Complete minimal surfaces and harmonic functions

Abstract

We prove that for any open Riemann surface $\mathcal{N}$ and any non-constant harmonic function $h:\mathcal{N}\to \mathbb{R}$, there exists a complete conformal minimal immersion $X:\mathcal{N}\to {\mathbb{R}}^3$ whose third coordinate function coincides with $h$.

As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are constructed.