Uniform approximation by complete minimal surfaces of finite total curvature in ℝ³

Abstract

We prove that any compact minimal surface in R 3 \mathbb {R}^3 can be uniformly approximated by complete minimal surfaces of finite total curvature in R 3 \mathbb {R}^3 . This Mergelyan type 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 regions of finite conformal type. We deal only with the orientable case.