Non-convergence and instability in the asymptotic behaviour of curves evolving by curvature

Andrews, Benjamin

doi:10.4310/cag.2002.v10.n2.a8

Cited by 20 publications

(48 citation statements)

References 19 publications

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“…Theorem 1 ( Ben Andrews [AN1], [AN3], [AN4]) For m = 1 and any α > 0, the curve γ t contracts to a point in finite time. If 0 < α < 1/3, then for generic initial data there is no limit of the curves w λ (x) [w xx (x) + w (x)] = 1, x ∈ S 1 (1)…”

Section: Introductionmentioning

confidence: 99%

Contracting convex immersed closed plane curves with slow speed of curvature

Lin

Poon

Tsai

2012

Trans. Amer. Math. Soc.

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We study the contraction of a convex immersed plane curve with speed 1 α k α , where α ∈ (0, 1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α = 1), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson [GR]). * AMS Subject Classifications: 35K15, 35K55. We thank Ben Andrews for giving us this summary.rescaled about the final point (the isoperimetric ratio approaches infinity); and the exceptional ones where the isoperimetric ratio remains bounded converge to homothetic solutions, which have been classified. For α > 1/3, the rescaled solutions converge to circles; and for α = 1/3, they converge to ellipses.Remark 2 As a consequence of Theorem 1, we have the following interesting elliptic result. For 0 < λ < 3 (here λ = 1/α), the only positive 2π-periodic solution to the equation

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Section: Introductionmentioning

confidence: 99%

Contracting convex immersed closed plane curves with slow speed of curvature

Lin

Poon

Tsai

2012

Trans. Amer. Math. Soc.

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“…Traditionally, differential geometry has been the study of curved spaces or shapes in which, for the most part, time did not play a role. In the last few decades, on the other hand, geometers and geometric analysts have made great strides in understanding shapes that evolve in time; see, for instance, [1][2][3][4][5][6]8,10,[12][13][14][15][16]18,24,25], etc. Among them, perhaps, the simplest case (but already a very subtle one) is the curve-shortening flow in the plane studied by Gage and Hamilton [12] and Grayson [13].…”

Section: Introductionmentioning

confidence: 99%

On a non-local curve evolution problem in the plane

Jiang¹,

Pan²

2008

Communications in Analysis and Geometry

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This paper deals with a new curvature flow for closed convex plane curves which shortens the length of the evolving curve but expands the area it bounds and makes the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle (as the time t goes to infinity). This flow is determined by a coupled system concerning both local and global geometric quantities of the evolving curve.

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“…For the case α = 1 of curve-shortening flow, the curve converges to a point and the limiting shape is a circle; this was proven for convex curves by Gage [32,31] and non-convex curves by Grayson [34], implying also that the isoperimetric quotient Q converges to 1. Andrews examined how this limit depends on the exponent 0 < α [4,6,7]. He showed that for 1 3 < α any smooth convex curve converges to a point, the limiting curve is a circle and, hence, the isoperimetric quotient converges to 1.…”

Section: Illustration Of Known Resultsmentioning

confidence: 99%

“…which is often referred to as the (planar) Bloore flow [21]. Moreover, the curveshortening flow (1) has been generalized in a purely mathematical context, dominantly through the work of Andrews [3,7,8], to the study of flows of the form…”

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Tracking the critical points of curves evolving under planar curvature flows

Fehér¹,

Domokos²,

Krauskopf³

2021

JCD

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<p style='text-indent:20px;'>We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function <inline-formula><tex-math id="M1">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function <inline-formula><tex-math id="M2">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> and of the curvature <inline-formula><tex-math id="M3">\begin{document}$ \kappa(\varphi) $\end{document}</tex-math></inline-formula> (characterized by <inline-formula><tex-math id="M4">\begin{document}$ dr/d\varphi = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ d\kappa /d\varphi = 0 $\end{document}</tex-math></inline-formula>, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.</p><p style='text-indent:20px;'>We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically — in the spirit of experimental mathematics — we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.</p>

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Non-convergence and instability in the asymptotic behaviour of curves evolving by curvature

Abstract: We consider curvature-driven evolution equations for curves in the plane, and prove that the isoperimetric ratios of the evolving curves generically approach infinity if the speed of motion is proportional to curvature to a power less than 1/3.

Cited by 20 publications

References 19 publications

Contracting convex immersed closed plane curves with slow speed of curvature

Contracting convex immersed closed plane curves with slow speed of curvature

On a non-local curve evolution problem in the plane

Tracking the critical points of curves evolving under planar curvature flows

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