Curve shortening on surfaces

Gage, Michael E.

doi:10.24033/asens.1603

Cited by 44 publications

(38 citation statements)

References 1 publication

(1 reference statement)

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“…The study of curvature flow of closed curves in the plane was generalized to certain surfaces by Grayson [Gra89] and Gage [Gag90]. On surfaces, a smooth embedded curve collapses to a round point in finite time as in the plane case, or exists for all time, converging to a closed geodesic.…”

Section: Previous Work and Backgroundmentioning

confidence: 99%

Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

Bryan

Louie

2015

J Geom Anal

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We prove that the only closed, embedded ancient solutions to the curve shortening flow on S 2 are equators or shrinking circles, starting at an equator at time t = −∞ and collapsing to the north pole at time t = 0. To obtain the result, we first prove a Harnack inequality for the curve shortening flow on the sphere. Then an application of the Gauss-Bonnet, easily allows us to obtain curvature bounds for ancient solutions leading to backwards smooth convergence to an equator. To complete the proof, we use an Aleksandrov reflection argument to show that maximal symmetry is preserved under the flow.

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Section: Previous Work and Backgroundmentioning

confidence: 99%

Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

Bryan

Louie

2015

J Geom Anal

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“…In particular, its total geodesic curvature γt √ κ 2 + 1 ds tends to 2π, and so, is less than 3π for t large enough. This ensures that the two conditions of Theorem 5.1 in [Ga90] are satisfied. Therefore, we conclude that the curvature flow (γ t ) converges to an equator as unparametrized curves.…”

Section: Preliminariesmentioning

confidence: 92%

“…We summarize the properties of the curvature flow on the canonical round two-sphere that we will need in this article as follows. [Ga90], [Gr89]). The curvature flow (γ t ) of an embedded closed curve γ on the canonical round two-sphere satisfies the following properties:…”

Section: Preliminariesmentioning

confidence: 99%

Strong deformation retraction of the space of Zoll Finsler projective planes

Sabourau

2019

Journal of Symplectic Geometry

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We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zoll Finsler projective plane to the geodesic flow of the round metric through a family of smooth free circle actions induced by the curvature flow of the canonical round projective plane. This construction provides a description of the geodesics of the Zoll Finsler metrics along the retraction.2010 Mathematics Subject Classification. Primary 53D25; Secondary 53C20, 53C44.

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“…The behavior of geodesic curvature flows on surfaces, however, are much more complicated than geodesic active contours on Euclidean planes because the surface geometry will also affect the curve evolution [20,21,22]. More recently, a level set formulation of geodesic curvature flow on surfaces was discussed in [23,24,25].…”

Section: Geodesic Curvature Flowsmentioning

confidence: 99%

Automated corpus callosum extraction via Laplace-Beltrami nodal parcellation and intrinsic geodesic curvature flows on surfaces

Lai

Shi

Sicotte

et al. 2011

2011 International Conference on Computer Vision

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Corpus callosum (CC) is an important structure in human brain anatomy. In this work, we propose a fully automated and robust approach to extract corpus callosum from T1-weighted structural MR images. The novelty of our method is composed of two key steps. In the first step, we find an initial guess for the curve representation of CC by using the zero level set of the first nontrivial LaplaceBeltrami (LB) eigenfunction on the white matter surface. In the second step, the initial curve is deformed toward the final solution with a geodesic curvature flow on the white matter surface. For numerical solution of the geodesic curvature flow on surfaces, we represent the contour implicitly on a triangular mesh and develop efficient numerical schemes based on finite element method. Because our method depends only on the intrinsic geometry of the white matter surface, it is robust to orientation differences of the brain across population. In our experiments, we validate the proposed algorithm on 32 brains from a clinical study of multiple sclerosis disease and demonstrate that the accuracy of our results.

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Curve shortening on surfaces

Cited by 44 publications

References 1 publication

Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

Strong deformation retraction of the space of Zoll Finsler projective planes

Automated corpus callosum extraction via Laplace-Beltrami nodal parcellation and intrinsic geodesic curvature flows on surfaces

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