Brendan Guilfoyle
A Converging Lagrangian Flow in the Space of Oriented Line
Guilfoyle, Brendan; Klingenberg, Wilhelm
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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