Harmonic mappings and conformal minimal immersions of Riemann surfaces into ${\mathbb{R}}^n$

We prove that for any open Riemann surface $N,$ natural number $n\ge 3,$ non-constant harmonic map $h:N\to {\mathbb{R}}^{n-2}$ and holomorphic 2-form $H$ on $N,$ there exists a weakly complete harmonic map $X=(X_j){j=1,\ldots,n}:N \to $\mathbb{R}$^n$withHopfdifferential$H$and$(X_j){j=3,\ldots,n}=h.$ In particular, there exists a complete conformal minimal immersion $Y=(Y_j{)}_{j=1,\dots ,n}:N\to {\mathbb{R}}^n$ such that $(Y_j{)}_{j=3,\dots ,n}=h\mathrm{.}$ As a consequence of these results, complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect $n$ hyperplanes of ${\mathbb{C}\mathbb{P}}^{n-1}$ in general position are constructed. Moreover, complete non-proper embedded minimal surfaces in ${\mathbb{R}}^n,$ $\mathrm{\forall }n>3,$ are exhibited.