Applications of a completeness lemma in minimal surface theory to various classes of surfaces

TL;DR

This paper explores the use of a completeness lemma in minimal surface theory to analyze various classes of surfaces, establishing new criteria for completeness and weak completeness, especially for CMC-1 surfaces in de Sitter space.

Contribution

It applies a classical completeness lemma to new classes of surfaces, providing criteria linking completeness, weak completeness, and surface end behavior.

Findings

Completeness and weak completeness coincide for certain classes of surfaces.

A CMC-1 surface in de Sitter space is complete iff it is weakly complete with compact singular set.

Ends of such surfaces are conformally equivalent to punctured disks.

Abstract

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.