### Abstract

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 S_{1}^{3}. For example, the singular set of a given CMC-1 surface in S_{1}^{3} is contained in the singular set of the associated extended hyperbolic metric. We then classify all catenoids in S_{1}^{3} (i.e. weakly complete constant mean curvature 1 surfaces in S_{1}^{3} of genus zero with two regular ends whose hyperbolic Gauss map is of degree one). Such surfaces are called S_{1}^{3}-catenoids. Since there is a bijection between the moduli space of S_{1}^{3}-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.).

Original language | English |
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Pages (from-to) | 1-47 |

Number of pages | 47 |

Journal | Springer Proceedings in Mathematics and Statistics |

Volume | 26 |

DOIs | |

Publication status | Published - Sep 9 2013 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Springer Proceedings in Mathematics and Statistics*,

*26*, 1-47. https://doi.org/10.1007/978-1-4614-4897-6_1