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

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 K\"ahler 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.