### 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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Title of host publication | Springer Proceedings in Mathematics and Statistics |

Pages | 1-47 |

Number of pages | 47 |

Volume | 26 |

DOIs | |

Publication status | Published - 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Springer Proceedings in Mathematics and Statistics*(Vol. 26, pp. 1-47) https://doi.org/10.1007/978-1-4614-4897-6_1

**Hyperbolic metrics on Riemann surfaces and space-like CMC-1 surfaces in de Sitter 3-space.** / Fujimori, Shoichi; Kawakami, Yu; Kokubu, Masatoshi; Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Springer Proceedings in Mathematics and Statistics.*vol. 26, pp. 1-47. https://doi.org/10.1007/978-1-4614-4897-6_1

}

TY - CHAP

T1 - Hyperbolic metrics on Riemann surfaces and space-like CMC-1 surfaces in de Sitter 3-space

AU - Fujimori, Shoichi

AU - Kawakami, Yu

AU - Kokubu, Masatoshi

AU - Rossman, Wayne

AU - Umehara, Masaaki

AU - Yamada, Kotaro

PY - 2013

Y1 - 2013

N2 - 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.).

AB - 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.).

UR - http://www.scopus.com/inward/record.url?scp=84883346242&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883346242&partnerID=8YFLogxK

U2 - 10.1007/978-1-4614-4897-6_1

DO - 10.1007/978-1-4614-4897-6_1

M3 - Chapter

VL - 26

SP - 1

EP - 47

BT - Springer Proceedings in Mathematics and Statistics

ER -