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arXiv · 1310.4402

A Converging Lagrangian Curvature Flow in the Space of Oriented Lines

Brendan Guilfoyle·Wilhelm Klingenberg
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Abstract

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is determined by the initial surface. 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 form the set of oriented lines through a point. The coordinates of this centre 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.

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Brendan Guilfoyle, Wilhelm Klingenberg. 2013-10-16. A Converging Lagrangian Curvature Flow in the Space of Oriented Lines. https://doi.org/10.2206/kyushujm.70.343

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