Hyperbolic metrics on Riemann surfaces and space-like CMC-1 surfaces in de Sitter 3-space

We introduce a new notion called the extended hyperbolic metrics, as a hyperbolic metric (i.e. metric of constant curvature - 1) with certain kinds of singularities defined on a Riemann surface, and we give several fundamental properties of such metrics. Extended hyperbolic metrics are closely related to space-like surfaces of constant mean curvature one (i.e. CMC-1 surfaces) in de Sitter 3-space S₁³. For example, the singular set of a given CMC-1 surface in S₁³ is contained in the singular set of the associated extended hyperbolic metric. We then classify all catenoids in S₁³ (i.e. weakly complete constant mean curvature 1 surfaces in S₁³ of genus zero with two regular ends whose hyperbolic Gauss map is of degree one). Such surfaces are called S₁³-catenoids. Since there is a bijection between the moduli space of S₁³-catenoids and the moduli space of co-orientable extended hyperbolic metrics with two regular singularities, a classification of such hyperbolic metrics is also given. (Co-orientability of extended hyperbolic metrics is defined in this paper.).