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.
Comment: Math. Z., in press. The official version is available on Springerlink at http://link.springer.com/article/10.1007%2Fs00209-015-1586-5