A Converging Lagrangian Flow in the Space of Oriented Line

Guilfoyle, Brendan; Klingenberg, Wilhelm

doi:10.2206/kyushujm.70.343

A Converging Lagrangian Flow in the Space of Oriented Line

Guilfoyle, Brendan; Klingenberg, Wilhelm

Authors

Brendan Guilfoyle

Dr Wilhelm Klingenberg wilhelm.klingenberg@durham.ac.uk
Associate Professor

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