Embedded minimal surfaces in ${\mathbb{R}}^n$

In this paper, we prove that every confomal minimal immersion of an open Riemann surface into ${\mathbb{R}}^n$ for $n\ge 5$ can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into ${\mathbb{R}}^5$. One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to ${\mathbb{R}}^n$ for any $n\ge 3$ which is also proved in the paper.