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 (that is, 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 disc.

我们给出了奥瑟曼（Osserman）所使用的一个关于完备性的引理的若干应用，他用此引理证明了具有有限总曲率的完备极小曲面的魏尔斯特拉斯（Weierstrass）数据的亚纯性。我们对几类允许有奇点的曲面定义了完备性和弱完备性。完备性引理是研究这些类曲面完备性的一个有用工具。特别地，我们证明了在德西特（de Sitter）3维空间中，一个常平均曲率为1（即CMC - 1）的曲面是完备的当且仅当它是弱完备的，奇点集是紧的，并且所有的端在共形等价意义下都等同于一个去心圆盘。