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# Communications in Analysis and Geometry

## Volume 17 (2009)

### Number 3

### Spacelike mean curvature one surfaces in de Sitter $3$-space

Pages: 383 – 427

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n3.a1

#### Authors

#### Abstract

The first author studiedspacelike constant mean curvature one (CMC-$1$)surfaces in the de Sitter 3-space $S^3_1$when the surfaces have no singularities except within somecompact subsets and are of finite total curvature onthe complement of this compact subset.However, there are many CMC-$1${} surfaces whosesingular sets are not compact.In fact, such examples have already appeared in theconstruction of trinoids given by Lee and the last authorvia hypergeometric functions.

In this paper, we improve the Osserman-type inequalitygiven by the first author.Moreover, we shall develop a fundamental framework that allowsthe singular set to be non-compact, and thenwill use it to investigate the global behavior of CMC-$1${} surfaces.