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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 puntured disk.Comment: 9 page
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