Physical Sciences and Mathematics Commons^™

An Indefinite Kähler Metric On The Space Of Oriented Lines, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space T of oriented affine lines in R3 with the tangent bundle of S2. Thus the round metric on S2 induces a Kähler structure on T which turns out to have a metric of neutral signature. It is shown that the identity component of the isometry group of this metric is isomorphic to the identity component of the isometry group of the Euclidean metric on R3.

The geodesics of this metric are either planes or helicoids in R3. The signature …

A Converging Lagrangian Flow In The Space Of Oriented Lines, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

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 …

Geodesic Flow On The Normal Congruence Of A Minimal Surface, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We study the geodesic flow on the normal line congruence of a minimal surface in ℝ3 induced by the neutral Kähler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description of minimal surfaces in ℝ3 and relate it to the classical Weierstrass representation.