Complete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$

Abstract

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to {ℝ}^n$ with $\overline{X\left(M\right)}={ℝ}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to {ℝ}^n$ endowed with the compact-open topology. Moreover, we show that every domain in ${ℝ}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.

Our method of proof can be adapted to give analogous results for nonorientable minimal surfaces in ${ℝ}^n\left(n\ge 3\right)$, complex curves in ${ℂ}^n\left(n\ge 2\right)$, holomorphic null curves in ${ℂ}^n\left(n\ge 3\right)$, and holomorphic Legendrian curves in ${ℂ}^{2n+1}\left(n\in \mathbb{N}\right)$.