On maximal surfaces in the space of oriented geodesics of hyperbolic 3-space

Abstract

We study area-stationary, or maximal, surfaces in the space $\mathbb{L}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in $\mathbb{L}({\mathbb{H}}^3)$ is a maximal surface. We then classify Lagrangian maximal surfaces $\Sigma$ in $\mathbb{L}({\mathbb{H}}^3)$ and prove that the family of parallel surfaces in ${\mathbb{H}}^3$ orthogonal to the geodesics $\gamma \in \Sigma$ form a family of equidistant tubes around a geodesic.

16 pages, AMS-Latex

References in corpus (1)

• On area-stationary surfaces in certain neutral Kaehler 4-manifolds