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

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Published/Copyright: January 7, 2011
Journal für die reine und angewandte Mathematik
From the journal Volume 2011 Issue 653

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.

Received: 2009-10-17
Revised: 2009-11-01
Published Online: 2011-01-07
Published in Print: 2011-April

© Walter de Gruyter Berlin · New York 2011

Articles in the same Issue

  1. Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions over number fields
  2. Wandering vectors and the reflexivity of free semigroup algebras
  3. On the equivariant main conjecture for imaginary quadratic fields
  4. Fixed point properties and second bounded cohomology of universal lattices on Banach spaces
  5. On the existence of certain affine buildings of type E6 and E7
  6. Endotrivial modules for finite group schemes
  7. Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem
  8. On a class of fully nonlinear flows in Kähler geometry
  9. Colocalizing subcategories of D(R)
https://doi.org/10.1515/crelle.2011.026

Articles in the same Issue

  1. Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions over number fields
  2. Wandering vectors and the reflexivity of free semigroup algebras
  3. On the equivariant main conjecture for imaginary quadratic fields
  4. Fixed point properties and second bounded cohomology of universal lattices on Banach spaces
  5. On the existence of certain affine buildings of type E6 and E7
  6. Endotrivial modules for finite group schemes
  7. Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem
  8. On a class of fully nonlinear flows in Kähler geometry
  9. Colocalizing subcategories of D(R)
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