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

We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature $1$ ($\mathrm{\text{CMC-1}}$) in the three-dimensional hyperbolic space ${ℍ}^3$. We go further and establish a jet interpolation theorem for complete conformal $\mathrm{\text{CMC-1}}$ immersions $M\to {ℍ}^3$. As a consequence, we show the existence of complete densely immersed $\mathrm{\text{CMC-1}}$ surfaces in ${ℍ}^3$ with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in ${ℂ}^2\times {ℂ}^{\ast }$ which is also established in this paper.