We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.

Durham Research Online
You are in:

A converging Lagrangian flow in the space of oriented line.

Guilfoyle, Brendan and Klingenberg, Wilhelm (2016) 'A converging Lagrangian flow in the space of oriented line.', Kyushu journal of mathematics., 70 (2). pp. 343-351.


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.

Item Type:Article
Full text:(VoR) Version of Record
Download PDF
Publisher Web site:
Date accepted:30 April 2016
Date deposited:07 November 2016
Date of first online publication:13 October 2016
Date first made open access:No date available

Save or Share this output

Look up in GoogleScholar