On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space - Georgiou, Nikos (Mathematica Scandinavica, 2012)
We study area-stationary surfaces in the space $L(H^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in $L(H^3)$ is an area-stationary surface. We then classify Lagrangian area-stationary surfaces $Σ$ in $L(H^3)$ and prove that the family of parallel surfaces in $H^3$ orthogonal to the geodesics $γ\in Σ$ 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.