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

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.

9 pages

References in corpus (1)

• Complete bounded holomorphic curves immersed in C^2 with arbitrary genus

Cited by in corpus (7)

• Zero mean curvature entire graphs of mixed type in Lorentz-Minkowski 3-space
• On the maximal number of exceptional values of Gauss maps for various classes of surfaces
• Value distribution of the Gauss map of improper affine spheres
• The Gauss images of complete minimal surfaces of genus zero of finite total curvature
• Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded
• Calabi-Yau Problem for Legendrian curves in C^3 and applications
• New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter 3-space