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

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

2009-08-19

arxiv.org/abs/0908.2682v1

Mathematics

Differential Geometry

8 pages, no figures, amsart document class

Scientific paper

A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2\pi$. 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.

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