Spacelike mean curvature one surfaces in de Sitter 3-space

The first author studied spacelike constant mean curvature one (CMC-1) surfaces in the de Sitter 3-space S-1(3) when the surfaces have no singularities except within some compact subsets and are of finite total curvature on the complement of this compact subset. However, there are many CMC-1 surfaces whose singular sets are not compact. In fact, such examples have already appeared in the construction of trinoids given by Lee and the last author via hypergeometric functions. In this paper, we improve the Osserman-type inequality given by the first author. Moreover, we shall develop a fundamental framework that allows the singular set to be non-compact, and then will use it to investigate the global behavior of CMC-1 surfaces.