Existence of proper minimal surfaces of arbitrary topological type

Consider a domain DD in R3R3 which is convex (possibly all R3R3) or which is smooth and bounded. Given any open surface MM, we prove that there exists a complete, proper minimal immersion f:M→Df:M→D. Moreover, if DD is smooth and bounded, then we prove that the immersion f:M→Df:M→D can be chosen so that the limit sets of distinct ends of MM are disjoint connected compact sets in ∂D∂D.