In our recent paper we develop the theory of approximation for holomorphic null curves in the special linear group SL(2,C). In particular, we establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for the family of holomorphic null immersions M -> SL(2,C) from any open Riemann surface M. Our results include jet interpolation of Weierstrass type and approximation by embeddings, as well as global conditions on the approximating curves. As application, we show that every open Riemann surface admits a proper holomorphic null embedding into SL(2,C), and hence also a proper conformal immersion of constant mean curvature 1 into hyperbolic 3-space. This settles a problem posed by Alarcón and Forstneric in 2015.

In a recent work with A.Alarcón and I.Castro-Infantes we show that every open Riemann surface admits a complete conformal CMC-1 (constant mean curvature one) immersion in the three dimensional hyperbolic space. In this talk I aim to explain the main ideas in the proof of this result, which relies on the holomorphic representation of CMC-1 surfaces given by Robert Bryant in 1987, and applies modern complex analysis techniques.