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 ${\mathbb{H}}^3$. We go further and establish a jet interpolation theorem for complete conformal CMC-1 immersions $M\to {\mathbb{H}}^3$. As a consequence, we show the existence of complete densely immersed CMC-1 surfaces in ${\mathbb{H}}^3$ with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in ${\mathbb{C}}^2\times {\mathbb{C}}^{\ast }$ which is also established in this paper.

To appear in Commun. Contemp. Math