Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space

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.)

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

$${h}_{\lambda }{(d{z}_{\lambda })}^{2} - {h}_{ \mu }{(d{z}_{\mu })}^{2} = {S}_{{ z}_{\lambda }}({z}_{\mu }){(d{z}_{\lambda })}^{2}$$

(46)

on ${U}_{\lambda } \cap {U}_{\mu }$. Here ${h}_{\lambda }$ is a meromorphic function in ${z}_{\lambda }$ for each $\lambda \in \Lambda$, and ${S}_{{z}_{\lambda }}({z}_{\mu })$ denotes the Schwarzian derivative of ${z}_{\mu } = {z}_{\mu }({z}_{\lambda })$ with respect to the coordinate ${z}_{\lambda }$.

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

$$S(G) :={ \left \{{S}_{{z}_{\lambda }}(G){(d{z}_{\lambda })}^{2}\right \}}_{ \lambda \in \Lambda }.$$

Then S(G) gives a projective connection on M. In this sense, our definition of the Schwarzian derivative S(G) as in Eq. (15) should be considered as a projective connection.

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

$$g \circ {T}^{-1} = \rho (T) \star g\qquad \left (T \in {\pi }_{ 1}(M)\right ).$$

The map g is called a developing map of P. The representationρ is called the monodromy representation of P.

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

$${C}_{\Gamma } :=\{ \sigma \in \text{ PSL}(2,\mathbf{C})\,;\,\sigma \Gamma {\sigma }^{-1} \subset \text{PSU}(1,1)\}.$$

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

$${I}_{\Gamma } := \text{PSU}(1,1)\setminus {C}_{\Gamma },$$

the structure of the set ${C}_{\Gamma }$ is completely determined. Define a map $\tilde{\varphi } : {C}_{\Gamma } \rightarrow {S}_{1}^{3}$ by

$$\tilde{\varphi }(\sigma ) := {\sigma }^{{_\ast}}{e}_{ 3}\sigma,$$

where ${S}_{1}^{3}$ is the de Sitter 3-space expressed by ${S}_{1}^{3} :=\{ a{e}_{3}{a}^{{_\ast}}\,;\,a \in \text{ PSL}(2,\mathbf{C})\}$. Then it induces an injective map $\varphi : {I}_{\Gamma } \rightarrow {S}_{1}^{3}$ such that $\varphi \circ \pi =\tilde{ \varphi }$, where $\pi : {C}_{\Gamma } \rightarrow {I}_{\Gamma }$ is the canonical projection. So we can identify I _Γ with a subset $\varphi ({I}_{\Gamma }) =\tilde{ \varphi }({C}_{\Gamma })$ of the de Sitter 3-space S ₁ ³. The following assertion holds.

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

$${C}_{\gamma } :=\{ \sigma \in \text{ PSL}(2,\mathbf{C})\,;\,\sigma \gamma {\sigma }^{-1} \in \text{PSU}(1,1)\}.$$

Then we have

$${C}_{\Gamma } :={ \bigcap \limits_{\gamma \in \Gamma }}{C}_{\gamma }.$$

(47)

The condition $\sigma \gamma {\sigma }^{-1} \in \text{PSU}(1,1)$ is rewritten as ${e}_{3}{\sigma }^{{_\ast}}{e}_{3} \cdot \gamma = \gamma {e}_{3} \cdot {\sigma }^{{_\ast}}{e}_{3} \cdot \sigma$, that is ${e}_{3}{\sigma }^{{_\ast}}{e}_{3}\sigma \in {Z}_{\gamma }$. So we have

$$\tilde{\varphi }({C}_{\gamma }) = {S}_{1}^{3} \cap {e}_{ 3}{Z}_{\gamma },$$

(48)

where Z _γ is the center of $\gamma \in \Gamma$ in { PSL}(2, C). In the following discussions, Γ can be considered as a subgroup of { SU}(1, 1) by ignoring the ± -ambiguity.

If $\Gamma \subset \{\pm {e}_{0}\}$, then obviously

$$\varphi ({I}_{\Gamma }) = {S}_{1}^{3}.$$

So we may assume that $\Gamma \not\subset \{ \pm {e}_{0}\}$. Take an element $\gamma \in \Gamma$ such that $\gamma \neq \pm {e}_{0}$. Then γ is conjugate to Λ _e or Λ _h or Λ _p in { SU}(1, 1).

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

$$E := \left (\begin{array}{c@{\quad }c} \alpha \quad & 0\\ 0\quad &{\alpha }^{-1} \end{array} \right )\qquad (\alpha \in \mathbf{C}\setminus \{1,0\}),$$

$$H := \left (\begin{array}{c@{\quad }c} \alpha \quad &\beta \\ \beta \quad &\alpha \end{array} \right )\qquad ({\alpha }^{2}-{\beta }^{2} = 1,\,\,\alpha,\beta \in \mathbf{C}),$$

$$P := \left (\begin{array}{c@{\quad }c} 1 + \mathrm{i}\beta \quad & \beta \\ \beta \quad &1 -\mathrm{i}\beta \end{array} \right )\qquad (\beta \in \mathbf{C}\setminus \{0\}).$$

Then the centers of them in $\text{ SL}(2,\mathbf{C})$ are given as follows:

$$\begin{array}{rcl} {Z}_{E}& =& \left \{\left (\begin{array}{c@{\quad }c} {e}^{z}\quad & 0 \\ 0 \quad &{e}^{-z} \end{array} \right )\,;\,z \in \mathbf{C}\right \}, \\ {Z}_{H}& =& \left \{\pm \left (\begin{array}{c@{\quad }c} \cosh z\quad &\sinh z\\ \sinh z\quad &\cosh z\end{array} \right )\,;\,z \in \mathbf{C}\right \}, \\ {Z}_{P}& =& \left \{\left (\begin{array}{c@{\quad }c} 1 + \mathrm{i}z\quad & z\\ z \quad &1 -\mathrm{i}z \end{array} \right )\,;\,z \in \mathbf{C}\right \}.\end{array}$$