Hyperbolic metrics on Riemann surfaces and space-like CMC-1 Surfaces in de Sitter 3-Space in "Recent Trends in Lorentzian Geometry"
Hyperbolic metrics on Riemann surfaces and space-like CMC-1 Surfaces in de Sitter 3-Space in "Recent Trends in Lorentzian Geometry"
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“洛伦兹几何的最新趋势”中德西特 3 空间中黎曼曲面和类空间 CMC-1 曲面上的双曲度量
DOI：
10.1007/978-1-4614-4897-6_1
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Masaaki Umehara and Kotaro Yamada
中科院分区：
文献类型：
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作者：
Shoichi Fujimori;Yu Kawakami;Masatoshi Kokubu;Wayne Rossman;Masaaki Umehara and Kotaro Yamada
引入了广义双曲度量的概念，即定义在黎曼曲面上的具有某种奇点的双曲度量(即常曲率度量−1)，并给出了这种度量的几个基本性质。推广的双曲度量与De Sitter空间S13中常平均曲率为1的类空曲面(即CMC-1曲面)密切相关。例如，S13中给定的CMC-1曲面的奇异集包含在相关的扩展双曲度规的奇异集中。然后，我们对S13中的所有链状曲面(即S13中具有两个正则两端且双曲高斯映射为一次的弱完备常平均曲率1曲面)进行分类。这样的表面被称为S13-链状表面。由于S13-链状的模空间与具有两个正则奇点的可共定向扩张双曲度量的模空间之间存在双射，因此还给出了这种双曲度量的分类。(定义了扩展双曲度量的同方向性。)
We introduce a new notion called theextended 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-spaceS13. For example, the singular set of a given CMC-1 surface inS13is contained in the singular set of the associated extended hyperbolic metric. We then classify all catenoids inS13(i.e. weakly complete constant mean curvature 1 surfaces inS13of genus zero with two regular ends whose hyperbolic Gauss map is of degree one). Such surfaces are calledS13-catenoids. Since there is a bijection between the moduli space ofS13-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.)