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

Ben Andrews*, Paul Bryan

*Corresponding author for this work

Research output: Contribution to journalArticle

10 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)179-187
Number of pages9
JournalJournal fur die Reine und Angewandte Mathematik
Volume2011
Issue number653
DOIs
Publication statusPublished - 2011
Externally publishedYes

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