Embedded minimal surfaces in Rn

In this paper, we prove that every conformal minimal immersion of an open Riemann surface into R-n for n >= 5 can be approximated uniformly on compacts by conformal minimal embeddings (see Theorem 1.1). Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into R-5 (see Theorem 1.2). One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to R-n for any n >= 3 which is also proved in the paper (see Theorem 5.3).