In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize “ideal cones” (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.
"Isometric immersions of the hyperbolic plane into the hyperbolic space." Tohoku Math. J. (2) 64 (2) 171 - 193, 2012. https://doi.org/10.2748/tmj/1341249370