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

TL;DR

This paper proves that any open Riemann surface can be realized as a complete constant mean curvature 1 surface in hyperbolic space, introducing new approximation and interpolation techniques for such surfaces.

Contribution

It establishes the existence of complete CMC-1 surfaces with arbitrary complex structures and develops a jet interpolation theorem for complete conformal CMC-1 immersions.

Findings

Every open Riemann surface is the complex structure of a complete CMC-1 surface in hyperbolic space.

Develops a uniform approximation theorem with jet interpolation for holomorphic null curves.

Constructs complete densely immersed CMC-1 surfaces with arbitrary complex structures.

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 $\mathbb{C}^2\times\mathbb{C}^*$ which is also established in this paper.