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

Abstract

We prove that every open Riemann surface M is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the 3-dimensional hyperbolic space H3. We go further and establish a jet interpolation theorem for complete conformal CMC-1 immersions M→H3. As a consequence, we show the existence of complete densely immersed CMC-1 surfaces in H3 with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in C2×C∗ which is also established in this paper.