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 $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology. Moreover, we show that every domain in $\mathbb{R}^n$ 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 non-orientable minimal surfaces in $\mathbb{R}^n$ $(n\ge 3)$, complex curves in $\mathbb{C}^n$ $(n\ge 2)$, holomorphic null curves in $\mathbb{C}^n$ $(n\ge 3)$, and holomorphic Legendrian curves in $\mathbb{C}^{2n+1}$ $(n\in\mathbb{N})$.