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.

We shall discuss some properties of minimal surfaces in Euclidean space which hold generically in Baire category sense. Based on joint work with Francisco J. López.

A complex manifold is called an Oka manifold if every holomorphic map from a convex set in a Euclidean space to the manifold is a limit of entire maps. We show that every Oka manifold admits families of noncompact complex curves of any given topological type, endowed with a prescribed family of conformal structures, with approximation on compact Runge subsets. A similar result holds for families of minimal surfaces and null holomorphic curves in Euclidean spaces.