# 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… Expand

#### 11 Citations

The evolution of Jordan curves on $\mathbb{S}^2$ by curve shortening flow

- Mathematics
- 2016

In this paper we prove that if $\gamma$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $\gamma$ in $\mathcal{C}^0$ as $t\to 0^+… Expand

Mean curvature flow of arbitrary co-dimensional Reifenberg sets

- Mathematics
- Calculus of Variations and Partial Differential Equations
- 2018

We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called k-dimensional $$(\varepsilon ,R)$$(ε,R) Reifenberg flat sets in… Expand

Mean Curvature Flow Of Reifenberg Sets

- Mathematics
- 2014

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $\mathbb{R}^{n+1}$ starting from any $n$-dimensional $(\varepsilon,R)$-Reifenberg flat set with… Expand

The fattening phenomenon for level set solutions of the mean curvature flow

- Mathematics
- 2017

Level set solutions are an important class of weak solutions to the mean curvature flow which allow the flow to be extended past singularities. Unfortunately, when singularities do develop it is… Expand

The Level-Set Flow of the Topologist’s Sine Curve is Smooth

- Mathematics
- The Journal of Geometric Analysis
- 2018

In this note we prove that the level-set flow of the topologist’s sine curve is a smooth closed curve. In Lauer (Geom Funct Anal 23(6): 1934–1961, 2013) it was shown by the second author that under… Expand

Nonconvex Surfaces which Flow to Round Points

- Mathematics
- 2019

In this article we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We will construct certain classes of mean convex and non-mean convex hypersurfaces that… Expand

LECTURES ON CURVE SHORTENING FLOW

- 2016

These are the lecture notes for the last three weeks of my PDE II course from Spring 2016. The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to… Expand

Solitons of Curve Shortening Flow and Vortex Filament Equation

- Physics, Mathematics
- 2017

In this paper we explore the nature of self-similar solutions of the Curve Shortening Flow and the Vortex Filament Equation, also known as the Binormal Flow. We explore some of their fundamental… Expand

UROP + Final Paper , Summer 2017 Bernardo

- 2017

In this paper we explore the nature of self-similar solutions of the Curve Shortening Flow and the Vortex Filament Equation, also known as the Binormal Flow. We explore some of their fundamental… Expand

#### References

SHOWING 1-10 OF 18 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… Expand

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… Expand

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… Expand

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

- Mathematics
- 1989

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… Expand

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],… Expand

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… Expand

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… Expand