# A New Length Estimate for Curve Shortening Flow and Low Regularity Initial Data

@article{Lauer2011ANL, title={A New Length Estimate for Curve Shortening Flow and Low Regularity Initial Data}, author={Joseph Lauer}, journal={Geometric and Functional Analysis}, year={2011}, volume={23}, pages={1934-1961} }

In this paper we introduce a geometric quantity, the r-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow (CSF). The length estimates we obtain are used to prove results about the level set flow in the plane. If K is locally-connected, connected and compact, then the level set flow of K either vanishes instantly, fattens instantly or instantly becomes a smooth closed curve. If the compact set in question is a Jordan curve J, then the proof proceeds…

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## References

SHOWING 1-10 OF 14 REFERENCES

### The size of the singular set in mean curvature flow of mean-convex sets

- Mathematics
- 2000

In this paper, we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that…

### THE NATURE OF SINGULARITIES IN MEAN CURVATURE FLOW OF MEAN-CONVEX SETS

- Mathematics
- 2002

Let K be a compact subset of R, or, more generally, of an (n+1)-dimensional riemannian manifold. We suppose that K is mean-convex. If the boundary of K is smooth and connected, this means that the…

### Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

- Mathematics
- 1989

This paper treats degenerate parabolic equations of second order
$$u_t + F(\nabla u,\nabla ^2 u) = 0$$
(14.1)
related to differential geometry, where ∇ stands for spatial derivatives of u =…

### Motion of level sets by mean curvature. I

- Mathematics
- 1991

We continue our investigation of the “level-set” technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is…

### Flow by mean curvature of convex surfaces into spheres

- Mathematics
- 1984

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],…

### The heat equation shrinks embedded plane curves to round points

- Mathematics
- 1987

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 zero set of a solution of a parabolic equation.

- Mathematics
- 1988

On etudie l'ensemble nul d'une solution u(t,x) de l'equation u t =a(x,t)u xx +b(x,t)u x +C(x,t)u, sous des hypotheses tres generales sur les coefficients a, b, et c

### The heat equation shrinking convex plane curves

- Mathematics
- 1986

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

### PARTIAL DIFFERENTIAL EQUATIONS

- Mathematics
- 1941

Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear…