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.
|Full text:||(VoR) Version of Record|
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|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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