Curve shortening on surfaces

@article{Gage1990CurveSO,
  title={Curve shortening on surfaces},
  author={Michael E. Gage},
  journal={Annales Scientifiques De L Ecole Normale Superieure},
  year={1990},
  volume={23},
  pages={229-256}
}
  • 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
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TLDR
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
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Curve shortening flow on Riemann surfaces with possible ambient conic singularities
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  • 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
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References

SHOWING 1-8 OF 8 REFERENCES
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