On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space

We study area-stationary surfaces in the space L(H³) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in L(H³) is an area-stationary surface. We then classify Lagrangian area-stationary surfaces Σ in L(H³) and prove that the family of parallel surfaces in H³ orthogonal to the geodesics γ ∞ Σ form a family of equidistant tubes around a geodesic. Finally we find an example of a two parameter family of rotationally symmetric area-stationary surfaces that are neither Lagrangian nor holomorphic.