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.)
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The authors thank Sadayoshi Kojima and Shingo Kawai for their valuable comments.
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1 Projective Connections
To define extended hyperbolic metrics, we recall the definition of projective connections on Riemann surfaces: Throughout this section, we fix a connected Riemann surface M.
Definition A.1 (cf. [17]).
Let \(\{{({U}_{\lambda },{z}_{\lambda })\}}_{\lambda \in \Lambda }\) be a covering of M consisting of local complex coordinates. A family of meromorphic sections \(P :=\{ {h}_{\lambda }\,{(d{z}_{\lambda }){}^{2}\}}_{\lambda \in \Lambda }\) is called a projective connection if it satisfies
We now give a typical example.
Example A.2.
Let G be a meromorphic function on M and \(\{{({U}_{\lambda },{z}_{\lambda })\}}_{\lambda \in \Lambda }\) a covering of M consisting of local complex coordinates. We set
The difference between two projective connections is a meromorphic 2-differential of M. Let \(P :=\{ {h}_{\lambda }\,{(d{z}_{\lambda }){}^{2}\}}_{\lambda \in \Lambda }\) be a projective connection on M. A point p ∈ M is called a singularity of P if it is a pole of \({h}_{\lambda }{(d{z}_{\lambda })}^{2}\) for λ satisfying \(p \in {U}_{\lambda }\). If P has no singularities, it is called a non-singular projective connection or holomorphic projective connection.
The order m of the pole of \({h}_{\lambda }\,{(d{z}_{\lambda })}^{2}\) at a singularity p is independent of the choice of the indices λ because of Eq. (46). The integer m( ≥ 1) is called the order of the singularity at p. If m ≤ 2, p is called a regular singularity of P.
The following fact is well known:
Fact A.3 ([8]).
Let \(P :=\{ {h}_{\lambda }{(d{z}_{\lambda }){}^{2}\}}_{\lambda \in \Lambda }\) be a projective connection on M, which is free of singularities. Then there exists a meromorphic function g on \(\widetilde{M}\) such that \({S}_{{z}_{\lambda }}(g) = {h}_{\lambda }\) holds on \({U}_{\lambda }\) for each index \(\lambda \in \Lambda \) , where \(\widetilde{M}\) is the universal covering of M. Moreover, there exists a group representation \(\rho : {\pi }_{1}(M) \rightarrow \text{ PSL}(2,\mathbf{C})\) such that
A developing map g of a given projective connection is not uniquely determined. For each \(a \in \text{ SL}(2,\mathbf{C})\), \(a \star g\) is also a developing map. The developing maps of P are determined up to such an ambiguity of the action of \(\text{ SL}(2,\mathbf{C})\).
1 A Property of Subgroups in PSU(1,1)
This appendix is an analogue of the appendix of [19], where the PSU(2) case was treated. Let Γ be a subgroup of \(\text{PSU}(1,1) = \text{ SU}(1,1)/\{ \pm 1\}\). We prove a property of a set of groups conjugate to Γ in \(\text{ PSL}(2,\mathbf{C})\) defined by
If \(\sigma \in {C}_{\Gamma }\), it is obvious that \(a\sigma \in {C}_{\Gamma }\) for all a ∈ PSU(1, 1). So if we consider the left quotient space
Proposition B.1.
The subset \(\varphi ({I}_{\Gamma })\) is a point, a geodesic line, or all of \({S}_{1}^{3}\) .
Proof.
For each \(\gamma \in \Gamma \), we set
If \(\Gamma \subset \{\pm {e}_{0}\}\), then obviously
Firstly, we consider the case that Γ is abelian. In these three cases, Lemma B.2 implies that Γ must be a subgroup of \({Z}_{{\Lambda }_{e}}\) or \({Z}_{{\Lambda }_{h}}\) or \({Z}_{{\Lambda }_{p}}\). Then \(\varphi ({I}_{\Gamma })\) consists of a geodesic.
Next we suppose that Γ is not abelian. Then there exists an element \(\gamma ^\prime \in \Gamma \) such that \(\gamma \gamma ^\prime\neq \gamma ^\prime\gamma \). We take \(a \in {Z}_{\gamma } \cap {Z}_{\gamma ^\prime}\) arbitrarily and suppose that \(a\neq \pm {e}_{0}\). Since \(a \in {Z}_{\gamma }\), a belongs to \({Z}_{{\Lambda }_{e}}\) or \({Z}_{{\Lambda }_{h}}\) or \({Z}_{{\Lambda }_{p}}\). Then Lemma B.2 yields that \({Z}_{a} = {Z}_{\gamma }\). Moreover, since \(a\gamma ^\prime = \gamma ^\prime a\), we have \(\gamma ^\prime \in {Z}_{a}\). Since \({Z}_{a} = {Z}_{\gamma }\), we have \(\gamma \gamma ^\prime = \gamma ^\prime\gamma \), a contradiction. Thus \(a = \pm {e}_{0}\) and \(\varphi ({I}_{\Gamma })\) consists of a point.
In the proof of Proposition B.1, we applied the following assertion, which can be proved easily.
Lemma B.2.
We set
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Fujimori, S., Kawakami, Y., Kokubu, M., Rossman, W., Umehara, M., Yamada, K. (2012). Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_1
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