# On the saddle point property of Abresch–Langer curves under the curve shortening flow

@article{Au2001OnTS, title={On the saddle point property of Abresch–Langer curves under the curve shortening flow}, author={Thomas Kwok-keung Au}, journal={Communications in Analysis and Geometry}, year={2001}, volume={18}, pages={1-21} }

In the study of the curve shortening flow on general closed curves, Abresch and Langer posed a conjecture that the homothetic curves can be regarded as saddle points between multi-folded circles and some singular curves. In other words, these homothetic curves are the watershed between curves with a nonsingular future and those with singular future along the flow. In this article, we provide an affirmitive proof to this conjecture.

## 17 Citations

### Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

- Mathematics
- 2018

We prove that a closed immersed plane curve with total curvature $2\pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the…

### Evolution of locally convex closed curves in the area-preserving and length-preserving curvature flows

- MathematicsCommunications in Analysis and Geometry
- 2020

We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge…

### On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

- Mathematics
- 2019

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a…

### Area-preserving evolution of nonsimple symmetric plane curves

- Mathematics
- 2014

The area-preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be nonsimple. For highly symmetric convex curves, the flows converge to m-fold circles,…

### Area-preserving evolution of nonsimple symmetric plane curves

- MathematicsJournal of Evolution Equations
- 2014

The area-preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be nonsimple. For highly symmetric convex curves, the flows converge to m-fold circles,…

### A note on the Abresch—Langer conjecture

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2014

A saddle-point property of the self-similar solutions in the curve shortening flow was conjectured by Abresch and Langer and confirmed by Au. An improvement on Au's solution is presented.

### Evolution of highly symmetric curves under the shrinking curvature flow

- Mathematics, Environmental Science
- 2017

Under the shrinking curvature flow with inner normal velocity V = kα(α > 1), it is shown that highly symmetric, locally convex initial curves evolve into a point asymptotically like an multi‐circles.…

### The stability of m-fold circles in the curve shortening problem

- Mathematics
- 2011

The stability of m-fold circles in the curve shortening problem (CSP) is studied in this paper. It turns out that a suitable perturbation of m-fold circle will shrink to a point asymptotically like…

### Length‐preserving evolution of non‐simple symmetric plane curves

- Mathematics, Environmental Science
- 2014

The length‐preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be non‐simple. It turns out that for certain classes of symmetric curves, the flows…

### A Embedding equations in Riemannian geometry 66 B Deforming curves and integrability 68 C Resistive diffusion of magnetic fields 69 1

- Physics
- 2008

The mean curvature flow describes the parabolic deformation of embedded branes in Riemannian geometry driven by their extrinsic mean curvature vector, which is typically associated to surface tension…

## References

SHOWING 1-10 OF 17 REFERENCES

### The affine curve-lengthening flow

- Mathematics
- 1999

Abstract The motion of any smooth closed convex curve in the plane in the direction of steepest increase of its affine arc length can be continued smoothly for all time. The evolving curve remains…

### Singularities of the curve shrinking flow for space curves

- Mathematics
- 1991

Singularities for space curves evolving by the curve shrinking flow are studied. Asymptotic descriptions of regions of the curve where the curvature is comparable to the maximum of the curvature are…

### The normalized curve shortening flow and homothetic solutions

- Mathematics
- 1986

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 Curve Shortening Problem

- Mathematics
- 2001

BASIC RESULTS Short Time Existence Facts from Parabolic Theory Evolution of Geometric Quantities INVARIANT SOLUTIONS FOR THE CURVE SHORTENING FLOW Travelling Waves Spirals The Support Function of a…

### Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature

- Mathematics
- 1990

This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature…

### A stable manifold theorem for the curve shortening equation

- Mathematics
- 1987

On presente une famille de solutions homothetiques de l'equation pour une courbe planaire ∂X/∂τ=KN et on demontre l'existence de varietes non lineaires stables et instables autour de telles solutions

### Parabolic equations for curves on surfaces Part II. Intersections, blow-up and generalized solutions

- Mathematics
- 1991

We describe a theory for parabolic equations for immersed curves on surfaces, which generalizes the curve shortening or flow-by-mean-curvature problem, as well as several models in the theory of…

### 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 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

### A CERTAIN PROPERTY OF SOLUTIONS OF PARABOLIC EQUATIONS WITH MEASURABLE COEFFICIENTS

- Mathematics
- 1981

In this paper Harnack's inequality is proved and the Holder exponent is estimated for solutions of parabolic equations in nondivergence form with measurable coefficients. No assumptions are imposed…