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

Abstract

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 $X=(X_j{)}_{j=1,\dots ,n}:N\to {\mathbb{R}}^n$ with Hopf differential $H$ and $(X_j{)}_{j=3,\dots ,n}=h.$ In particular, there exists a complete conformal minimal immersion $Y=(Y_j{)}_{j=1,\dots ,n}:N\to {\mathbb{R}}^n$ such that $(Y_j{)}_{j=3,\dots ,n}=h.$ 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 ${\mathbb{CP}}^{n-1}$ in general position are constructed. Moreover, complete non-proper embedded minimal surfaces in ${\mathbb{R}}^n,$ $\forall n>3,$ are exhibited.

arXiv comment14 pages, 2 figures, new title