Applications of a Completeness Lemma in Minimal Surface Theory to Various Classes of Surfaces

We give several applications of a lemma on completeness used by Osserman to show the meromorphicity of Weierstrass data for complete minimal surfaces with finite total curvature. Completeness and weak completeness are defined for several classes of surfaces which admit singular points. The completeness lemma is a useful machinery for the study of completeness in these classes of surfaces. In particular, we show that a constant mean curvature one (i.e. CMC-1) surface in de Sitter 3-space is complete if and only if it is weakly complete, the singular set is compact and all the ends are conformally equivalent to a punctured disk.