Limit sets for complete minimal immersions

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G \subset {\mathbb{R}}^3$$\end{document} with the boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial G \subset C^2.$$\end{document} We prove that the second fundamental form of the surface ∂G is nonnegatively defined at every point of the limit set of such immersions.