Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface Formula: see text is the complex structure of a complete surface of constant mean curvature Formula: see text (Formula: see text‒1) in the three-dimensional hyperbolic space Formula: see text. We go further and establish a jet interpolation theorem for complete conformal Formula: see text‒1 immersions Formula: see text. As a consequence, we show the existence of complete densely immersed Formula: see text‒1 surfaces in Formula: see text with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in Formula: see text which is also established in this paper.