# 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.

#### 14 Citations

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#### References

SHOWING 1-10 OF 18 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… Expand

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… Expand

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… Expand

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… Expand

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… 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 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… Expand