Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure Alarcón López, Antonio Castro Infantes, Ildefonso Hidalgo, Jorge Constant mean curvature surface Bryant surface Regular end 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. 2024-05-16T11:01:46Z 2024-05-16T11:01:46Z 2024-02-15 journal article Published version: Communications in Contemporary Mathematics (2024) 245001 [https://dx.doi.org/10.1142/S0219199724500111] https://hdl.handle.net/10481/91878 10.1142/S0219199724500111 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional