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

TL;DR

This paper demonstrates that complete minimal surfaces with dense images can be densely approximated within any domain in ^n, extending to various types of complex and minimal surfaces with arbitrary topology.

Contribution

It proves the density of complete conformal minimal immersions with dense images in the space of all such immersions for open Riemann surfaces in ^n and constructs dense minimal surfaces with arbitrary topology in any domain.

Findings

Complete minimal surfaces with dense images are dense in the space of all conformal minimal immersions.

Every domain in ^n contains complete minimal surfaces with arbitrary topology.

The method applies to non-orientable minimal surfaces, complex curves, null curves, and Legendrian curves.

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 $\mathbb{C}^{2n+1}$ $(n\in\mathbb{N})$.