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
• Public URL: https://durham-repository.worktribe.com/output/1386003