COMPACT COMPLETE MINIMAL IMMERSIONS IN R3

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface M, an open domain M subset of M with the fixed topological type, and a conformal complete minimal immersion X : M -> R-3 which can be extended to a continuous map X : (M) over bar -> R-3, such that X-vertical bar partial derivative M is an embedding and the Hausdorff dimension of X(partial derivative M) is 1. We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in R-3, endowed with the topology of the Hausdorff distance.