The Curve Shortening Flow

@inproceedings{Epstein1987TheCS,
  title={The Curve Shortening Flow},
  author={Charles L. Epstein and Michael E. Gage},
  year={1987}
}
This is an expository paper describing the recent progress in the study of the curve shortening equation $${X_{{t\,}}} = \,kN $$ (0.1) Here X is an immersed curve in ℝ2, k the geodesic curvature and N the unit normal vector. We review the work of Gage on isoperimetric inequalities, the work of Gage and Hamilton on the associated heat equation and the work of Epstein and Weinstein on the stable manifold theorem for immersed curves. Finally we include a new proof of the Bonnesen… 
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Figure Captions
1. Geometry of the geodesic curvature vector. 2. Geometry of the constant velocity projection. 3. A simple planar example. The nal geodesic curve is as straight line as expected. (a) Top view (the ^
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References

SHOWING 1-10 OF 25 REFERENCES
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
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
Flow by mean curvature of convex surfaces into spheres
The motion of surfaces by their mean curvature has been studied by Brakke [1] from the viewpoint of geometric measure theory. Other authors investigated the corresponding nonparametric problem [2],
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations
where Vu is the (spatial) gradiant of u. Here VM/|VW| is a unit normal to a level surface of u, so div(Vw/|Vw|) is its mean curvature unless Vu vanishes on the surface. Since ut/\Vu\ is a normal
Algorithms Based on Hamilton-Jacobi Formulations
TLDR
New numerical algorithms, called PSC algorithms, are devised for following fronts propagating with curvature-dependent speed, which approximate Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws.
Deforming metrics in the direction of their Ricci tensors
In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive
Fast reaction, slow diffusion, and curve shortening
The reaction-diffusion problem \[ u_1 = \varepsilon \Delta u - \varepsilon ^{ - 1} V_n ( u ),\quad u( {x,0,\varepsilon } ) = g( x ),\quad\partial _n u = 0\text{ on }\partial \Omega \] for a vector
BONNESEN-STYLE ISOPERIMETRIC INEQUALITIES
Because of Property 1, any Bonnesen inequality implies the isoperimetric inequality (1). From Property 2, it follows that equality can hold in (1) only when C is a circle. The effect of Property 3 is
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
An isoperimetric inequality with applications to curve shortening
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