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

Shoichi Fujimori, Yu Kawakami, Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, Kotaro Yamada

Research output: Chapter in Book/Report/Conference proceedingChapter


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.).

Original languageEnglish
Title of host publicationSpringer Proceedings in Mathematics and Statistics
Number of pages47
Publication statusPublished - 2013


ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Fujimori, S., Kawakami, Y., Kokubu, M., Rossman, W., Umehara, M., & Yamada, K. (2013). Hyperbolic metrics on Riemann surfaces and space-like CMC-1 surfaces in de Sitter 3-space. In Springer Proceedings in Mathematics and Statistics (Vol. 26, pp. 1-47) https://doi.org/10.1007/978-1-4614-4897-6_1