Interpolation by complete minimal surfaces whose Gauss map misses two points

TL;DR

This paper proves the existence of complete conformal minimal immersions from open Riemann surfaces into Euclidean spaces, with prescribed values on discrete sets and Gauss maps avoiding certain hyperplanes or points, extending interpolation theory.

Contribution

It establishes new existence results for minimal surfaces with prescribed boundary data and Gauss map constraints, including cases where the Gauss map omits two points on the sphere.

Findings

Existence of complete conformal minimal immersions with prescribed values on discrete sets.

Construction of minimal surfaces whose Gauss map omits two antipodal points.

Extension of interpolation theorems for conformal minimal immersions.

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.