TY - JOUR

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

N1 - Funding Information:
The authors were partially supported by the Grant-in-Aid for Young Scientists (B) No. 21740052 (Fujimori), (B) No. 21740053 (Kawakami), and by the Grant-in-Aid for Scientific Research (C) No. 22540100 (Kokubu), (B) No. 20340012 (Rossman), (A) No. 22244006 (Umehara) and (B) No. 21340016 (Yamada), Japan Society for the Promotion of Science.

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.).

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

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

M3 - Article

AN - SCOPUS:84883346242

VL - 26

SP - 1

EP - 47

JO - Springer Proceedings in Mathematics and Statistics

JF - Springer Proceedings in Mathematics and Statistics

SN - 2194-1009

ER -