On a non-local curve evolution problem in the plane

Jiang, Lishang; Pan, Shengliang

doi:10.4310/cag.2008.v16.n1.a1

Cited by 43 publications

(29 citation statements)

References 22 publications

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“…The same result in the one-dimensional case was obtained by Gage [5]. Recently, a similar convergence theorem for a modified curve flow along which the isoperimetric ratio also decreases was proved by Jiang and Pan [11]. Based on center manifold analysis, Escher and Simonett [4] showed that if the initial hypersurface is sufficiently close to a fixed sphere, then the flow will converge exponentially to a round sphere.…”

Section: Introductionsupporting

confidence: 62%

Volume-preserving mean curvature flow of hypersurfaces in space forms

Leng

Zhao

2014

Int. J. Math.

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The purpose of this paper is to investigate the convergence of the volume-preserving mean curvature flow of closed hypersurfaces in space forms. Assume that the initial hypersurface satisfies suitable integral curvature pinching conditions. We prove that along the volume-preserving mean curvature flow the initial hypersurface will be deformed to a totally umbilical sphere.The basic property of flow (1.1) is its isoperimetric nature. Namely, in the case that M t is embedded along the flow, the (n + 1)-dimensional volume |Ω t | of the region Ω t enclosed by M t is preserved:while the n-dimensional volume of M t strictly decreases unless the mean curvature is constant:1450021-2 Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by KANSAS STATE UNIVERSITY on 06/25/14. For personal use only.Volume-preserving mean curvature flow of hypersurfaces in space forms then the volume-preserving mean curvature flow with M as initial hypersurface will converge exponentially fast to a round sphere.When the ambient space is the hyperbolic space, we obtain the following convergence theorem. Theorem 1.2. Let M be an n-dimensional smooth closed and orientable hypersurface immersed in H n+1 . Suppose the volume Vol(M ), the second fundamental form A and the averaged mean curvature h satisfy Vol(M ) ≤ V, |A| ≤ Λ, h ≥ n + γ for positive constants V, Λ, γ. There is a positive constant ε = ε(n, V, Λ, γ) depending only on n, V, Λ and γ such that if M |Å| 2 dµ < ε, (1.3)then the volume-preserving mean curvature flow with M as initial hypersurface will converge exponentially fast to a round sphere.

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Section: Introductionsupporting

confidence: 62%

Volume-preserving mean curvature flow of hypersurfaces in space forms

Leng

Zhao

2014

Int. J. Math.

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“…Such a flow was first investigated by Jiang and Pan in . Similarly, as Mu and Zhu in , they employed the Gauss parameterization for evolution of convex curves and so results of can be applied to convex curves. In what follows, we show that some of their results (e.g., area increasing property) can be directly generalized to the case of evolution of Jordan curves by using the arc‐length parameterization.…”

Section: Evolution Of Plane Curves With a Nonlocal Normal Velocitymentioning

confidence: 99%

Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity

Ševčovič

Yazaki

2012

Math Methods in App Sciences

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In this paper, we investigate a time‐dependent family of plane closed Jordan curves evolving in the normal direction with a velocity that is assumed to be a function of the curvature, tangential angle, and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so‐called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations, we prove local existence, uniqueness, and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities such as the length, enclosed area, or total elastic energy of a curve. We also propose a stable numerical approximation scheme on the basis of the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper. Copyright © 2012 John Wiley & Sons, Ltd.

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“…Following his work, Green and Osher [8] obtained a generalised formula with respect to the curvature of all C 2 convex curves in the plane. These inequalities play a critical role in the curve evolution problem (see, for example, [5,13]). For a fixed convex domain E, Böröczky et al [3] rediscovered the generalised case of (1.4) in relative geometry, that is,…”

Section: Introductionmentioning

confidence: 99%