We will show that any (non-constant) harmonic map on an arbitrary open Riemannian N surface can be realized as the third coordinate of a complete minimal immersion of $N$ in $\mathbb{R}^3$. As a consequence we will prove that any open Riemann surface admits a complete conformal minimal immersion whose Gauss map misses two points.
This is a joint work with F.J. López and A. Alarcón.