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

  title={The shape of a figure-eight under the curve shortening flow},
  author={M. Grayson},
  journal={Inventiones mathematicae},
  • M. Grayson
  • Published 1 February 1989
  • Mathematics
  • Inventiones mathematicae
Curve shortening flow on singular Riemann surfaces
In this paper, we study curve shortening flow on Riemann surfaces with singular metrics. It turns out that this flow is governed by a degenerate quasilinear parabolic equation. Under naturalExpand
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We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flow; this includes coupled mean curvature/Kähler-Ricci flow in the sense ofExpand
Some New Results in Geometric Analysis
This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preservingExpand
Legendrian curve shortening flow in R 3
This gives a partial answer to a conjecture of Grayson [12], which states that all figure-eight curves with zero signed area should shrink to a point under curve shortening flow. In particular, ourExpand
Curve shortening flow in a 3-dimensional pseudohermitian manifold
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In this paper, we introduce a curve shortening flow in a 3-dimensional pseudohermitian manifold with vanishing torsion. The flow preserves the Legendrian condition and decreases the length of curves.Expand
Existence of solution and asymptotic behavior for a class of parabolic equations
We prove existence and uniqueness of a positive solution for a class of quasilinear parabolic equations. We also show some maximum principles on the derivatives of the solution and study theExpand
The Affine Shape of a Figure-Eight under the Curve Shortening Flow
We consider the curve shortening flow applied to a natural class of figureeight curves, those with dihedral symmetry and some monotonicity assumptions on the curvature and its derivatives. We proveExpand
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A stable manifold theorem for the curve shortening equation
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
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) estExpand
The heat equation shrinking convex plane curves
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 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 ZΛgradientExpand
Curve shortening makes convex curves circular