A Converging Lagrangian Flow in the Space of Oriented Lines

Authors

BRENDAN GUILFOYLE, School of Science, Technology, Engineering and Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.Follow
Wilhelm Klingenberg, Department of Mathematical Sciences University of Durham Durham DH1 3LE UK.

Document Type

Article

Disciplines

Mathematics

Publication Details

Kyushu Journal of Mathematics

© 2016 Faculty of Mathematics, Kyushu University.

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.

Recommended Citation

Guilfoyle, Brendan & Klingenberg, Wilhelm. (2013). A Converging Lagrangian Curvature Flow in the Space of Oriented Lines. Kyushu Journal of Mathematics. 70. 10.2206/kyushujm.70.343.