published Online: 2011-01-07

Published in Print: 2011-04-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2011, Issue 653, Pages 179–187, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle.2011.026.

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