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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.

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.

Item Type:Article
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Status:Peer-reviewed
Publisher Web site:http://doi.org/10.2206/kyushujm.70.343
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

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