The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends

In this paper, we show that if R is a compact Riemann surface and M = R \ boolean OR(i) D-i is a domain in R whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs, D-i, then there is a complete conformal minimal immersion X : M -> R-3, extending to a continuous map X : (M) over bar -> R-3, such that X (bM) = boolean OR X-i (bD(i)) is a union of pairwise disjoint Jordan curves. In particular, M is the complex structure of a complete bounded minimal surface in R-3. This extends a recent result for finite bordered Riemann surfaces.