Complete minimal surfaces densely lying in arbitrary domains of R-n

In this paper we prove that, given an open Riemann surface M and an integer n >= 3, the set of complete conformal minimal immersions M -> R-n with <(X(M))over bar> = R-n forms a dense subset in the space of all conformal minimal immersions M -> R-n endowed with the compact-open topology. Moreover, we show that every domain in 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 nonorientable minimal surfaces in R-n (n >= 3), complex curves in C-n (n >= 2 ), holomorphic null curves in C-n (n >= 3), and holomorphic Legendrian curves in C2n+1 (n is an element of N).