M ar 2 00 9 Uniform Approximation by Algebraic Minimal Surfaces in R 3 Francisco

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in R is obtained. This result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of finite conformal type. We deal only with the orientable case. As a consequence, complete minimal surfaces in R of weak finite total curvature with exotic geometry are produced. More specifically, surfaces with arbitrarily prescribed conformal structure and flux map, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.