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A Converging Lagrangian Flow in the Space of Oriented Line

Guilfoyle, Brendan; Klingenberg, Wilhelm

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Authors

Brendan Guilfoyle



Abstract

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the three-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is the Steiner point of the initial surface, which remains constant under the flow. To prove this we show that the oriented normal lines, considered as a surface in the space of all oriented lines, evolve by a parabolic flow which preserves the Lagrangian condition.Moreover, this flow converges to a holomorphic Lagrangian section, which forms the set of oriented lines through a point. The coordinates of the Steiner point are projections of the support function into the first non-zero eigenspace of the spherical Laplacian and are given by explicit integrals of initial surface data.

Citation

Guilfoyle, B., & Klingenberg, W. (2016). A Converging Lagrangian Flow in the Space of Oriented Line. Kyushu journal of mathematics, 70(2), 343-351. https://doi.org/10.2206/kyushujm.70.343

Journal Article Type Article
Acceptance Date Apr 30, 2016
Online Publication Date Oct 13, 2016
Publication Date Sep 1, 2016
Deposit Date Apr 27, 2016
Publicly Available Date Nov 7, 2016
Journal Kyushu journal of mathematics.
Print ISSN 1340-6116
Electronic ISSN 1883-2032
Publisher Faculty of Mathematics, Kyushu University
Peer Reviewed Peer Reviewed
Volume 70
Issue 2
Pages 343-351
DOI https://doi.org/10.2206/kyushujm.70.343

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