Curve shortening on surfaces

  title={Curve shortening on surfaces},
  author={Michael E. Gage},
  journal={Annales Scientifiques De L Ecole Normale Superieure},
  • M. Gage
  • Published 1990
  • Mathematics
  • Annales Scientifiques De L Ecole Normale Superieure
What happens when a simple closed curve on a surface M is allowed to move so that the instant instantaneous velocity at each point is proportional to the gepdesic curvature k of the curve at that point? This evolution is suggested by the equation of motion of an idealized elastic band along the surface (assuming that the friction term is relatively large and the mass relatively small) 
An application of the curve shortening flow on surfaces
As an application of the curve shortening flow, this paper will show an inequality for the maximum curvature of a smooth simple closed curve on surfaces.
Curve shortening flows on rotational surfaces generated by monotone convex functions
In this paper, we study curve shortening flows on rotational surfaces in R. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the
The Dirichlet problem for curve shortening flow
We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we
This paper deals with a non-local curve evolution problem in the plane which will increase both the length of the evolving curve and the area it bounds and make the evolving curve more and more
Curve Flows on Ruled Surfaces
Special flows of curves on ruled surfaces are studied using a discretization of the ruled surfaces, the curves on them, and the curvatures to study closed curves on closed ruled surfaces.
Soliton solutions to the curve shortening flow on the sphere
It is shown that a curve on the unit sphere is a soliton solution to the curve shortening flow if and only if its geodesic curvature is proportional to the inner product between its tangent vector
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 natural
Curve shortening flow on Riemann surfaces with possible ambient conic singularities
  • Biao Ma
  • Mathematics
    Differential Geometry and its Applications
  • 2022
Continuous dependence of curvature flow on initial conditions
We study the evolution of a Jordan curve on the 2-sphere by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the


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
Ordinary Differential Equations.
together with the initial condition y(t0) = y0 A numerical solution to this problem generates a sequence of values for the independent variable, t0, t1, . . . , and a corresponding sequence of values
ANGENENT, The Zeroset of a Solution of a Parabolic Equation
  • (J. reine angew. Math.,
  • 1988
in Wave motion: theory
  • modelling, and computation. Proceedings of a Conference in honor of 'the 60th birthday of Peter D. Lax, MSRI Publications, Springer-Verlag,
  • 1987