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.