Interpolation by complete minimal surfaces whose Gauss map misses two points

Abstract

Let $M$ be an open Riemann surface and let $\Lambda \subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to {\mathbb{R}}^n$, $n\ge 3$, with prescribed values on $\Lambda$ and whose generalized Gauss map $M\to {\mathbb{CP}}^{n-1}$, $n\ge 3$, avoids $n$ hyperplanes of ${\mathbb{CP}}^{n-1}$ located in general position. In case $n=3$, we obtain complete nonflat conformal minimal immersions whose Gauss map $M\to {\mathbb{S}}^2$ omits two (antipodal) values of the sphere. This result is deduced as a consequence of an interpolation theorem for conformal minimal immersions $M\to {\mathbb{R}}^n$ into the Euclidean space ${\mathbb{R}}^n$, $n\ge 3$, with $n-2$ prescribed components.

J. Geom. Anal., in press