The shape of a figure-eight under the curve shortening flow

@article{Grayson1989TheSO,
  title={The shape of a figure-eight under the curve shortening flow},
  author={M. Grayson},
  journal={Inventiones mathematicae},
  year={1989},
  volume={96},
  pages={177-180}
}
  • M. Grayson
  • Published 1 February 1989
  • Mathematics
  • Inventiones mathematicae

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Curve shortening makes convex curves circular

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On presente une famille de solutions homothetiques de l'equation pour une courbe planaire ∂X/∂τ=KN et on demontre l'existence de varietes non lineaires stables et instables autour de telles solutions

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Soient M et M' des varietes de Riemann et F:M→M' une application reguliere. Si M est une courbe convexe plongee dans le plan R 2 , l'equation de la chaleur contracte M a un point

The heat equation shrinks embedded plane curves to round points

Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est

The normalized curve shortening flow and homothetic solutions

The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient