A Geometric Characterization of the Symmetrized Bidisc

TL;DR

This paper provides a geometric characterization of the symmetrized bidisc, highlighting its unique properties related to complex geodesics and automorphisms, and establishing conditions that uniquely identify it among complex manifolds.

Contribution

It introduces a set of geometric conditions that uniquely characterize the symmetrized bidisc within the class of complex manifolds.

Findings

Existence of a unique invariant complex geodesic under all automorphisms.

Foliation of the bidisc by specific complex geodesics meeting the invariant geodesic.

Characterization of the bidisc through geometric properties and automorphism actions.

Abstract

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\} \] has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic $\mathcal{R}$ in $G$ that is invariant under all automorphisms of $G$. Moreover, $G$ is foliated by those complex geodesics that meet $\mathcal{R}$ in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of $G$, characterize the symmetrized bidisc in the class of complex manifolds.

Equations

$$G\stackrel{\mathrm{def}}{=}\{(z+w,zw):|z|<1,|w|<1\}$$

$${\mathbb{A}}_q\stackrel{\mathrm{def}}{=}\{z\in \mathbb{C}:q<|z|<q^{-1}\}$$

$$\{z:|z|=t\}\cup \{z:|z|=t^{-1}\}$$

$$G\stackrel{\mathrm{def}}{=}\{(z+w,zw):|z|<1,|w|<1\}$$

$$(z+w,zw)\mapsto (m(z)+m(w),m(z)m(w))$$

$$1-x^{1}z-x^{2}w+x^{3}zw\ne 0$$

$$F_s\cap \mathcal{R}=\{s\}.$$

$$B_{\alpha }(z)=\frac{z-\alpha }{1-\bar{\alpha }z}$$

$${\mathrm{e}}^{2P(\mu )}(\Theta (B_{it})(\mu )-\mu )=i\left(\Theta (B_t)(\mu )-\mu \right)+o(t)$$

$$\frac{\partial {\xi }_i}{\partial \bar{\beta }}=z\frac{\partial {\xi }_i}{\partial \beta },i=1,2$$

$$\frac{\partial {\xi }_i}{\partial \bar{z}}=0,i=1,2$$

$$\frac{\partial {\xi }_i}{\partial \bar{\beta }}=z\frac{\partial {\xi }_i}{\partial \beta }\text{mbox}and\frac{\partial {\xi }_i}{\partial \bar{z}}=0,i=1,2,$$

$$U_{\Omega }(K,O)\stackrel{\mathrm{def}}{=}\{g\in \mathrm{Aut}\Omega :g(K)\subset O\},$$

$${\gamma }_m(z+w,zw)=(m(z)+m(w),m(z)m(w))$$

$$\gamma (m)={\gamma }_m$$

$$\pi (\overline{{\mathbb{D}}^2})=\{(z+w,zw):|z|\le 1,|w|\le 1\}$$

$$t\cdot \Gamma =\pi (t\overline{{\mathbb{D}}^2})=\{(z+w,zw):|z|\le t,|w|\le t\}$$

$$U_{\mathbb{D}}(t\bar{\mathbb{D}},O)\text{mbox}fort\in (0,1)\text{mbox}andO\text{mbox}openin\mathbb{D}.$$

$$U_G(t\cdot \Gamma ,O)\text{mbox}wheret\in (0,1)\text{mbox}andO\text{mbox}isopeninG$$

$$\gamma (m)\in U_G(t\cdot \Gamma ,O)$$

$$\iff \pi (m(z),m(w))\in O\text{mbox}forallz,w\in t\bar{\mathbb{D}}$$

$$\iff m(t\bar{\mathbb{D}})\times m(t\bar{\mathbb{D}})\subset {\pi }^{-1}(O).$$

$$m_0(t\bar{\mathbb{D}})\times m_0(t\bar{\mathbb{D}})\subset {\pi }^{-1}(O).$$

$$e_s:\mathrm{Aut}\mathbb{D}\to G\text{mbox}givenby{e}_s(m)={\gamma }_m(s)$$

$$\mathcal{R}$$

$$=\{(2z,z^2):z\in \mathbb{D}\}$$

$$R(z)=(2z,z^2)\text{mbox}forz\in \mathbb{D}.$$

$${\gamma }_m(R(z))=R(m(z))\text{mbox}forz\in \mathbb{D}\text{mbox}andm\in \mathrm{Aut}\mathbb{D}.$$

$$r_b(z)=(2b(z),b(z{)}^2),\text{mbox}forz\in \mathbb{D},$$

$$\gamma_{m}(r_{b}(z))=r_{b}\big{(}b^{-1}\circ m\circ b(z)\big{)},\qquad z\in\mathbb{D}.$$

$$r(z)=(2z,z^2),z\in \mathbb{D}$$

$${\gamma }_m(r(z))=r(m(z)),z\in \mathbb{D},m\in \mathrm{Aut}\mathbb{D}.$$

Full text

[TABLE]

has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic $\mathcal{R}$ in $G$ that is invariant under all automorphisms of $G$. Moreover, $G$ is foliated by those complex geodesics that meet $\mathcal{R}$ in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of $G$, characterize the symmetrized bidisc in the class of complex manifolds.

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We also prove a second characterization of $G$ in terms of the existence of global co-ordinates ranging over the bidisc and satisfying certain partial differential equations. These co-ordinates are related to the flat geodesics in $G$.

Partially supported by National Science Foundation Grants DMS 1361720 and 1665260, a Newcastle URC Visiting Professorship and the Engineering and Physical Sciences Research Council grant EP/N03242X/1

1. 1 The action of automorphisms on $G$

2. 1.1 The action of $\operatorname{Aut}\mathbb{D}$ on $G$

3. 1.2 The action of $\operatorname{Aut}G$ on the royal variety

4. 1.3 Orbits in $G$ as manifolds

5. 1.4 The sharp direction in $G$

6. 1.5 Flat geodesics and the action of $\operatorname{Aut}G$

7. 1.6 Synchrony in $G$

8. 2 Royal manifolds

9. 2.1 Regularity properties of royal manifolds

10. 2.2 Flat fibrations over royal discs

11. 2.3 Synchrony in $\Omega$

12. 2.4 The sharp direction in $\Omega$

13. 2.5 Sharpness of the action of $\operatorname{Aut}\Omega$

14. 2.6 A characterization of $G$

15. 3 A characterization of $G$ via flat co-ordinates

16. 4 Asymmetry of domains

Introduction

By a domain we mean a connected open set in $\mathbb{C}^{n}$ for some integer $n\geq 1$.

A domain is homogeneous

if the automorphisms of the domain act transitively. It is symmetric

if every point of the domain is an isolated fixed point of an involutive automorphism of the domain.

The nature of a bounded symmetric homogeneous domain in $\mathbb{C}^{n}$ is captured by the great classification theorem of Élie Cartan [8],

an early triumph of the theory of several complex variables [14, 16].

It states that any such domain is isomorphic to a product of domains, each of which is isomorphic to a domain of one of six concrete types. The theorem is fundamental to the complex geometry and function theory of bounded symmetric homogeneous domains.

In this paper we are interested in irreducible domains $\Omega$ which narrowly miss being homogeneous, in the sense that the action of the automorphisms of $\Omega$ splits the domain into a one-parameter family of orbits.

Such domains are said to have cohomogeneity $1$, and have an extensive theory [15, 11] in both the mathematical and physics literatures.

One familiar domain that has cohomegeneity $1$ is the annulus

[TABLE]

where $0<q<1$. The orbits here are the sets

[TABLE]

where $q<t\leq 1$.

For a higher-dimensional example, consider the domain

[TABLE]

in $\mathbb{C}^{2}$, known as the symmetrized bidisc.

The automorphisms of $G$ are the maps of the form

[TABLE]

for some automorphism $m$ of the unit disc $\mathbb{D}$. The orbits in $G$ are therefore generically $3$-dimensional real manifolds, and there is a one-parameter family of them.

Another domain, now in $\mathbb{C}^{3}$, having a one-parameter family of orbits is the tetrablock,

which comprises the points $(x^{1},x^{2},x^{3})\in\mathbb{C}^{3}$ such that

[TABLE]

for all $z,w\in\mathbb{C}$ such that $|z|\leq 1$ and $|w|\leq 1$.

An ambitious project would be to classify bounded domains in $\mathbb{C}^{n}$, and more generally, complex manifolds, for which the orbits under the automorphisms of the manifold comprise a one-parameter family. By way of a start we shall here characterize in geometric terms our archetypal example $G$ defined in equation (0.3). This domain has been studied by numerous authors over the past 20 years, and has proved to be a domain with a very rich complex geometry and function theory: see, besides many other papers, [4, 10, 13, 17, 22, 19, 25, 2]. $G$ is significant for the theory of invariant distances [18], because it has Lempert’s property, that the Carathéodory and Kobayashi metrics coincide [21], despite the fact that $G$ is not convex (nor even biholomorphic to a convex domain [10]).

It plays a role in operator theory [7, 23] and even has applications to a problem in the theory of robust control (for example, [27]); indeed the control application was the original motivation for the study of $G$. In an earlier paper [3] we characterized $G$ in terms of the Carathéodory extremal functions that it admits. Here we give another characterization, this time in terms of its complex geodesics and automorphisms.

An automorphism of a complex manifold is a bijective holomorphic self-map of the manifold; such a map automatically has a holomorphic inverse.

For any complex manifold $\Omega$ we denote by $\operatorname{Aut}\Omega$ the automorphism group of $\Omega$ with the compact-open topology. A complex geodesic of $G$ can be defined as the range of an analytic map $f:\mathbb{D}\to G$ that has an analytic left inverse, where $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$.

We draw attention to two striking geometric properties of $G$.

1. (1)

There exists a unique complex geodesic $\mathcal{R}$ in $G$ that is invariant under all automorphisms of $G$. Moreover, every automorphism of $\mathcal{R}$ extends to a unique automorphism of $G$; 2. (2)

for every $s\in\mathcal{R}$ there exists a unique geodesic $F_{s}$ in $G$ having a nontrivial stabilizer in $\operatorname{Aut}G$ and such that

[TABLE]

Moreover, the geodesics $\{F_{s}:s\in\mathcal{R}\}$ foliate $G$.111That is, every point of $G$ lies in some $F_{s}$ and no point of $G$ lies in two distinct $F_{s}$

We call $\mathcal{R}$ the royal variety and the sets $F_{s}$ the flat geodesics of $G$.

Could it be that properties (1) and (2) suffice to characterize $G$? In the present paper we show that the answer is yes under some further geometric hypotheses, which we now describe.

We say that a properly embedded analytic disc222not assumed to be a geodesic $D$ in a complex manifold $\Omega$ is a royal disc if it has properties analogous to those of $\mathcal{R}$ in (1),

that is, $D$ is invariant under every automorphism of $\Omega$, and every automorphism of $D$ extends to a unique automorphism of $\Omega$. A royal manifold is a pair $(\Omega,D)$ where $\Omega$ is a complex manifold and $D$ is a royal disc in $\Omega$.

If $(\Omega,D)$ is a royal manifold then a collection $\mathcal{E}=\{E_{\lambda}:\lambda\in D\}$ of properly embedded analytic discs333again, not assumed to be geodesics in $\Omega$ is a flat fibration over $D$ if it has properties similar to those of $\{F_{s}:s\in\mathcal{R}\}$ in (2), that is, $E_{\lambda}\cap D=\{\lambda\}$ for every $\lambda\in D$, $\mathcal{E}$ is a partition of $\Omega$ and, for every automorphism $\theta$ of $\Omega$ and every $\lambda\in D$, $\theta(E_{\lambda})=E_{\theta(\lambda)}$. The triple $(\Omega,D,\mathcal{E})$ is then called a flatly fibered royal manifold.

The orbits in $(\Omega,D,\mathcal{E})$ have a natural parametrization by $[0,\infty)$. For any $\mu\in\Omega$ there is a unique $\lambda\in D$ such that $\mu\in E_{\lambda}$; we define the Poincaré parameter $P(\mu)$

to be the Poincaré distance from $\mu$ to $\lambda$ in the disc $E_{\lambda}$ (see Definition 2.26). Two points $\mu_{1},\mu_{2}$ in $\Omega$ lie in the same orbit if and only if $P(\mu_{1})=P(\mu_{2})$.

Flatly fibered royal manifolds can enjoy two geometric properties: synchrony and sharpness. Synchrony

is a condition which relates the actions of $\operatorname{Aut}\Omega$ on $D$ and on the discs in $\mathcal{E}$. To be precise, if $\theta$ is an automorphism of $\Omega$ which fixes a point $\lambda\in D$, then it follows easily from the definition of a flat fibration over a royal manifold that the eigenspaces of the operator $\theta^{\prime}(\lambda)$ on the tangent space $T_{\lambda}\Omega$ to $\Omega$ at $\lambda$

are the tangent spaces $T_{\lambda}D$ and $T_{\lambda}E_{\lambda}$. We say that $(\Omega,D,\mathcal{E})$ is synchronous if, for every $\lambda\in D$, the eigenvalue of $\theta^{\prime}(\lambda)$ corresponding to $T_{\lambda}E_{\lambda}$ is the square of the eigenvalue of $\theta^{\prime}(\lambda)$ corresponding to $T_{\lambda}D$.

Sharpness

is a condition on the action of $\operatorname{Aut}\mathbb{D}$ on $\Omega$ in a flatly fibered royal manifold $(\Omega,D,\mathcal{E})$. The definition of $(\Omega,D,\mathcal{E})$ implies that every $m\in\operatorname{Aut}\mathbb{D}$ induces an automorphism $\Theta(m)$ of $\Omega$. For $\alpha\in\mathbb{D}$, let $B_{\alpha}$ denote the automorphism

[TABLE]

of $\mathbb{D}$.

We say that $\operatorname{Aut}\Omega$* acts sharply* at a point $\mu\in\Omega\setminus D$ if, in local co-ordinates,

[TABLE]

as $t\to 0$ in $\mathbb{R}$.

The geometric content of the sharpness condition at $\mu$ relates to the derivative at zero of the map $\alpha\mapsto\Theta(B_{\alpha})(\mu)$ from $\mathbb{D}$ to $\Omega$. This map is a priori a real-linear map from $T_{0}\mathbb{D}$ to $T_{\mu}\Omega$; now $T_{0}\mathbb{D}(=\mathbb{C})$ and $T_{\mu}\Omega$ are both complex vector spaces, and the sharpness condition is equivalent to the statement that the derivative at zero is also a complex linear map.

If we denote by $\mu^{\sharp}$ the range of this complex-linear derivative, then it is easy to see that $\mu^{\sharp}$

is the unique nonzero complex linear subspace of $T_{\mu}\Omega$ that is contained in the $3$-dimensional real tangent space at $\mu$ to the orbit of $\mu$ in $\Omega$.

The sharp direction

$\mu^{\sharp}$ is a covariant line bundle over $\Omega$ which has interesting geometric properties. For example, in $G$, the sharp direction $\mu^{\sharp}$ is characterized by the fact that the complex geodesic $\mathcal{C}$ through $\mu$ with direction $\mu^{\sharp}$ has the closest point property, meaning that, for any point $\lambda\in\mathcal{C}$, if $F_{s},\,s\in\mathcal{R}$, is the flat geodesic containing $\mu$, then the closest point to $\lambda$ in $F_{s}$ is $\mu$.

Our main result, Theorem 2.39 in the body of the paper, gives a precise version of the following statement, which holds under suitable regularity conditions.

Theorem A. Let $\Omega$ be a complex manifold. $\Omega$ is isomorphic to $G$ if and only if there exist a royal disc $D$ in $\Omega$ and a flat fibration $\mathcal{E}$ of $\Omega$ over $D$ such that $(\Omega,D,\mathcal{E})$ is a synchronous flatly fibered royal manifold and $\operatorname{Aut}{\Omega}$ acts sharply on $E_{\lambda}\setminus\{\lambda\}$ for some $\lambda\in D$.

Formal definitions of synchrony and sharp action are given in Subsections 2.3 and 2.5. The appropriate notion of regularity is described in Subsection 2.1.

Remarkably, Theorem A implies that if $(\Omega,D,\mathcal{E})$ is a synchronous flatly fibered royal domain with suitable regularity, and $\operatorname{Aut}\Omega$ acts sharply, then both $D$ and the leaves in $\mathcal{E}$ are complex geodesics of $\Omega$. It suggests that $G$ might be characterized also in terms of the properties of its complex geodesics, and in a future paper we shall show that it is so.

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Theorem A also opens the door to the use of co-ordinates related to flat fibrations, and in Theorem 3.2, we show the following. Theorem B. If $\Omega$ is a domain in $\mathbb{C}^{2}$, then $\Omega$ is biholomorphic to $G$ if and only if there exists a differentiable homeomorphism $\Xi=(\xi_{1},\xi_{2})$ from $\mathbb{D}^{2}$ onto $\Omega$ satisfying

[TABLE]

and

[TABLE]

at all $(\beta,z)\in\mathbb{D}^{2}$.

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In Section 3 we give a characterization of $G$ via ‘flat coordinates’, which are related to the flat geodesics in property (2) of $G$ described above. In Theorem 3.2 we show that a domain $\Omega$ in $\mathbb{C}^{2}$ is biholomorphic to $G$ if and only if there exists a differentiable homeomorphism $\Xi=(\xi_{1},\xi_{2})$ from $\mathbb{D}^{2}$ onto $\Omega$ satisfying

[TABLE]

at all $(\beta,z)\in\mathbb{D}^{2}$.

In Section 3 we give in Theorem 3.2 a characterization of $G$ in terms of the existence of global co-ordinates ranging over the bidisc and satisfying certain partial differential equations. These co-ordinates are related to the flat geodesics in $G$.

In a short final section we discuss the relevance of the notion of symmetric space to the question of classification and show that the annulus, the symmetrized bidisc and the tetrablock, besides being inhomogeneous, also fail to be symmetric in É. Cartan’s sense.

If $U$ and $\Omega$ are complex manifolds, we denote by $\Omega(U)$ the set of holomorphic mappings from $U$ into $\Omega$.

We have used the expression properly embedded analytic disc in a complex manifold $\Omega$. By this phrase we mean a proper injective analytic map $k:\mathbb{D}\to\Omega$ such that $k^{\prime}(z)\neq 0$ for all $z\in\mathbb{D}$. The range of such a map $k$ will also be called a properly embedded analytic disc.

1. The action of automorphisms on $G$

In this section we study the orbit structure of $G$ under the action of $\operatorname{Aut}G$.

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$\operatorname{Aut}\Omega$ carries a natural topology, the compact-open topology, with respect to which it is a topological group. This topology is defined to be the coarsest topology containing all sets of the form

[TABLE]

where $K$ is compact and $O$ is open in $\Omega$. Clearly the compact-open topology on $\operatorname{Aut}\Omega$ has a countable base. A theorem of H. Cartan [9]

states that if $\Omega$ is irreducible (that is, not isomorphic to a product of domains)

then $\operatorname{Aut}\Omega$ is a Lie group and the group action $(\operatorname{Aut}\Omega)\times\Omega\to\Omega$ is a real-analytic map of real manifolds.

1.1. The action of $\operatorname{Aut}\mathbb{D}$ on $G$

As we stated in the introduction (see equation (0.4)), every automorphism $m$ of $\mathbb{D}$ induces a map $\gamma_{m}:G\to G$ via the formula

[TABLE]

for $z,w\in\mathbb{D}$.

It is easy to check that this formula defines a map $\gamma_{m}\in G(G)$ and that $\gamma_{m}\in\operatorname{Aut}G$.

Proposition 1.1**.**

The map $\gamma:\operatorname{Aut}\mathbb{D}\to\operatorname{Aut}G$ given by

[TABLE]

for $m\in\operatorname{Aut}\mathbb{D}$ is a continuous isomorphism of topological groups.

The fact that $\gamma$ is an isomorphism of groups is proved in [6, Theorem 5.1] or [18]. It is routine to show that $\gamma$ is continuous with respect to the compact-open topologies on $\operatorname{Aut}\mathbb{D}$ and $\operatorname{Aut}G$.

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We shall denote by $\Gamma$ the closed symmetrised bidisc

[TABLE]

and by $t\cdot\Gamma$ the compact set

[TABLE]

for $t>0$.

Observe that a base for the topology of $\operatorname{Aut}\mathbb{D}$ is furnished by the sets of the form

[TABLE]

Similarly, the sets

[TABLE]

constitute a base for the topology of $\operatorname{Aut}G$.

Consider a basic open set $U_{G}(t\cdot\Gamma,O)$ in $\operatorname{Aut}G$ for some $t\in(0,1)$ and open subset $O$ of $G$. For any $m\in\operatorname{Aut}\mathbb{D}$,

[TABLE]

Consider any point $m_{0}\in\operatorname{Aut}\mathbb{D}$ and any basic open set $U_{G}(t\cdot\Gamma,O)$ of $\gamma(m_{0})$. By the statement (1.4),

[TABLE]

The following lemma is evident.

If $C$ is a compact subset of $\mathbb{D}$ and $W$ is an open neighborhood of $C\times C$ in $\mathbb{D}^{2}$ then there is an open neighborhood $V$ of $C$ in $\mathbb{D}$ such that $V\times V\subset W$.

It follows from equation (1.5) that there exists an open neighborhood $V$ of $m_{0}(t\bar{\mathbb{D}})$ in $\mathbb{D}$ such that $\pi(V\times V)\subset O$. Clearly $N\stackrel{{\scriptstyle\rm def}}{{=}}U_{\mathbb{D}}(t\bar{\mathbb{D}},V)$ is an open neighborhood of $m_{0}$ in $\operatorname{Aut}\mathbb{D}$ such that $\gamma(N)\subset U_{G}(t\cdot\Gamma,O)$. Hence $\gamma$ is continuous at $m_{0}$.

The theorem of Cartan [9] on continuous transformation groups mentioned above implies the following (Cartan’s statement is slightly more general). Let $\Omega$ be a domain in $\mathbb{C}^{d}$. A topological group $A$ is said to act continuously on $\Omega$ if a continuous homomorphism $\Phi:A\to\operatorname{Aut}\Omega$ is given. The pair $(A,\Phi)$ is called a continuous transformation group on $\Omega$. The statement is (see, for example, [kaup67, page 45]):

If $(A,\Phi)$ is a continuous transformation group on $\Omega$, $A$ is locally compact, $\Phi$ is injective and $\Omega$ is irreducible, then $A$ is a Lie group and the map $A\times\Omega\to\Omega$ given by $(g,\lambda)\mapsto\Phi(g)(\lambda)$ is real-analytic.

Note that $G$ is irreducible. Otherwise $G$ is isomorphic to a product $U\times V$ of one-dimensional domains. These domains are clearly connected and simply connected, hence are isomorphic to $\mathbb{D}$. This implies that $G$ is isomorphic to the bidisc $\mathbb{D}^{2}$, which is easily seen to be false (for example, $\mathbb{D}^{2}$ is homogeneous while $G$ is not).

Here is an immediate consequence.

The following statements are elementary.

Proposition 1.2**.**

1. (1)

$\operatorname{Aut}\mathbb{D}$* and $\operatorname{Aut}G$ are Lie groups.* 2. (2)

For any $s\in G$ the map

[TABLE]

is real-analytic.

The map $e_{s}$, where $s\in G$, will be called the evaluation map at $s$ on $\operatorname{Aut}\mathbb{D}$.

1.2. The action of $\operatorname{Aut}G$ on the royal variety

The royal variety in $G$ is defined to be the set

[TABLE]

(we use superscripts to denote the components of a point in $\mathbb{C}^{d}$). Thus $\mathcal{R}=R(\mathbb{D})$ where

[TABLE]

Clearly

[TABLE]

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How does $\gamma_{m}$ act on $\mathcal{R}$? If $\delta$ is a datum that contacts $\mathcal{R}$ then the general solution to $\operatorname{{\mathrm{K}ob}}({\delta})$ has the form

[TABLE]

where $b$ is an element of $\operatorname{Aut}(G)$. It then follows from equation (1.2) that

[TABLE]

Thus, if one views $r_{b}$ as a co-ordinate for $\mathcal{R}$, then the action of $\gamma_{m}$ on $\mathcal{R}$ corresponds to the action of $b^{-1}\circ m\circ b$ on $\mathbb{D}$. In the sequel, we shall set $r=r_{b}$ in the special case that $b={\mathrm{id}}{\mathbb{D}}$. Thus

[TABLE]

and

[TABLE]

The observations (1.3) and (1.8) have three consequences, summarized in the following proposition.

Proposition 1.3**.**

1. (1)

Every automorphism of $G$ leaves $\mathcal{R}$ invariant. 2. (2)

Every automorphism of $G$ is uniquely determined by its values on $\mathcal{R}$. 3. (3)

Every automorphism of $\mathcal{R}$ has a unique extension to an automorphism of $G$.

In statement (3), automorphisms of $\mathcal{R}$ are with respect to the structure of $\mathcal{R}$ as a complex manifold.

We can summarize these three statements by saying that the restriction map $\gamma\mapsto\gamma|\mathcal{R}$ is an isomorphism from $\operatorname{Aut}G$ to $\operatorname{Aut}\mathcal{R}$. The following commutative diagram describes the situation, where $\iota_{\mathcal{R}}$ denotes the injection of $\mathcal{R}$ into $G$ and $m\in\operatorname{Aut}\mathbb{D}$.

[TABLE]

1.3. Orbits in $G$ as manifolds

For any complex manifold $U$ and any $\lambda\in U$, we denote by $\operatorname{{\rm Orb}}_{U}(\lambda)$

the orbit of $\lambda$ under the action of the group of automorphisms of $U$:

[TABLE]

Consider the case that $U=G$ and $\lambda=s\in G$.

In view of Proposition 1.1, for any $s\in G$,

[TABLE]

so that $\operatorname{{\rm Orb}}_{G}(s)$ is the range of the evaluation map $e_{s}$ of equation (1.6).

$\operatorname{Aut}\mathbb{D}$ is a $3$-dimensional real-analytic manifold, for which we shall need local co-ordinates.

Lemma 1.4**.**

For $(r,\alpha)\in\mathbb{R}\times\mathbb{D}$ let $m_{r,\alpha}\in\operatorname{Aut}\mathbb{D}$ be given by the formula

[TABLE]

Let

[TABLE]

and define

[TABLE]

and similarly for $\varphi_{2}:U_{2}\to(0,2\pi)\times\mathbb{D}$. Then $(U_{1},\varphi_{1})$ and $(U_{2},\varphi_{2})$ are charts in $\operatorname{Aut}\mathbb{D}$ which together comprise a real-analytic atlas $\mathcal{A}$ for the group manifold. The identity automorphism $\mathrm{id}_{\mathbb{D}}=m_{0,0}$ belongs to $U_{1}$.

Proof.

The automorphisms of $\mathbb{D}$ consist of the maps $m_{r,\alpha}$ for $r\in[-2\pi,2\pi]$ and $\alpha\in\mathbb{D}$, and therefore $\operatorname{Aut}\mathbb{D}=U_{1}\cup U_{2}$. If $-\pi<r<0$ then

[TABLE]

and similarly when $0<r<\pi$. The transition map is therefore real-analytic from $\varphi_{1}(U_{1}\cap U_{2})$ to $\varphi_{2}(U_{1}\cap U_{2})$. ∎

Proposition 1.5**.**

If $s\not\in\mathcal{R}$ then the evaluation map $e_{s}:\operatorname{Aut}\mathbb{D}\to\operatorname{{\rm Orb}}_{G}(s)$ is a local homeomorphism and a two-to-one covering map, given explicitly by

[TABLE]

Proof.

Consider a point $s=(z+w,zw)\in G$ where $z,w\in\mathbb{D}$ and $z\neq w$. Let $\upsilon$ be the unique automorphism of $\mathbb{D}$ that maps $z$ to $w$ and $w$ to $z$. Note that $\upsilon$ is not the identity automorphism ${\mathrm{id}}_{\mathbb{D}}$ since $z\neq w$. For $m_{1},m_{2}\in\operatorname{Aut}\mathbb{D}$,

[TABLE]

Thus $e_{s}:m\mapsto\gamma_{m}(s)$ is two-to-one from $\operatorname{Aut}\mathbb{D}$ to $\operatorname{{\rm Orb}}_{G}(s)$.

To prove that $e_{s}$ is a local homeomorphism, choose any point $e_{s}(\beta)$ of $\operatorname{{\rm Orb}}_{G}(s)$, where $\beta\in\operatorname{Aut}\mathbb{D}$. Choose a neighborhood $U$ of $\mathrm{id}_{\mathbb{D}}$ such that

[TABLE]

We claim that $\beta\circ U$ is a neighborhood of $\beta$ on which $e_{s}$ is injective. Certainly it is a neighborhood of $\beta$, and if $e_{s}|\beta\circ U$ is not injective then there exist distinct points $m_{1},m_{2}\in U$ such that $e_{s}(\beta\circ{m_{1}})=e_{s}(\beta\circ{m_{2}})$. That is, $\gamma_{\beta}\circ\gamma_{m_{1}}(s)=\gamma_{\beta}\circ\gamma_{m_{2}}(s)$, and therefore $\gamma_{m_{1}}(s)=\gamma_{m_{2}}(s)$. Hence, by the equivalence (1.22), $m_{2}^{-1}\circ m_{1}=\upsilon$. This equation contradicts the statement (1.23). Thus $e_{s}$ is locally injective on $\operatorname{Aut}{\mathbb{D}}$.

Choose a compact neighborhood $V$ of $\mathrm{id}_{\mathbb{D}}$ contained in $U$. Since a continuous bijective map from a compact space to a Hausdorff space is a homeomorphism, $e_{s}|V$ is a homeomorphism onto its range. It follows by homogeneity that $e_{s}$ is a local homeomorphism on $\operatorname{Aut}\mathbb{D}$. Indeed, consider any $m\in\operatorname{Aut}\mathbb{D}$ and its neighborhood $m\circ V$. Define $L_{m}:\operatorname{Aut}\mathbb{D}\to\operatorname{Aut}\mathbb{D}$ by $L_{m}(\theta)=m^{-1}\circ\theta$ for $\theta\in\operatorname{Aut}\mathbb{D}$. In the commutative diagram

[TABLE]

the map $e_{s}|m\circ V$ is expressed as the composition of three homeomorphisms, and so is itself a homeomorphism. Thus $e_{s}$ is a local homeomorphism.

The formula for $e_{s}(m_{r,\alpha})$ is a simple calculation. ∎

For any $s\in G$, the map $e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})$ is a real-linear map from the tangent space $T_{\mathrm{id}_{\mathbb{D}}}\operatorname{Aut}\mathbb{D}$ to $T_{s}\operatorname{{\rm Orb}}_{G}(s)$. The space $T_{\mathrm{id}_{\mathbb{D}}}\operatorname{Aut}\mathbb{D}$ is the Lie algebra of $\operatorname{Aut}\mathbb{D}$, so we shall denote it by $\operatorname{\mathrm{Lie}(\mathrm{Aut}~{}\mathbb{D})}$

(though we shall not use its Lie structure, only its real-linear structure).

For every $s\in G$ we define a real-linear subspace $\mathcal{V}(s)$

of $\mathbb{C}^{2}$ by

[TABLE]

Theorem 1.6**.**

1. (1)

If $s\in\mathcal{R}$, then $\operatorname{{\rm Orb}}_{G}(s)$ is a one-dimensional complex manifold properly embedded in $G$. 2. (2)

If $s\in G\setminus\mathcal{R}$, then $\operatorname{{\rm Orb}}_{G}(s)$ is a three-dimensional real-analytic manifold properly embedded in $G$.

Moreover, in either case, the tangent space to $\operatorname{{\rm Orb}}_{G}(s)$ at $s$ is $\mathcal{V}(s)$ and

[TABLE]

{comment}

In both cases

[TABLE]

In the sequel the notation $T_{s}\operatorname{{\rm Orb}}_{G}(s)$ denotes the complex tangent space if $s\in\mathcal{R}$ and the real tangent space if $s\notin\mathcal{R}$. Thus, for all $s\in G$,

[TABLE]

Proof.

Consider $s\in G$. We shall calculate the rank of the real linear operator $e_{s}^{\prime}(m)$ for $m\in\operatorname{Aut}\mathbb{D}$. Let $\iota:\operatorname{{\rm Orb}}_{G}(s)\to\mathbb{C}^{2}_{r}$ denote the inclusion map.

Lemma 1.7**.**

For any tangent vector $(r,\alpha)$ at $(0,0)$ to $(-\pi,\pi)\times\mathbb{D}$, let $v_{r,\alpha}(s)$ denote the tangent vector $(e_{s}\circ\varphi_{1}^{-1})^{\prime}(0,0)(r,\alpha)$ in $T_{s}G\subset\mathbb{C}^{2}$. Then

[TABLE]

Proof.

We have $(r,\alpha)\in\mathbb{R}\times\mathbb{C}$. Define a path $\kappa(t)=(tr,t\alpha)$ in $(-\pi,\pi)\times\mathbb{D}$ for $|t|<\varepsilon$, where $\varepsilon$ is small enough.

Then let

[TABLE]

and define $v_{r,\alpha}(s)\in\mathbb{C}^{2}$ by the formula

[TABLE]

From equation (1.15), for any $z\in\mathbb{D}$,

[TABLE]

Hence, by equations (1.2) and (1.31), if $s=(z_{1}+z_{2},z_{1}z_{2})$,

[TABLE]

∎

Continuing the proof of Theorem 1.6, by the Chain Rule we have, from equation (1.30),

[TABLE]

Thus the range of the real linear map $f_{s}^{\prime}(0,0):\mathbb{R}\times\mathbb{C}\to\mathbb{C}^{2}$ is the set

[TABLE]

On taking $(r,\alpha)$ to be successively $(1,0),\,(0,-1)$ and $(0,-i)$ we find that, for any $s\in G$,

[TABLE]

the real vector space introduced in equation (1.25). Thus

[TABLE]

for all $s\in G$. In the sequel we shall suppress the inclusion map $\iota^{\prime}(s)$ and regard $\operatorname{ran}e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})$ as a subspace of $\mathbb{C}^{2}_{r}$.

Now consider $s\in G\setminus\mathcal{R}$. By Lemma 1.8 below, $\dim_{\mathbb{R}}\mathcal{V}(s)=3$, and so, by equation (1.34), $e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})$ has rank $3$. We claim that $e_{s}^{\prime}(m)$ has rank $3$ for all $m\in\operatorname{Aut}\mathbb{D}$. Indeed, on differentiating the relation

[TABLE]

we find (since $e_{s}(\mathrm{id}_{\mathbb{D}})=s$) that

[TABLE]

Since $\gamma_{m}$ is an automorphism of $G$, $\gamma_{m}^{\prime}(s)$ is a nonsingular real linear transformation of $\mathbb{C}^{2}$. Thus

[TABLE]

for every $m\in\operatorname{Aut}\mathbb{D}$.

We wish to deduce that $\operatorname{{\rm Orb}}_{G}(s)$ is a real $3$-dimensional $C^{\infty}$-manifold which (modulo the identification map $\iota^{\prime}(s)$) lies in $\mathbb{C}^{2}$. The following statement is [24, Theorem 5.2].

A subset $M$ of $\mathbb{R}^{n}$ is a $k$-dimensional manifold if and only if, for every point $s\in M$ there exist an open neighborhood $V$ of $s$ in $\mathbb{R}^{n}$, an open set $W$ in $\mathbb{R}^{k}$ and an injective differentiable function $f:W\to\mathbb{R}^{n}$ such that

1. (1)

$f(W)=M\cap V$, 2. (2)

$f^{\prime}(y)$* has rank $k$ for every $y\in W$.*

We shall apply this criterion in the case $n=4,\ k=3,\ M=\operatorname{{\rm Orb}}_{G}(s)$. Consider any point $e_{s}(m)\in\operatorname{{\rm Orb}}_{G}(s)$, where $m\in\operatorname{Aut}\mathbb{D}$, say $m\in U_{j},\,j=1$ or $2$. By Proposition 1.5, $e_{s}$ is a local homeomorphism, and so we may choose an open neighborhood $N$ of $m$ in $U_{j}$ such that $e_{s}|N$ is a homeomorphism from $N$ to an open subset of $\operatorname{{\rm Orb}}_{G}(s)$. Since $\operatorname{{\rm Orb}}_{G}(s)$ has the relative topology induced by $G$, there is an open set $V$ in $\mathbb{R}^{4}$ such that $e_{s}(N)=V\cap\operatorname{{\rm Orb}}_{G}(s)$.

Let $W=\varphi_{j}(N)$. Then the map $f=e_{s}\circ\varphi_{j}^{-1}$ satisfies conditions (1) and (2). It follows that $\operatorname{{\rm Orb}}_{G}(s)$ is a real $3$-dimensional $C^{\infty}$ manifold in $\mathbb{C}^{2}$.

The linear map $(e_{s}\circ\varphi_{1}^{-1})^{\prime}(0,0)$ maps the tangent space $T_{(0,0)}(-\pi,\pi)\times\mathbb{D}$ into the tangent space $T_{s}\operatorname{{\rm Orb}}_{G}(s)\subset\mathbb{C}^{2}$. We have seen that the range of $(e_{s}\circ\varphi_{1}^{-1})^{\prime}(0,0)$ is $\mathcal{V}(s)$. Hence

[TABLE]

Since both spaces have real dimension $3$, the inclusion holds with equality.

In the case that $s\in\mathcal{R}$, say $s=(2\zeta,\zeta^{2})$ for some $\zeta\in\mathbb{D}$,

[TABLE]

which is a one-dimensional complex manifold properly embedded in $G$ by the map $R:\mathbb{D}\to G$. The complex tangent space to $\mathcal{R}$ at $s$ is $\mathbb{C}(1,\zeta)$, and, by equation (1.25),

[TABLE]

Thus $T_{s}\operatorname{{\rm Orb}}_{G}(s)=\mathcal{V}(s)$ in the sense of complex manifolds. ∎

Lemma 1.8**.**

For any $s\in G$, the real vector space $\mathcal{V}(s)$ defined by equation (1.25) satisfies

[TABLE]

Proof.

It is clear from the definition (1.25) that $\mathcal{V}(s)$ is a real vector subspace of $\mathbb{C}^{2}$ of real dimension at most $3$.

Suppose that scalars $\lambda,\mu,\nu\in\mathbb{R}$ satisfy

[TABLE]

Multiply on the left by the row matrix $\begin{pmatrix}2s^{2}&-s^{1}\end{pmatrix}$ to obtain

[TABLE]

Consider the first case in equation (1.35), namely, that $(s^{1})^{2}\neq 4s^{2}$ (equivalently, $s\notin\mathcal{R}$). By equation (1.37)

[TABLE]

whence

[TABLE]

Since $s\in G$, we have $|s^{2}|<1$, and so necessarily $\mu=\nu=0$. Since $(s^{1})^{2}\neq 4s^{2}$, at least one of $s^{1},s^{2}$ is nonzero, and so, by equation (1.36), $\lambda=0$. Hence the three spanning vectors for $\mathcal{V}(s)$ in equation (1.25) are linearly independent. We have shown that $\dim_{\mathbb{R}}\mathcal{V}(s)=3$ when $s\notin\mathcal{R}$.

Next consider a point $s\in\mathcal{R}$. On substituting $s^{2}=\tfrac{1}{4}(s^{1})^{2}$ in equation (1.25) we obtain

[TABLE]

Since each of these vectors is a complex scalar multiple of the vector $\begin{pmatrix}2&s^{1}\end{pmatrix}^{T}$, it follows that $\dim_{\mathbb{R}}\mathcal{V}(s)\leq 2$.

In fact $\dim_{\mathbb{R}}\mathcal{V}(s)=2$. For otherwise the second and third spanning vectors for $\mathcal{V}(s)$ are linearly dependent over $\mathbb{R}$, and so there exist $\mu,\nu\in\mathbb{R}$, not both zero, such that

[TABLE]

and consequently

[TABLE]

Thus $|s^{1}|=2$, contrary to choice of $s\in G$. Therefore

[TABLE]

∎

1.4. The sharp direction in $G$

By Theorem 1.6, for any $s\in G\setminus\mathcal{R}$, the tangent space $\mathcal{V}(s)$ at $s$ to the orbit $\operatorname{{\rm Orb}}_{G}(s)$ is a real $3$-dimensional subspace of $\mathbb{C}^{2}$. Accordingly $\mathcal{V}(s)$ contains a unique $2$-real-dimensional subspace that is also a one-dimensional complex subspace of $\mathbb{C}^{2}$, equal to $\mathcal{V}(s)\cap i\mathcal{V}(s)$. On the other hand, for $s\in\mathcal{R}$, the tangent space $\mathcal{V}(s)$ is already a complex subspace of $\mathbb{C}^{2}$.

Definition 1.9**.**

*For any $s\in G$, the sharp direction at $s$

is the unique nonzero complex subspace of $\mathcal{V}(s)$ in $\mathbb{C}^{2}$ and is denoted by $s^{\sharp}$. Thus*

[TABLE]

The sharp direction is covariant with automorphisms of $G$, in the following sense.

Proposition 1.10**.**

If $\gamma\in\operatorname{Aut}G$ and $s\in G$, then

[TABLE]

Proof.

Since $\gamma$ is a differentiable self-map of $\operatorname{{\rm Orb}}_{G}(s)$, its derivative $\gamma^{\prime}(s)$ is a real-linear map between the tangent spaces $\mathcal{V}(s)$ and $\mathcal{V}(\gamma(s))$. Since furthermore $\gamma^{\prime}(s)$ is a nonsingular complex linear map from $T_{s}G\sim\mathbb{C}^{2}$ to $T_{\gamma(s)}\sim\mathbb{C}^{2}$, it maps the complex subspace $s^{\sharp}$ of $\mathbb{C}^{2}$ to a nonzero complex subspace of $\mathbb{C}^{2}$. Hence $\gamma^{\prime}(s)s^{\sharp}$ is a nonzero complex subspace of $\mathcal{V}(\gamma(s))$. Hence $\gamma^{\prime}(s)s^{\sharp}=\gamma(s)^{\sharp}$. ∎

1.5. Flat geodesics and the action of $\operatorname{Aut}G$

In the introduction we defined the flat geodesics of $G$ to be the geodesics that meet the royal geodesic $\mathcal{R}$ exacly once and are stabilized by a nontrivial automorphism of $G$.

This definition has the merit that it is geometrical in character, but in practice (for example, to show that the flat geodesics foliate $G$) it is often simpler to use the fact that the flat geodesics in $G$ are the sets of the form

[TABLE]

for some $\beta\in\mathbb{D}$.

One can check that the point $s\in G$ lies on the unique $F^{\beta}$ with

[TABLE]

More details can be found in [18, 10, 5] and [2, Appendix A].

Let us at least sketch a proof that the set $F^{\beta}$ is indeed a flat geodesic according to the definition in the introduction. Firstly, a straightforward calculation shows that any automorphism of $G$ maps $F^{\beta}$ to a set of the form $F^{\beta^{\prime}}$ for some $\beta^{\prime}\in\mathbb{D}$. Clearly $F^{\beta}$ is a complex geodesic in $G$: for any $\beta\in\mathbb{D}$ the co-ordinate function $s^{2}$ is a holomorphic left inverse of the properly embedded analytic disc $z\mapsto(\beta+\bar{\beta}z,z)$ in $G$. It is simple to check that $F^{\beta}$ meets $\mathcal{R}$ exactly once, say at the point $s(\beta)\in\mathcal{R}$. Choose a nontrivial automorphism $\theta$ of the analytic disc $\mathcal{R}$ that fixes $s(\beta)$, and let $\gamma$ be the unique extension of $\theta$ to a (necessarily nontrivial) automorphism of $G$. Then $\gamma(F^{\beta})=F^{\beta^{\prime}}$, and since $\gamma$ fixes $s(\beta)$, it follows that $F^{\beta^{\prime}}$ meets $\mathcal{R}$ at $s(\beta)$. Distinct sets $F^{\beta}$ are disjoint, and therefore $\beta=\beta^{\prime}$. That is, $F^{\beta}$ is stabilized by a nontrivial automorphism of $G$.

The converse statement, that every flat geodesic is an $F^{\beta}$, follows from the classification into five types of the complex geodesics in $G$ given in [2, Chapter 7].

We summarize the main geometric properties of flat geodesics.

Proposition 1.11**.**

1. (1)

*Through each point $s$ in $G$ there passes a unique flat geodesic $F_{s}$.

---

2. (2)

Every flat geodesic intersects the royal geodesic $\mathcal{R}$ in exactly one point. 3. (3)

Automorphisms of $G$ carry flat geodesics to flat geodesics.

The following lemma is a reformulation of the first two of these facts.

Lemma 1.12**.**

The family

[TABLE]

is a partition of $G$.

Definition 1.13**.**

*For any $s\in G$, the complex tangent space at $s$ to the unique flat geodesic through $s$ will be called the flat direction at $s$, and will be denoted by $s^{\flat}$.

Thus, if $s^{1}=\beta+\bar{\beta}s^{2}$, then

[TABLE]

which is a one-dimensional complex subspace of $\mathbb{C}^{2}$. The map $s\mapsto s^{\flat}$ is a covariant line bundle which is a sub-bundle of $TG$.

Facts (1)-(3) in Proposition 1.11 imply the following description of the action of $\operatorname{Aut}G$ on $\mathcal{F}$.

Lemma 1.14**.**

If $\gamma\in\operatorname{Aut}G$ and $s\in\mathcal{R}$, then $\gamma(F_{s})=F_{\gamma(s)}$.

Proof.

Fix $\gamma\in\operatorname{Aut}G$ and $s\in\mathcal{R}$. By Fact 3, there exists $t\in\mathcal{R}$ such that $\gamma(F_{s})=F_{t}$, and Condition (i) in Proposition 1.3 implies that $\gamma(s)\in\mathcal{R}$. Therefore $\gamma(s)\in\mathcal{R}\cap F_{t}$. Hence by Fact 2, $t=\gamma(s)$. ∎

We shall call $\{F_{s}:s\in\mathcal{R}\}$ the flat fibration of $G$.

Proposition 1.15**.**

For all $s\in G$ the spaces $s^{\sharp}$ and $s^{\flat}$ are unequal.

This statement will follow from explicit formulae for the sharp and flat directions. We already know that, for $s\in F^{\beta}$, $s^{\flat}$ is given by equation (1.40).

Proposition 1.16**.**

For any $\beta\in\mathbb{D}$ and any $s\in F^{\beta}$,

[TABLE]

Proof.

Consider $s\in G\setminus\mathcal{R}$. Let

[TABLE]

By Theorem 1.6, $\{v_{1},v_{2},v_{3}\}$ constitutes a basis for $T_{s}\operatorname{{\rm Orb}}_{G}(s)$. Let

[TABLE]

and note that $c_{2},c_{3}$ are nonzero. One finds that $c_{1}v_{1}+c_{2}v_{2}+c_{3}v_{3}=0$, and therefore

[TABLE]

Hence

[TABLE]

and

[TABLE]

Further calculation yields the formula

[TABLE]

in agreement with equation (1.41). This proves the proposition in the case that $s\in G\setminus\mathcal{R}$. For $s=(2z,z^{2})\in\mathcal{R}$ equation (1.41) is easily checked. ∎

The fact that $s^{\sharp}\neq s^{\flat}$ can now be verified by a simple comparison of equations (1.41) and (1.40).

Corollary 1.17**.**

The tangent bundle of $G$ is the direct sum of the sharp bundle and the flat bundle:

[TABLE]

1.6. Synchrony in $G$

There is a subtle relationship between the action of an automorphism of $G$ on the royal variety and its action on any flat geodesic.

For any complex manifold $U$ and $\lambda$ in $U$, denote by $\operatorname{Aut}_{\lambda}U$

the stabilizer

of $\lambda$ in $\operatorname{Aut}U$ (also known as the isotropic subgroup of $\operatorname{Aut}U$ at $\lambda$). For any $s_{0}\in\mathcal{R}$, the sets $\mathcal{R}$ and $F_{s_{0}}$ are embedded analytic discs in $G$ that intersect transversally at the point $s_{0}$. Every $\theta$ in $\operatorname{Aut}_{s_{0}}G$ determines an automorphism of the analytic variety $\mathcal{R}\cup F_{s_{0}}$. For an automorphism of a general variety there need be no connection between the action on two leaves beyond what is implied by the condition that the restrictions of the automorphism to the two leaves must agree at any common point. However, in the context of the domain $G$, in the light of Condition (ii) in Proposition 1.3, the action of $\theta$ on $F_{s_{0}}$ is uniquely determined by the action of $\theta$ on $\mathcal{R}$. The following propositions describe this dependence explicitly.

We denote the unit circle $\{z:|z|=1\}$ in the complex plane by $\mathbb{T}$.

For $\eta\in\mathbb{T}$ let $\rho_{\eta}$ denote the element of $\operatorname{Aut}_{0}\mathbb{D}$ defined by $\rho_{\eta}(z)=\eta z$.

Clearly $\operatorname{Aut}_{0}\mathbb{D}=\{\rho_{\eta}:\eta\in\mathbb{T}\}$.

Proposition 1.18**.**

If $s_{0}\in\mathcal{R}$ and $\theta\in\operatorname{Aut}_{s_{0}}G$, then $\theta^{\prime}(s_{0})$ has eigenspaces $T_{s_{0}}\mathcal{R}$ and $T_{s_{0}}F_{s_{0}}$ with corresponding eigenvalues $\eta$ and $\eta^{2}$ for some $\eta\in\mathbb{T}$.

Proof.

Since $\theta$ leaves invariant both $\mathcal{R}$ and $F_{s_{0}}$, it follows that $\theta^{\prime}(s_{0})$ leaves invariant the tangent spaces $T_{s_{0}}\mathcal{R}$ and $T_{s_{0}}F_{s_{0}}$. These two one-dimensional tangent spaces are thus eigenspaces of $\theta^{\prime}(s_{0})$.

Observe that $\gamma_{\rho_{\eta}}$ is the restriction to $G$ of the linear operator on $\mathbb{C}^{2}$ with matrix $\operatorname{diag}\{\eta,\eta^{2}\}$, and hence

[TABLE]

Let $s_{0}=(2\alpha,\alpha^{2})$. Since $\theta\in\operatorname{Aut}_{s_{0}}G$, $\theta=\gamma_{m}$ for some $m\in\operatorname{Aut}\mathbb{D}$ such that $m(\alpha)=\alpha$. Therefore $b_{\alpha}\circ m\circ b_{-\alpha}\in\operatorname{Aut}_{0}\mathbb{D}$, and so there exists $\eta\in\mathbb{T}$ such that

[TABLE]

Since $m\mapsto\gamma_{m}$ is an isomorphism,

[TABLE]

It follows by the chain rule that

[TABLE]

where $X=\gamma_{b_{-\alpha}}^{\prime}{(0,0)}$. But $\operatorname{diag}\{\eta,\eta^{2}\}$ has eigenspaces $\mathbb{C}\oplus 0$ and $0\oplus\mathbb{C}$ with corresponding eigenvalues $\eta$ and $\eta^{2}$. Therefore, $\theta^{\prime}(s_{0})$ has eigenspaces $X(\mathbb{C}\oplus 0)$ and $X(0\oplus\mathbb{C})$ with corresponding eigenvalues $\eta$ and $\eta^{2}$. We have

[TABLE]

Hence

[TABLE]

and therefore

[TABLE]

Thus $T_{s_{0}}\mathcal{R}$ and $T_{s_{0}}F_{s_{0}}$ are eigenspaces of $\theta^{\prime}(s_{0})$ with corresponding eigenvalues $\eta,\eta^{2}$ respectively. ∎

Proposition 1.19**.**

Let $s_{0}=(2\alpha,\alpha^{2})$ for some $\alpha\in\mathbb{D}$ and let $m\in\operatorname{Aut}_{\alpha}\mathbb{D}$. If $g$ is any proper embedding of $\mathbb{D}$ into $G$ such that $g(\mathbb{D})=F_{s_{0}}$ and $g(\alpha)=s_{0}$, then

[TABLE]

Proof.

Note that

[TABLE]

This equation implies that

[TABLE]

which is to say that $m^{\prime}(\alpha)$ is the eigenvalue of $\gamma_{m}^{\prime}(s_{0})$ corresponding to the eigenspace $T_{s_{0}}\mathcal{R}$. Consequently, by Lemma 1.18,

[TABLE]

Since $g$ is a proper embedding and $\gamma_{m}(F_{s_{0}})=F_{s_{0}}$, there exists $b\in\operatorname{Aut}{\mathbb{D}}$ such that

[TABLE]

As $b(\alpha)=\alpha$, this equation implies that

[TABLE]

that is, $b^{\prime}(\alpha)$ is the eigenvalue of $\gamma_{m}^{\prime}(s_{0})$ corresponding to the eigenspace $T_{s_{0}}F_{s_{0}}$. Therefore, statement (1.6) implies that

[TABLE]

Since $b,\,m\circ m\in\operatorname{Aut}\mathbb{D}$, $b(\alpha)=\alpha=(m\circ m)(\alpha)$, and $b^{\prime}(\alpha)=(m\circ m)^{\prime}(z_{0})$, it follows that $b=m\circ m$. ∎

We describe the phenomena described in Propositions 1.18 and 1.19 as the synchrony property of $G$.

2. Royal manifolds

Perhaps the most far-reaching feature of the complex geometry of $G$ is the existence of the special variety $\mathcal{R}$ with the properties described in Proposition 1.3. We formalize these properties in order to characterize $G$ up to isomorphism.

Definition 2.1**.**

*Let $\Omega$ be a complex manifold. We say that $D$ is a royal disc in $\Omega$

if $D$ is a properly embedded analytic disc in $\Omega$ and $D$ satisfies the three conditions of Proposition 1.3*, that is,

1. (1)

every automorphism of $\Omega$ leaves $D$ invariant, 2. (2)

every automorphism of $\Omega$ is uniquely determined by its values on $D$, 3. (3)

every automorphism of $D$ has an extension to an automorphism of $\Omega$.

A royal manifold

is an ordered pair $(\Omega,D)$

where $\Omega$ is a complex manifold and $D$ is a royal disc in $\Omega$.

The following lemma is straightforward.

Lemma 2.2**.**

If $\Omega$ is a complex manifold and $\Lambda:G\to\Omega$ is a biholomorphic map, then $\Lambda(\mathcal{R})$ is a royal disc in $\Omega$ and $(\Omega,\Lambda(\mathcal{R}))$ is a royal manifold.

The next proposition spells out the analog of formula (1.8) on a general royal manifold.

Proposition 2.3**.**

Let $(\Omega,D)$ be a royal manifold. Then $\operatorname{Aut}\Omega$ is isomorphic to $\operatorname{Aut}\mathbb{D}$. Furthermore, if $d:\mathbb{D}\to\Omega$ is a properly embedded analytic disc such that $d(\mathbb{D})=D$, then there exists a unique isomorphism $\Theta:\operatorname{Aut}\mathbb{D}\to\operatorname{Aut}\Omega$ such that

[TABLE]

The counterpart of the commutative diagram (1.13) is

[TABLE]

where $\iota_{D}$ is the injection of $D$ into $\Omega$.

Proof.

Fix a properly embedded analytic disc $d$ such that $d(\mathbb{D})=D$. For each $\tau\in\operatorname{Aut}\Omega$, Condition (i) in Definition 2.1 implies that there exists a function $\varphi_{\tau}:\mathbb{D}\to\mathbb{D}$ such that

[TABLE]

Clearly, since $\tau$ is an automorphism of $\Omega$ and $d$ is a properly embedded analytic disc, $\varphi_{\tau}\in\operatorname{Aut}\mathbb{D}$.

If $\tau_{1},\tau_{2}\in\operatorname{Aut}\Omega$, then for each $z\in\mathbb{D}$ we see using equation (2.3) that

[TABLE]

This relation proves that the map $\Psi:\operatorname{Aut}\Omega\to\operatorname{Aut}\mathbb{D}$ given by

[TABLE]

is a homomorphism of automorphism groups.

If $\tau_{1},\tau_{2}\in\operatorname{Aut}\Omega$ and $\varphi_{\tau_{1}}(z)=\varphi_{\tau_{2}}(z)$ for all $z\in\mathbb{D}$, then equation (2.3) implies that $\tau_{1}(d(z))=\tau_{2}(d(z))$ for all $z\in\mathbb{D}$, which is to say that $\tau_{1}$ and $\tau_{2}$ agree on $D$. Hence, by Condition (ii) in Definition 2.1, $\tau_{1}=\tau_{2}$. This proves that $\Psi$ is injective.

Consider any $b\in\operatorname{Aut}\mathbb{D}$. The map

[TABLE]

is an automorphism of the complex manifold $D$. Condition (iii) in Definition 2.1 implies that there exists $\tau\in\operatorname{Aut}\Omega$ such that $\tau(d(z))=d\circ b(z)$ for all $z\in\mathbb{D}$. But then

[TABLE]

for all $z\in\mathbb{D}$, so that $\varphi_{\tau}=b$. This proves that $\Psi$ is surjective from $\operatorname{Aut}\Omega$ onto $\operatorname{Aut}\mathbb{D}$.

We have shown that $\Psi$ is an isomorphism of groups. In particular, the first assertion of Proposition 2.3 (that $\operatorname{Aut}\Omega$ is isomorphic to $\operatorname{Aut}\mathbb{D}$) is proven. To define an isomorphism $\Theta$ satisfying the second assertion of the proposition, let $\Theta=\Psi^{-1}$. Then $\Theta$ is an isomorphism from $\operatorname{Aut}\mathbb{D}$ onto $\operatorname{Aut}\Omega$, and equation (2.1) follows from the relation (2.3).

To see that $\Theta$ is unique consider $m\in\operatorname{Aut}\mathbb{D}$ and observe that if $\Theta_{1}$ and $\Theta_{2}$ are isomorphisms satisfying equation (2.3), then $\Theta_{1}(m)(d(z))=\Theta_{2}(m)(d(z))$ for all $z\in\mathbb{D}$. Since $\Theta_{1}(m)$ and $\Theta_{2}(m)$ agree on $D$, Condition (ii) in Definition 2.1 imples that $\Theta_{1}(m)=\Theta_{2}(m)$. As $m$ is arbitrary, $\Theta_{1}=\Theta_{2}$. ∎

In the light of Proposition 2.3 we adopt the following definition.

Definition 2.4**.**

*Let $(\Omega,D)$ be a royal manifold. We say that $(d,\Theta)$ is a concomitant pair for $(\Omega,D)$

if $d:\mathbb{D}\to\Omega$ is a proper analytic embedding, $d(\mathbb{D})=D$, and $\Theta:\operatorname{Aut}\mathbb{D}\to\operatorname{Aut}\Omega$ is an isomorphism of groups that satisfies, for all $m\in\operatorname{Aut}\mathbb{D}$,*

[TABLE]

as in equation (2.1).

In other words, $(d,\Theta)$ is a concomitant pair for $(\Omega,D)$ if the diagram (2.2) commutes for every $m\in\operatorname{Aut}\mathbb{D}$.

Remark 2.5**.**

Concomitant pairs are essentially unique in the following sense. If $(\Omega,D)$ is a royal manifold and $(d_{0},\Theta_{0})$ is a concomitant pair for $(\Omega,D)$, then $(d,\Theta)$ is a concomitant pair for $(\Omega,D)$ if and only if there exists $b\in\operatorname{Aut}\mathbb{D}$ such that $d=d_{0}\circ b$ and $\Theta=\Theta_{0}\circ I_{b}$, where $I_{b}$ denotes the inner automorphism of $\operatorname{Aut}\mathbb{D}$ defined by $I_{b}(m)=b\circ m\circ b^{-1}$.

As a companion to Lemma 2.2 we have the following equally straightforward lemma.

Lemma 2.6**.**

If $\Omega$ is a complex manifold, $\Lambda:G\to\Omega$ is a biholomorphic map, $d=\Lambda\circ R$ and

[TABLE]

then $(d,\Theta)$ is a concomitant pair for $(\Omega,\Lambda(\mathcal{R}))$.

Definition 2.7**.**

*A concomitant pair $(d,\Theta)$ for a royal manifold $(\Omega,D)$ is consistent with

a bijective map $\Lambda:G\to\Omega$ if $d=\Lambda\circ R$ and $\Theta(m)=\Lambda\circ\gamma_{m}\circ\Lambda^{-1}$ for all $m\in\operatorname{Aut}\mathbb{D}$.*

{comment}

Continuity properties of the isomorphism $\Theta$ appearing in Definition 2.4 will be important in the sequel.

Recall that a domain is said to be taut

if the set $\Omega(\mathbb{D})$ is normal, that is, if every sequence $(f_{n})$ in $\Omega(\mathbb{D})$ either has a locally uniformly convergent subsequence or has a locally uniformly divergent subsequence.

{comment}

We make two observations about continuity.

Remark 2.8**.**

If $(\Omega,D)$ is a royal domain, $(d,\Theta)$ is a concomitant pair for $(\Omega,D)$ and $D$ is an analytic retract in $\Omega$, then $d$ has an analytic left inverse. For we may choose an idempotent $\rho\in\Omega(\Omega)$ such that $\operatorname{ran}\rho=D$; then $\Phi\stackrel{{\scriptstyle\rm def}}{{=}}d^{-1}\circ\rho\in\mathbb{D}(\Omega)$ and $\Phi\circ d=\mathrm{id}_{\mathbb{D}}$. It follows from equation (2.1) that

[TABLE]

Remark 2.9**.**

If $(\Omega,D)$ is a royal domain, $(d,\Theta)$ is a concomitant pair for $(\Omega,D)$ and $D$ is an analytic retract in $\Omega$, then $\Theta^{-1}$ is a continuous map from $\operatorname{Aut}\Omega$ to $\operatorname{Aut}\mathbb{D}$.

Proof.

Consider any point $g_{0}\in\operatorname{Aut}\Omega$ and any basic neighborhood $U_{\mathbb{D}}(K,O)$ of $\Theta^{-1}(g_{0})$ in $\operatorname{Aut}\mathbb{D}$, where $K,O$ are compact and open respectively in $\mathbb{D}$. By equation (2.6),

[TABLE]

By choice of $K$ and $O$,

[TABLE]

and therefore

[TABLE]

It follows that, if

[TABLE]

then $g_{0}\in N$. That is, $N$ is a basic open neighborhood of $g_{0}$ in $\operatorname{Aut}\Omega$. If $g\in N$ then

[TABLE]

which is to say that

[TABLE]

Therefore

[TABLE]

Hence $\Theta^{-1}$ is continuous on $\operatorname{Aut}\Omega$.

∎

{comment}

The next two results will enable us to show that $\Theta$ is continuous for any royal domain.

Lemma 2.10**.**

The composition map $\Omega(\Omega)\times\Omega(\Omega)\to\Omega(\Omega)$ is continuous for the compact-open topology.

Proof.

Consider $f_{0},g_{0}\in\Omega(\Omega)$. We prove that the map $(f,g)\mapsto f\circ g$ is continuous at $(f_{0},g_{0})$. Consider a basic neighborhood $U(K,O)\stackrel{{\scriptstyle\rm def}}{{=}}\{f:f(K)\subset O\}$ of $f_{0}\circ g_{0}$ in $\Omega(\Omega)$, where $K,O$ are compact and open respectively in $\Omega$. Then $f_{0}\circ g_{0}(K)\subset O$, and so $g_{0}(K)\subset f_{0}^{-1}(O)$. Choose an open neighborhood $O_{1}$ of $g_{0}(K)$ such that $\overline{O_{1}}$ is compact and

[TABLE]

Now if $f\in U(\overline{O_{1}},O)$ and $g\in U(K,O_{1})$ then

[TABLE]

and so $f\circ g(K)\subset O$. Hence $U(\overline{O_{1}},O)\times U(K,O_{1})$ is a neighborhood of $(f_{0},g_{0})$ whose image under composition is contained in $U(K,O)$. Thus composition is continuous at $(f_{0},g_{0})$. ∎

Lemma 2.11**.**

Let $\Omega$ be a taut domain. If $(h_{n})$ is a sequence in $\Omega(\Omega)$ converging to $h\in\bar{\Omega}(\Omega)$ then $h(\Omega)$ lies either in $\Omega$ or in $\partial\Omega$.

Proof.

Suppose not. Then $h^{-1}(\Omega)$ is a nonempty proper open subset of $\Omega$, and so its closure in $\Omega$ contains a point $\lambda_{1}\notin h^{-1}(\Omega)$. Choose an open ball $B$ centered at $\lambda_{1}$ contained in $\Omega$. Then $B$ contains a point $\lambda_{2}\in h^{-1}(\Omega)$ and an analytic disc $D$ such that $\lambda_{1},\lambda_{2}\in D$. We have $h(\lambda_{1})\in\partial\Omega$ and $h(\lambda_{2})\in\Omega$. However, $(h_{n}|D)$ is a sequence of analytic maps from the disc to $\Omega$, and so by tautness, the sequence either has a subsequence that converges locally uniformly in $\Omega(\mathbb{D})$ (in which case $h(D)\subset\Omega$) or diverges locally uniformly on $D$ (in which case $h(D)\subset\partial\Omega$). This is a contradiction. ∎

Proposition 2.12**.**

Let $(\Omega,D)$ be a royal domain and let $(d,\Theta)$ be a concomitant pair. If $\Omega$ is taut and $\operatorname{Aut}\Omega$ is locally bounded on $\Omega$, then the map $\Theta:\operatorname{Aut}\mathbb{D}\to\operatorname{Aut}\Omega$ is continuous.

Proof.

The compact-open topology on $\operatorname{Aut}\Omega$, which is the same as the topology of locally uniform convergence, is metrizable. It therefore suffices to prove the sequential continuity of $\Theta$.

Consider a sequence $(m_{n})$ in $\operatorname{Aut}\mathbb{D}$ converging to $\mathrm{id}_{\mathbb{D}}$. Note that, for any $\lambda\in D$, say $\lambda=d(z)$ where $z\in\mathbb{D}$,

[TABLE]

Since $\operatorname{Aut}\Omega$ is locally bounded, Montel’s Theorem applies to the sequence $(\Theta(m_{n}))$ in $\Omega(\Omega)$ to show that the sequence either has a subsequence that converges to an element of $\Omega(\Omega)$ or has a subsequence that diverges locally uniformly on $\Omega$. Statement (2.7) rules out the second alternative, and so, after passage to a subsequence, we may assume that $\Theta(m_{n})$ converges to a function $h\in\Omega(\Omega)$ which fixes every point of $D$. Similarly, after passage to a further subsequence, we may suppose that $\Theta(m_{n})^{-1}$ converges to some function $g\in\Omega(\Omega)$. Since the composition map $(h,g)\mapsto h\circ g$ is continuous for the compact-open topology,

[TABLE]

Likewise $g\circ h={\mathrm{id}}_{\Omega}$. Thus $h\in\operatorname{Aut}\Omega$. Since $h$ acts as the identity automorphism on $D$, it follows from property (ii) of Definition 2.1 that $h={\mathrm{id}}_{\Omega}$. We have shown that $\Theta$ is continuous at $\mathrm{id}_{\mathbb{D}}$. It follows easily that $\Theta$ is continuous at every point of $\operatorname{Aut}\mathbb{D}$. ∎

Corollary 2.13**.**

If $(d,\Theta)$ is a concomitant pair for a royal manifold $(\Omega,D)$, where $\Omega$ is isomorphic to a bounded taut domain, then $\Theta$ is continuous.

Proof.

Suppose $\Lambda:\Omega\to\Omega_{1}$ is a biholomorphic map and $\Omega_{1}$ is a bounded taut domain. Then $(\Omega_{1},\Lambda(D))$ is a royal domain for which $(\Lambda\circ d,\Theta_{1})$ is a concomitant pair, where $\Theta_{1}(m)=\Lambda\circ\Theta(m)\circ\Lambda^{-1}$ for every $m\in\operatorname{Aut}\mathbb{D}$. By Proposition 2.12, $\Theta_{1}$ is continuous. Therefore $\Theta(\cdot)=\Lambda^{-1}\circ\Theta_{1}(\cdot)\circ\Lambda$ is continuous. ∎

2.1. Regularity properties of royal manifolds

{comment}

Theorem 1.6 states some properties of the orbits in $G$ under $\operatorname{Aut}\mathbb{D}$. These properties are preserved by isomorphisms, and so they imply conditions on a royal manifold that are necessary for the complex manifold to be isomorphic to $G$.

For the extension of these notions to a general royal domain $(\Omega,D)$ it is desirable that the orbits $\operatorname{{\rm Orb}}_{\Omega}(s)$ be smooth manifolds. In particular, we want the analog of the vector $v_{r,\alpha}(s)$ in the last subsection to be well defined. We therefore formulate a definition. Note first that, according to Cartan’s Theorem, $\operatorname{Aut}\Omega$ is a Lie group and the action $(\operatorname{Aut}\Omega)\times\Omega\to\Omega$ is a real-analytic map of real manifolds. A fortiori, for any $\lambda\in\Omega$, the evaluation map $E_{\lambda}:\operatorname{Aut}\Omega\to\mathbb{C}^{2}$ given by

[TABLE]

is a differentiable map of real manifolds. We introduce the notation

[TABLE]

Thus $T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is a real subspace of the real tangent space $T_{\lambda}\mathbb{C}^{2}$.

Definition 2.14**.**

*Let $(\Omega,D)$ be a royal manifold and $(d,\Theta)$ a concomitant pair. We say that $(\Omega,D)$ is a regular royal manifold if *

1. (1)

$\Theta:\operatorname{Aut}\mathbb{D}\to\operatorname{Aut}\Omega$* is differentiable;* 2. (2)

for every $\lambda\in\Omega\setminus D$, the stabilizer of $\lambda$ in $\operatorname{Aut}\Omega$ is finite, and 3. (3)

for every $\lambda\in\Omega\setminus D$, $e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})$ is an invertible real-linear map, where $e_{\lambda}:\operatorname{Aut}\mathbb{D}\to\Omega$ is defined by

[TABLE]

Remark 2.15**.**

If the complex manifold $\Omega$ is isomorphic to a bounded taut domain [18], then $\Theta$ is automatically differentiable – indeed, by a theorem of H. Cartan [9], real analytic.

$e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})$ is a real-linear map between real tangent spaces,

[TABLE]

Conditions (1) to (3) are certainly necessary for $\Omega$ to be biholomorphic to $G$. They do not depend on the choice of concomitant pair for $(\Omega,D)$.

The following statement is simple to prove.

Proposition 2.16**.**

If $\Omega$ and $\Lambda$ are as in Lemma 2.6 then $(\Omega,\Lambda(\mathcal{R}))$ is a regular royal manifold.

There is an analog of Proposition 1.5 for $G$.

Proposition 2.17**.**

If $(\Omega,D)$ is a regular royal manifold then, for any $\lambda\in\Omega\setminus D$, the map $e_{\lambda}:\operatorname{Aut}\mathbb{D}\to\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is a local homeomorphism and an $N$-to-one covering map, where $N$ is the order of the stabilizer group of $\lambda$ in $\operatorname{Aut}\Omega$.

Proof.

Let $(d,\Theta)$ be a concomitant pair for $(\Omega,D)$ and let $H$ be the stabilizer of $\lambda$ in $\operatorname{Aut}\Omega$. By condition (2) in Definition 2.14, $H$ is a finite subgroup of $\operatorname{Aut}\Omega$. For any $m_{1},m_{2}\in\operatorname{Aut}\mathbb{D}$,

[TABLE]

Since $\Theta$ is bijective, $|\Theta^{-1}(H)|=N$. It follows that $e_{\lambda}$ is an $N$-to-one map.

To prove that $e_{\lambda}$ is a local homeomorphism, consider any point $e_{\lambda}(\beta)$ of $\operatorname{{\rm Orb}}_{\Omega}(\lambda)$, where $\beta\in\operatorname{Aut}\mathbb{D}$. Choose a neighborhood $U$ of $\mathrm{id}_{\mathbb{D}}$ such that

[TABLE]

Let $V$ be a compact neighborhood of $\mathrm{id}_{\mathbb{D}}$ contained in $U$. Then $\beta\circ V$ is a compact neighborhood of $\beta$ on which $e_{\lambda}$ is injective, and so $e_{\lambda}|\beta\circ V$ is a homeomorphism onto its range. Thus $e_{\lambda}$ is a local homeomorphism. ∎

Remark 2.18**.**

The natural analog of equation (1.35), to wit,

[TABLE]

is not required in Definition 2.14, since the condition holds automatically, as is clear from Proposition 2.20 below.

Proposition 2.19**.**

Let $(\Omega,D)$ be a regular royal manifold with concomitant pair $(d,\Theta)$.

1. (1)

If $\lambda\in D$ then $\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is a one-dimensional complex manifold properly embedded in $\Omega$. 2. (2)

If $\lambda\in\Omega\setminus D$, then $\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is a three-dimensional real manifold properly embedded in $\Omega$.

In either case,

[TABLE]

Proof.

(1) Let $\lambda\in D$. By conditions (1) and (3) in Definition 2.1, $\operatorname{{\rm Orb}}_{\Omega}(\lambda)=D$, which is by hypothesis a properly embedded analytic disc in $\Omega$ and therefore a one-dimensional complex manifold.

(2) The proof that $\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is a $3$-dimensional real manifold for any $\lambda\in\Omega\setminus D$ is almost identical to the proof of the corresponding statement for $G$ in Theorem 1.6, and so we omit it. ∎

{comment}

Let $(d,\Theta)$ be a concomitant pair for $\Omega$ and let $d=(d^{1},d^{2}):\mathbb{D}\to\Omega$.

From the relation

[TABLE]

it follows that

[TABLE]

and therefore, since $\Theta(m)^{\prime}$ is an invertible real linear map of real tangent spaces, the regularity assumption (LABEL:aut110) implies that

[TABLE]

Consider an arbitrary point $e_{\lambda}(m)\in\operatorname{{\rm Orb}}_{\Omega}(\lambda)$, where $m\in\operatorname{Aut}\mathbb{D}$. By Proposition 2.17 we may choose an open neighborhood $N$ of $m$ in $U_{j},\,j=1$ or $2$, (the chart defined in equation (1.16)) such that $e_{\lambda}|N$ is a homeomorphism from $N$ to an open neighborhood of $e_{\lambda}(m)$ in $\operatorname{{\rm Orb}}_{\Omega}(\lambda)$. Let $V$ be an open set in $\Omega$ such that $e_{\lambda}(N)=V\cap\operatorname{{\rm Orb}}_{\Omega}(\lambda)$.

Let $W=\varphi_{j}(N)$. The map $f=e_{\lambda}\circ\varphi_{j}^{-1}$ is an injective differentiable function on an open subset of $\mathbb{R}^{3}$ which satisfies the conditions

(1) $f(W)=\operatorname{{\rm Orb}}_{\Omega}(\lambda)\cap V$,

(2) $f^{\prime}(y)$ has rank $3$ for every $y\in W$.

It follows that $\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is a real $3$-dimensional real manifold in $\Omega_{r}$.

By definition, $e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})$ is a real linear map between the real tangent spaces $T_{\mathrm{id}_{\mathbb{D}}}\operatorname{Aut}\mathbb{D}$ and $T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$. Since both spaces have real dimension $3$ and $e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})$ has rank $3$, it follows that $\operatorname{ran}e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})=T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$.

For a domain $U$ in $\mathbb{C}^{n}$, when necessary we shall write $U_{r}$ for $U$ considered as a $2n$-dimensional real manifold and $U_{c}$ for $U$ as a complex manifold. For $p\in U$ the spaces $T_{p}U_{r},\,T_{p}U_{c}$ are respectively the real and complex tangent spaces to $U$ at $p$. We regard elements of $T_{p}U_{r}$ as point derivations at $p$ on the algebra $C^{1}_{p}(U)$ of germs at $p$ of real-valued $C^{1}$ functions on $U_{r}$. Elements of $T_{p}U_{c}$ are point derivations at $p$ on the algebra $\mathcal{O}_{p}(U)$ of germs at $p$ of holomorphic functions on $U_{c}$. We express the action of a point derivation $\delta$ on a germ $g$ of the appropriate type by the notation $\left\langle g,\delta\right\rangle$.

The complexification $(T_{p}M)_{\mathbb{C}}$ of the real tangent space at $p$ to a real manifold $M$ is the complex vector space comprising the point derivations at $p$ on the complex algebra $C^{1}_{p}(M,\mathbb{C})$ of germs at $p$ of complex-valued $C^{1}$ functions on $M$. If $\delta\in(T_{p}M)_{\mathbb{C}}$ then the functional $\operatorname{Re}\delta$ on $C^{1}_{p}(M)$ defined by

[TABLE]

is a point derivation, that is, a member of $T_{p}M$. We also define $\operatorname{Im}\delta\in T_{p}M$ to be $-\operatorname{Re}(i\delta)$. In the reverse direction, for a tangent vector $\delta\in T_{p}M$ we denote by $\delta_{\mathbb{C}}$ the complexification of $\delta$, so that, for any complex-valued $C^{1}$ function $h$ in a neighborhood of $p$,

[TABLE]

Then, for $\delta\in(T_{p}M)_{\mathbb{C}}$, the relation $\delta=(\operatorname{Re}\delta)_{\mathbb{C}}+i(\operatorname{Im}\delta)_{\mathbb{C}}$ holds. Note that, for $\delta\in T_{p}M$, we have $\operatorname{Re}(\delta_{\mathbb{C}})=\delta$.

Furthermore, since every holomorphic function on $U_{c}$ is a $\mathbb{C}$-valued $C^{1}$ function on $U_{r}$, every tangent vector $\delta\in(T_{p}U_{r})_{\mathbb{C}}$ determines by restriction an element $\delta|\mathcal{O}$ of $T_{p}U_{c}$.

We can summarize the various tangent spaces and their inclusions in the diagram

[TABLE]

The vector spaces in the bottom row are respectively real of dimension $2n$, complex of dimension $2n$ and complex of dimension $n$. The composition of $\cdot_{\mathbb{C}}$ and $\cdot|\mathcal{O}$ is a natural real-linear map

[TABLE]

For $\delta\in T_{p}U_{r}$, the complex tangent vector $\kappa\delta$ satisfies, for $g\in\mathcal{O}_{p}(U_{c})$,

[TABLE]

the last line by equation (2.12). In terms of the traditional co-ordinates $z^{j}=x^{j}+iy^{j}$ in a neighborhood of $p$,

[TABLE]

Therefore $\kappa$ is surjective, and since both domain and codomain have real dimension $2n$, it follows that $\kappa$ is a real linear isomorphism.

For $\lambda\in D$ the orbit $\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is the royal disc $D$, which is a properly embedded analytic disc under the complex structure induced by $\Omega$. Let the evaluation map $e_{\lambda}$ be as in Definition 2.14, so that $e_{\lambda}(m)=\Theta(m)(\lambda)$. The derivative $e^{\prime}_{\lambda}(\mathrm{id}_{\mathbb{D}})$ is then a real-linear map from $\operatorname{\mathrm{Lie}(\mathrm{Aut}~{}\mathbb{D})}$ to the real tangent space $T_{\lambda}D_{r}$, and so if $\kappa:T_{\lambda}D_{r}\to T_{\lambda}D_{c}$ is the natural embedding of real and complex tangent spaces, then

[TABLE]

is a real-linear map from a $3$-dimensional real space to a $1$-dimensional complex space. In fact this map is surjective.

Proposition 2.20**.**

Let $(\Omega,D)$ be a regular royal manifold and let $\lambda\in D$. Then

[TABLE]

Proof.

Let $(d,\Theta)$ be a concomitant pair for $(\Omega,D)$.

Since $\lambda\in D$, there exists $z_{0}\in\mathbb{D}$ such that $\lambda=d(z_{0})$. Consider a tangent vector $\delta$ to $\operatorname{Aut}\mathbb{D}$ at $\mathrm{id}_{\mathbb{D}}$. We shall calculate $e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})\delta$. For any germ $g$ of real-valued $C^{1}$ functions on $\Omega$ at $\lambda$,

[TABLE]

For $m\in\operatorname{Aut}\mathbb{D}$,

[TABLE]

Hence

[TABLE]

where $g\circ d\circ m(z_{0})$ is understood as a real-valued function of $m$, with $z_{0}$ fixed. Recall the local co-ordinates $r,\alpha$ for $\operatorname{Aut}\mathbb{D}$ introduced in equation (1.15). Here we shall write $\alpha=\xi+i\eta$, with $\xi,\eta\in\mathbb{R}$ and shall use the local co-ordinates $r,\xi,\eta$ for $\operatorname{Aut}\mathbb{D}$. Note that $\mathrm{id}_{\mathbb{D}}\in\operatorname{Aut}\mathbb{D}$ corresponds to the local co-ordinates $r=\xi=\eta=0$. By an elementary calculation,

[TABLE]

Here $(g\circ d)^{\prime}(z_{0})$ is a real linear functional on $T_{z_{0}}\mathbb{D}_{r}$. If

[TABLE]

for some real $\delta_{1},\delta_{2},\delta_{3}$, then

[TABLE]

Now we calculate $\kappa e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})\in T_{\lambda}D_{c}$. To this end consider any $h\in\mathcal{O}_{\lambda}(D)$. By equation (2.14),

[TABLE]

Thus, by equation (2.17),

[TABLE]

Since $d$ is only determined up to composition with an automorphism of $\mathbb{D}$, no generality is lost by the assumption that $z_{0}=0$. Hence

[TABLE]

On the other hand,

[TABLE]

and therefore

[TABLE]

Thus

[TABLE]

{comment}

The right hand side of this equation is a real number, the third factor is a complex number and the first two factors are real-linear mappings

[TABLE]

Let us write down the $1\times 2$ matrix of $g^{\prime}(\lambda)k^{\prime}(z_{0}):T_{z_{0}}\mathbb{D}_{r}\to\mathbb{R}$ in terms of the real variables $x,y$ where $z=x+iy\in\mathbb{D}$. Points of $\Omega$ will be written $(\lambda^{1},\lambda^{2})=(x^{1}+iy^{1},x^{2}+iy^{2})$, where $x^{j},y^{j}\in\mathbb{R}$. Then

[TABLE]

Let $k=(k^{1},k^{2})$ and write $k^{j}(x+iy)=u^{j}(x,y)+iv^{j}(x,y)$. Then

[TABLE]

where superscript $T$ denotes transposition. Therefore

[TABLE]

at $(x,y)$. Hence

[TABLE]

where $\beta_{1},\beta_{2}\in T_{\lambda}\Omega_{r}$ are given by

[TABLE]

[TABLE]

where, as usual,

[TABLE]

Since $k$ is by hypothesis a properly embedded analytic disc, $k^{\prime}(z)\neq 0$, and therefore $v\neq 0$.

It is immediate from equation (LABEL:striking) that $\operatorname{rank}_{\mathbb{R}}e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})$ is at most $2$. To show it is at least $2$ one can calculate $\left\langle\mathrm{ev}_{z},\delta_{\mathbb{C}}\right\rangle$ in local co-ordinates $r,\alpha$ as in equation (1.15), or better, $r,\xi,\eta\in\mathbb{R}$ where $\alpha=\xi+i\eta$. One finds that if

[TABLE]

where $\delta_{1},\delta_{2},\delta_{3}\in\mathbb{R}$, then

[TABLE]

Since $|z|<1$, the vectors $z^{2}-1$ and $-i(z^{2}+1)$ are linearly independent in the real vector space $\mathbb{C}$, and so

[TABLE]

Hence

[TABLE]

and $\operatorname{rank}_{\mathbb{R}}e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})=2$, as claimed.

∎

{comment}

I don’t think we need the following.

A comparison of formulas (1.31) and (LABEL:aut106) allows one to set up a linear correspondence between the spaces $\mathcal{V}$ and $\mathcal{W}$ defined in (LABEL:aut96) and (LABEL:aut108) respectively. Explicitly, for fixed $s\in G$ and $\lambda\in\Omega$ we define the real linear mapping

[TABLE]

by the formula

[TABLE]

Note here, that the notation has suppressed the dependence of $d\Gamma/d\gamma$ on $s$ and $\lambda$. In the sequel, when this is an issue, we shall employ the somewhat ungainly expression

[TABLE]

I think that

[TABLE]

for all $s\in G\setminus\mathcal{R}$ and $\lambda\in\Omega$.

2.2. Flat fibrations over royal discs

In this subsection we shall formalize the consequences for isomorphs of $G$ of the flat fibration of $G$ described in Subsection 1.5.

Definition 2.21**.**

*Let $(\Omega,D)$ be a royal manifold. If $\mathcal{E}=\left\{E_{\lambda}\right\}_{\lambda\in D}$ is a family of subsets of $\Omega$ indexed by $D$, then we say that $\mathcal{E}$ is a flat fibration of $\Omega$ over $D$

if*

1. (1)

for each $\lambda\in D$, $E_{\lambda}$ is a properly embedded analytic disc in $\Omega$ such that $E_{\lambda}\cap D=\{\lambda\}$; 2. (2)

$\mathcal{E}$* is a partition of $\Omega$, and* 3. (3)

if $\theta\in\operatorname{Aut}\Omega$ and $\lambda\in D$, then $\theta(E_{\lambda})=E_{\theta(\lambda)}$.

*We say that $(\Omega,D,\mathcal{E})$ is a flatly fibered royal manifold

if $(\Omega,D)$ is a royal manifold and $\mathcal{E}$ is a flat fibration of $\Omega$ over $D$. We define the flat direction

$\lambda^{\flat}$ at a point $\lambda$ in $\Omega$ to be the tangent space at $\lambda$ to $E_{\mu}$ where $\mu\in D$ and $\lambda\in E_{\mu}$.*

Clearly, if $(\Omega,D,\mathcal{E})$ is a flatly fibered royal manifold then $\Omega$ has complex dimension $2$.

Note that if $(\Omega,D)$ happens to be $(G,\mathcal{R})$ then the definition of the flat direction is consistent with that given earlier in Definition 1.13.

Lemma 2.22**.**

Let $\Omega$ be a complex manifold, let $\Lambda:G\to\Omega$ be a biholomorphic map, let $D=\Lambda(\mathcal{R})$ and let

[TABLE]

where $\{F_{s}:s\in\mathcal{R}\}$ is the flat fibration of $G$. Then

[TABLE]

is a flat fibration of the royal manifold $(\Omega,D)$ over $D$, and $(\Omega,D,\mathcal{E})$ is a flatly fibered royal manifold.

Proof.

By Lemma 2.2, $(\Omega,D)$ is a royal manifold. Since $\Lambda$ is a bijection from $\mathcal{R}$ to $D$, we may write $\mathcal{E}=\{E_{\lambda}:\lambda\in D\}$. Properties (1) and (2) of Definition 2.21 for the sets $E_{\lambda}$ follow from the corresponding properties of the sets $F_{s}$ for $G$. If $\theta\in\operatorname{Aut}\Omega$ then $\Lambda^{-1}\circ\theta\circ\Lambda\in\operatorname{Aut}G$. Consider any $s\in\mathcal{R}$ and $\lambda=\Lambda(s)\in D$. We have

[TABLE]

the last step by virtue of property (3) for the flat geodesics as a flat fibration of $(G,\mathcal{R})$. Write $\tilde{s}=\Lambda^{-1}\circ\theta(\lambda)$. Now

[TABLE]

Hence

[TABLE]

Thus the partition $\mathcal{E}$ has the property (3) of Definition 2.21, and so $(\Omega,D,\mathcal{E}$) is a flatly fibered royal manifold. ∎

2.3. Synchrony in $\Omega$

Lemma 1.19 suggests the following definition concerning the action of $\operatorname{Aut}\Omega$ on a flat fibration.

Definition 2.23**.**

Let $(\Omega,D,\mathcal{E})$ be a flatly fibered royal manifold with concomitant pair $(d,\Theta)$, let $\lambda_{0}\in D$ and let $\lambda_{0}=d(z_{0})$ for some $z_{0}\in\mathbb{D}$.

*We say that $(\Omega,D,\mathcal{E})$ is synchronous at $\lambda_{0}$

if, for some properly embedded analytic disc $f:\mathbb{D}\to\Omega$ such that $f(z_{0})=\lambda_{0}$ and $f(\mathbb{D})=E_{\lambda_{0}}$,*

[TABLE]

for all $m\in\operatorname{Aut}_{z_{0}}\mathbb{D}$.

Remark 2.24**.**

If $(\Omega,D,\mathcal{E})$ is as in the definition, then the synchrony of $(\Omega,D,\mathcal{E})$ at $\lambda_{0}$ depends neither on the choice of $(d,\Theta)$ nor the choice of $f$.

For suppose $(\Omega,D,\mathcal{E})$ is synchronous at $\lambda_{0}$ with respect to the concomitant pair $(d,\Theta)$ and let $(f_{1},\Theta_{1})$ be a second concomitant pair. By Remark 2.5 there exists $b\in\operatorname{Aut}\mathbb{D}$ such that $f_{1}=f\circ b$ and $\Theta_{1}=\Theta\circ I_{b}$, where $I_{b}(m)=b\circ m\circ b^{-1}$ for $m\in\operatorname{Aut}\mathbb{D}$. Let $f_{1}=f\circ b,\,z_{1}=b^{-1}(z_{0})$. Then $d_{1}(z_{1})=\lambda_{0}=f_{1}(z_{1})$. Consider $m\in\operatorname{Aut}_{z_{1}}\mathbb{D}$ and $\zeta\in\mathbb{D}$. Note that $I_{b}(m)\in\operatorname{Aut}_{z_{0}}\mathbb{D}$, and therefore, from equation (2.20) with $z=b(\zeta)$,

[TABLE]

Hence

[TABLE]

This shows that synchrony at $\lambda_{0}$ does not depend on the choice of concomitant pair.

Nor does it depend on the choice of the map $f$. For suppose that $f_{1}$ is a second properly embedded analytic disc of $\mathbb{D}$ in $\Omega$ such that $f_{1}(z_{0})=\lambda_{0}$ and $f_{1}(\mathbb{D})=E_{\lambda_{0}}$. Then there exists $b\in\operatorname{Aut}\mathbb{D}$ such that $f_{1}=f\circ b$ and $b(z_{0})=z_{0}$. Consider any $m\in\operatorname{Aut}_{z_{0}}\mathbb{D}$ and $\zeta\in\mathbb{D}$. By equation (2.20),

[TABLE]

Since $\operatorname{Aut}_{z_{0}}\mathbb{D}$ is conjugate in $\operatorname{Aut}\mathbb{D}$ to $\operatorname{Aut}_{0}\mathbb{D}$, it is an abelian group. Hence

[TABLE]

which is the desired relation for $f_{1}$.

Remark 2.25**.**

If $(\Omega,D,\mathcal{E})$ is as in the definition, then $(\Omega,D,\mathcal{E})$ is synchronous at a particular $\lambda_{0}\in D$ if and only if $(\Omega,D,\mathcal{E})$ is synchronous at $\lambda$ for every $\lambda\in D$. Consequently, it makes sense to say simply that $(\Omega,D,\mathcal{E})$ is synchronous.

For suppose $(\Omega,D,\mathcal{E})$ is synchronous at $\lambda_{0}$ with respect to the concomitant pair $(d,\Theta)$ where $\lambda_{0}=d(z_{0})$, and let $\lambda_{1}\in D$.

Suppose $\lambda_{1}=d(z_{1})$, for $z_{1}\in\mathbb{D}$, and $b(z_{0})=z_{1}$, for $b\in\operatorname{Aut}\mathbb{D}$. For every $m\in\operatorname{Aut}\mathbb{D}$, we have $\Theta(m)\circ d=d\circ m$. Hence

[TABLE]

Let

[TABLE]

Then $f_{1}(z_{1})=\lambda_{1}=d(z_{1})$. Consider $m\in\operatorname{Aut}_{z_{1}}\mathbb{D}$ and $\zeta\in\mathbb{D}$. Then

[TABLE]

Since $m$ fixes $z_{1}=b(z_{0})$, $b^{-1}\circ m\circ b$ fixes $z_{0}$. By assumption, $(\Omega,D,\mathcal{E})$ is synchronous at $\lambda_{0}$ with respect to the concomitant pair $(d,\Theta)$. Hence

[TABLE]

Therefore, by equations (2.3) and (2.3),

[TABLE]

Thus $(\Omega,D,\mathcal{E})$ is synchronous at $\lambda_{1}$ with respect to the concomitant pair $(d,\Theta)$.

In view of Remarks 2.24 and 2.25, the following statement follows easily from Lemma 1.19.

Lemma 2.26**.**

If $(\Omega,D,\mathcal{E})$ is as in Lemma 2.22, then $(\Omega,D,\mathcal{E}$) is synchronous.

2.4. The sharp direction in $\Omega$

For a regular royal manifold $(\Omega,D)$ we may define the sharp direction just as we did for $G$ in Definition 1.9. By Proposition 2.19, for $\lambda\in\Omega$ the space $T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$ is either a one-dimensional complex subspace (if $\lambda\in D$) or a $3$-dimensional real subspace (if $\lambda\in\Omega\setminus D$) of $T_{\lambda}\Omega$. We may therefore define the space $\lambda^{\sharp}$ to be the unique nonzero complex subspace of $T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$.

In either case

[TABLE]

Covariance of the sharp direction under automorphisms is proved in the same way as Proposition 1.10.

Proposition 2.27**.**

If $\theta\in\operatorname{Aut}\Omega$ and $\lambda\in\Omega$ then

[TABLE]

{comment}

Now let $\Lambda:G\to\Omega$ be a biholomorphic map, so that $(\Omega,\Lambda(\mathcal{R}))$ is a royal manifold with concomitant pair $(d,\Theta)$ as defined in Lemma 2.6. Fix $s\in G$ and $\lambda\in\Omega$ with $\Lambda(s)=\lambda$. If $r\in\mathbb{R}$ and $\alpha\in\mathbb{C}$, then

[TABLE]

i.e.,

[TABLE]

This formula has two implications which we record in the following lemmas. Recall that, for any regular royal manifold $(\Omega,D)$ with concomitant pair $(d,\Theta)$ and any $\lambda\in\Omega$,

[TABLE]

where $e_{\lambda}:\operatorname{Aut}\mathbb{D}\to\Omega$ is defined by $e_{\lambda}(m)=\Theta(m)(\lambda)$.

Proposition 2.28**.**

Let $\Lambda:G\to\Omega$ be a biholomorphic map and let $(d,\Theta)$ be the concomitant pair for $(\Omega,\Lambda(\mathcal{R}))$ consistent with $\Lambda$. If $s\in G$ and $\Lambda(s)=\lambda$, then

1. (1)

$\Lambda^{\prime}(s)T_{s}\operatorname{{\rm Orb}}_{G}(s)=T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda);$** 2. (2)

$\Lambda^{\prime}(s)s^{\sharp}=\lambda^{\sharp}$.

Moreover, if $s\notin\mathcal{R}$, then $e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})$ is invertible and

1. (3)

e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})^{-1}=\Lambda^{\prime}(s)\big{|}T_{s}\operatorname{{\rm Orb}}_{G}(s); 2. (4)

$e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})^{-1}:T_{s}\operatorname{{\rm Orb}}_{G}(s)\to T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$* is a real linear map whose restriction to $s^{\sharp}$ is complex linear and maps $s^{\sharp}$ to $\lambda^{\sharp}$.*

For $s\in\mathcal{R}$, the real linear map $e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})$ maps the $3$-dimensional space $\operatorname{\mathrm{Lie}(\mathrm{Aut}~{}\mathbb{D})}$ to $T_{s}\mathcal{R}$, which is $2$-dimensional, so we cannot form $e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})^{-1}$.

Proof.

(1) By assumption, $d=\Lambda\circ R$ and $\Theta(m)=\Lambda\circ\gamma_{m}\circ\Lambda^{-1}\in\operatorname{Aut}\Omega$ for every $m\in\operatorname{Aut}\mathbb{D}$. Hence

[TABLE]

That is, $e_{\lambda}=\Lambda\circ e_{s}$. Hence

[TABLE]

Therefore

[TABLE]

which is to say (by virtue of equations (1.28) and (2.11)) that

[TABLE]

(2) $s^{\sharp}$ is a nonzero complex subspace of $T_{s}\operatorname{{\rm Orb}}_{G}(s)$. Since $\Lambda^{\prime}(s)$ is a nonsingular complex linear map, $\Lambda^{\prime}(s)s^{\sharp}$ is a nonzero complex linear subspace of $T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$, hence is $\lambda^{\sharp}$.

(3) Consider $s\in G\setminus\mathcal{R}$. By Theorem 1.6, the real linear map $e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})$ has full rank between the $3$-dimensional spaces $\operatorname{\mathrm{Lie}(\mathrm{Aut}~{}\mathbb{D})}$ and $T_{s}\operatorname{{\rm Orb}}_{G}(s)$, and so is nonsingular. Hence, $e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})^{-1}$ exists and is a real linear map from $T_{s}\operatorname{{\rm Orb}}_{G}(s)$ to $T_{\lambda}\operatorname{{\rm Orb}}_{\Omega}(\lambda)$. By equation (2.24),

[TABLE]

(4) Since $\Lambda^{\prime}(s)$ is a complex linear map on $\mathbb{C}^{2}$, it follows that $e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})^{-1}$ is a complex linear map on the complex linear subspace $s^{\sharp}$ of $\mathbb{C}^{2}$. By (2) and equation (2.25), $e_{\lambda}^{\prime}(\mathrm{id}_{\mathbb{D}})e_{s}^{\prime}(\mathrm{id}_{\mathbb{D}})^{-1}s^{\sharp}=\lambda^{\sharp}$.

∎

{comment}

Definition 2.29**.**

Let $(\Omega,D,\mathcal{E})$ be a regular royal manifold, let $(d,\Theta)$ be a concomitant pair and let $f:\mathbb{D}\to\Omega$ be a properly embedded analytic disc of $\mathbb{D}$ with image $f(\mathbb{D})=E_{k(0)}$. We say that $\Theta$ acts linearly in the sharp direction on $E_{k(0)}$ if

[TABLE]

is complex linear for each $z\in\mathbb{D}$.

2.5. Sharpness of the action of $\operatorname{Aut}\Omega$

In this section, for a flatly fibered royal manifold $(\Omega,D,\mathcal{E})$, we shall show that sharp action of $\operatorname{Aut}\Omega$, as described in the introduction, is necessary for $\Omega$ to be isomorphic to $G$. In the next subsection we shall show that the condition is also sufficient. We first define sharpness more formally than in the introduction. Recall that, for a flatly fibered royal manifold $(\Omega,D,\mathcal{E})$, we defined the Poincaré parameter $P(\mu)$ for $\mu\in\Omega$ to be the Poincaré distance of $\mu$ from $\lambda$, where $\lambda\in D$ and $\mu\in E_{\lambda}$ and the distance is taken in the disc $E_{\lambda}$. That is, if $f:\mathbb{D}\to\Omega$ is a proper analytic embedding with range $E_{\lambda}$ and $f(z_{0})=\lambda,\,f(z)=\mu$, then

[TABLE]

It will be convenient to use also the pseudohyperbolic variant $C(\mu)$ of $P(\mu)$, defined for $\mu\in E_{\lambda}$ to be the pseudohyperbolic distance in $E_{\lambda}$ from $\mu$ to $\lambda$. In other words, if $f\in\Omega(\mathbb{D})$ has range $E_{\lambda}$ and $f(z_{0})=\lambda,\,f(z)=\mu$, then

[TABLE]

Thus $P$ and $C$ are related by the equations

[TABLE]

Remark 2.30**.**

Observe that $P(\cdot)$ and $C(\cdot)$ are invariant under isomorphisms which preserve foliations. If $(\Omega_{j},D_{j},\mathcal{E}_{j})$ is a flatly fibered royal manifold for $j=1,2$, if $\Lambda:\Omega_{1}\to\Omega_{2}$ is an isomorphism which maps the leaves of $\mathcal{E}_{1}$ to those of $\mathcal{E}_{2}$ and if $\mu\in\Omega_{1}$ then $C(\mu)=C(\Lambda(\mu))$.

Definition 2.31**.**

Let $(\Omega,D,\mathcal{E})$ be a regular flatly fibered royal manifold having a concomitant pair $(d,\Theta)$. Let $\mu\in\Omega\setminus D$ and let $(U,\psi)$ be a chart in $\Omega$ such that $\mu\in U$. We say that $\operatorname{Aut}\Omega$ acts sharply at $\mu$ with respect to $(d,\Theta)$ if

[TABLE]

as $t\to 0$ in $\mathbb{R}$.

The condition (2.29) states that the tangents $v_{\bf i}$ and $v_{\bf 1}\in\mathbb{C}^{2}$ at $t=0$ to the curves $\psi(\Theta(B_{{\bf i}t})(\mu))$ and $\psi(\Theta(B_{t})(\mu))$ in $\psi(U)$ satisfy

[TABLE]

where $\bf i$ is (temporarily, for this sentence) the imaginary unit. This property is clearly independent of the chart $\psi$ since the derivative of any transition function at a point is a complex-linear map.

We need to examine how sharpness depends on the concomitant pair $(d,\Theta)$.

Proposition 2.32**.**

With the notation of Definition 2.31, let $(d_{1},\Theta_{1})$ (for some $b\in\operatorname{Aut}\mathbb{D}$) be the concomitant pair

[TABLE]

Let $\mu_{1}=\Theta(b)(\mu)\in\Omega\setminus D$. Then $\operatorname{Aut}\Omega$ acts sharply at $\mu_{1}$ with respect to $(d_{1},\Theta_{1})$ if and only if $\operatorname{Aut}\Omega$ acts sharply at $\mu$ with respect to $(d,\Theta)$.

Proof.

We may choose the chart

[TABLE]

at $\mu_{1}$. For small real $t$,

[TABLE]

These equations, together with the analogous ones with $it$ replaced by $t$ and the fact that $P(\mu_{1})=P(\mu)$, imply the statement in the proposition. ∎

Definition 2.33**.**

Let $(\Omega,D,\mathcal{E})$ be a regular flatly fibered royal manifold having a concomitant pair $(d,\Theta)$. Let $\mu\in\Omega\setminus D$ and let $(U,\psi)$ be a chart in $\Omega$ such that $\mu\in U$. We say that $\operatorname{Aut}\Omega$ acts sharply on $\Omega$ if $\operatorname{Aut}\Omega$ acts sharply with respect to $(d,\Theta)$ at every point of $\Omega\setminus D$.

Remark 2.34**.**

(1) Propositions 2.5 and 2.32 show that the sharpness of the action of $\operatorname{Aut}\Omega$ on $\Omega$ does not depend on the choice of the concomitant pair $(d,\Theta)$.

(2) With respect to a fixed concomitant pair, for any $\theta\in\operatorname{Aut}\Omega$, automorphisms act sharply at $\mu\in\Omega\setminus D$ if and only if they act sharply at $\theta(\mu)$. Since every orbit in $\Omega$ meets every leaf in $\mathcal{E}$, to conclude that $\operatorname{Aut}\Omega$ acts sharply, it is enough to show that, for some $\lambda\in D$, automorphisms act sharply at every point of $E_{\lambda}\setminus\{\lambda\}$.

{comment}

A second necessary condition follows from Lemma 2.28. Let $(d,\Theta)$ be the concomitant pair consistent with $\Lambda$ and let $\lambda_{0}$ be a point in $D$. We make the following sequence of constructions.

1. (1)

Let $z_{0}$ be the point in $\mathbb{D}$ such that $d(z_{0})=\lambda_{0}$. 2. (2)

Choose an embedding $f:\mathbb{D}\to\Omega$ such that $f(z_{0})=\lambda_{0}$ and $f(\mathbb{D})=E_{\lambda_{0}}$. 3. (3)

Let $s_{0}=R(z_{0})$. 4. (4)

Let $g:\mathbb{D}\to G$ be an embedding such that $g(z_{0})=s_{0}$ and $g(\mathbb{D})=F_{s_{0}}$.

Observe that $\Lambda$ can be replaced by a new biholomorphic map $\Lambda^{\sim}:G\to\Omega$ satisfying $\Lambda^{\sim}\circ g=f$. After this replacement, Proposition 2.28(4) implies that

    for all $z\in\mathbb{D}\setminus\{z_{0}\}$, the real linear map

[TABLE]

maps $g(z)^{\sharp}$ into $f(z)^{\sharp}$ and is complex-linear on $g(z)^{\sharp}$.

Note that any flat geodesic meets $\mathcal{R}$ in a unique point, so that $g(z)\notin\mathcal{R}$ for $z\in\mathbb{D}\setminus\{z_{0}\}$.

Statement (2.30) makes sense for any flatly fibered royal manifold, independently of the existence of $\Lambda$. This fact prompts the following definition.

Definition 2.35**.**

*Let $(\Omega,D,\mathcal{E})$ be a regular flatly fibered royal manifold with concomitant pair $(d,\Theta)$ and let $\lambda_{0}\in D$. We say that *$\Theta$ acts sharply

over $\lambda_{0}$* if the statement (2.30) holds for all $z\in\mathbb{D}\setminus\{z_{0}\}$.*

Remark 2.36**.**

Firstly, the embeddings $g$ and $f$ are only determined up to composition on the right with an element of $\operatorname{Aut}_{z_{0}}\mathbb{D}$. Sharpness of the action of $\Theta$ over $\lambda_{0}$ does not depend on the choice of $g$ and $f$.

Secondly, $\Theta$ acts sharply over $\lambda_{0}$ for some $\lambda_{0}\in D$ if and only if $\Theta$ acts sharply over $\lambda$ for every $\lambda\in D$.

Thirdly, the sharpness condition does not depend on the choice of concomitant pair $(d,\Theta)$. Consequently, the sharpness condition depends only on the action of $\operatorname{Aut}\Omega$ on $\Omega$. For this reason we say $\operatorname{Aut}\Omega$ acts sharply when the conditions of Definition 2.35 hold.

Proposition 2.37**.**

Let $\Omega$ be a complex manifold and $\Lambda:G\to\Omega$ be a biholomorphic map. There exist a royal disc $D$ in $\Omega$, a flat fibration $\mathcal{E}$ of $\Omega$ over $D$ and a concomitant pair $(d,\Theta)$ such that $(\Omega,D,\mathcal{E})$ is a synchronous regular flatly fibered royal manifold, $(d,\Theta)$ is consistent with $\Lambda$ and $\operatorname{Aut}\Omega$ acts sharply on $\Omega\setminus D$.

Proof.

Let $D=\Lambda(\mathcal{R})$. By Lemmas 2.2 and 2.6, $(\Omega,D)$ is a royal manifold and there is a concomitant pair $(d,\Theta)$ for $(\Omega,D)$ such that

[TABLE]

for all $m\in\operatorname{Aut}\mathbb{D}$. Thus $(d,\Theta)$ is consistent with $\Lambda$. By Proposition 2.16, $(\Omega,D)$ is a regular royal manifold. Let $\mathcal{E}$ correspond under $\Lambda$ to the flat fibration of $G$. By Lemma 2.26, $(\Omega,D,\mathcal{E})$ is a synchronous regular flatly fibered royal manifold. It remains to show that $\operatorname{Aut}\Omega$ acts sharply on $\Omega\setminus D$.

Consider a point $\mu\in\Omega\setminus D$, say $\mu\in E_{\lambda}$, for some $\lambda\in D$. Let $s=\Lambda^{-1}(\mu)$. We may assume (by modifying $\Lambda$ and $\Theta$ and utilising Remark 2.34) that $s$ has the form $(0,p)$ for some $p\in(0,1)$. Then $s$ lies in the flat geodesic $F^{0}$, and so $\Lambda^{-1}(E_{\lambda})=F^{0}$. It follows that $\Lambda^{-1}(\lambda)=(0,0)$ and since isomorphisms preserve the Möbius distance, $p=C(\mu)$.

Let $(U,\psi)$ be a chart at $\mu$. For any $\alpha\in\mathbb{C}$ and all small enough real $t$, in view of equation (2.31),

[TABLE]

By Lemma 1.7, with $r=0$ and $s=(0,p)$,

[TABLE]

Let $A=-2(\psi\circ\Lambda)^{\prime}(s)$, so that $A$ is a complex-linear map. Taking successively $\alpha=i$ and $\alpha=1$ in equation (2.5) we obtain

[TABLE]

Hence

[TABLE]

Since $p=C(\mu)$, this is to say that $\operatorname{Aut}\Omega$ acts sharply at $\mu$. ∎

The next statement justifies the terminology of ‘sharp action’.

Lemma 2.38**.**

Let $(\Omega,D,\mathcal{E})$ be a regular flatly fibered royal manifold and suppose that $\operatorname{Aut}\Omega$ acts sharply at a point $\mu\in\Omega\setminus D$. For any $s\in G$ such that $C(s)=C(\mu)$ the map

[TABLE]

maps $s^{\sharp}$ to $\mu^{\sharp}$ and is a complex-linear map.

Proof.

Let

[TABLE]

$X$ is a real-linear map from $T_{s}G$ to $T_{\mu}\Omega$. We must show that $Xs^{\sharp}\subseteq\mu^{\sharp}$ and that $X$ is complex-linear on $s^{\sharp}$.

We can assume that $s=(0,p)$ where $0<p<1$. Clearly

[TABLE]

By equation (2.28),

[TABLE]

The sharpness hypothesis, according to Definition 2.38, is

[TABLE]

as $t\to 0$ in $\mathbb{R}$. By Proposition 1.16,

[TABLE]

We shall use the local co-ordinates $(r,\alpha)\in(-\pi,\pi)\times\mathbb{D}$ for a neighborhood of $\mathrm{id}_{\mathbb{D}}$ in $\operatorname{Aut}\mathbb{D}$, as in Lemma 1.4. By Lemma 1.7,

[TABLE]

which is in $s^{\sharp}$ if and only if $r=0$. Thus

[TABLE]

Moreover, for all $\alpha\in\mathbb{C}$,

[TABLE]

Note that $m_{0,\alpha}=B_{\alpha}$ in the notation of equation (0.6). Let $\psi$ be a chart on $\Omega$ at $\mu$. For any $\alpha\in\mathbb{C}$, as $t\to 0$ in $\mathbb{R}$,

[TABLE]

Take in succession $\alpha=1$ and $\alpha=i$ and use the real-linearity of $X$ to obtain the relations

[TABLE]

We have

[TABLE]

Since $\psi^{\prime}(\mu)$ is an invertible complex-linear map which identifies $T_{\mu}\Omega$ with $\mathbb{C}^{2}$, it follows that

[TABLE]

The vectors $(1,0)$ and $(i,0)$ span $s^{\sharp}$ over $\mathbb{R}$, and

[TABLE]

Equation (2.38) now shows both that $\operatorname{ran}X\subset\mu^{\sharp}$ and that $X$ is complex-linear on $s^{\sharp}$. ∎

2.6. A characterization of $G$

We have arrived at the main theorem of the paper.

Theorem 2.39**.**

A complex manifold $\Omega$ is isomorphic to $G$ if and only if there exist a royal disc $D$ in $\Omega$ and a flat fibration $\mathcal{E}$ of $\Omega$ over $D$ such that $(\Omega,D,\mathcal{E})$ is a synchronous regular flatly fibered royal manifold and $\operatorname{Aut}{\Omega}$ acts sharply on $\Omega$.

Proof.

Necessity is Proposition 2.37. We prove sufficiency. Let $(d,\Theta)$ be a concomitant pair for $(\Omega,D)$. Choose $z_{0}\in\mathbb{D}$ and let $s_{0}=R(z_{0})$ and $\lambda_{0}=d(z_{0})$. We shall construct a biholomorphic map $\Lambda:G\to\Omega$ satisfying $\Lambda(s_{0})=\lambda_{0}$. Figure 1 represents the construction.

Choose a properly embedded analytic disc $g$ of $\mathbb{D}$ into $G$ satisfying $g(\mathbb{D})=F_{s_{0}}$ and $g(z_{0})=s_{0}$. Choose also a properly embedded analytic disc $f:\mathbb{D}\to\Omega$ such that $f(z_{0})=\lambda_{0}$ and $f(\mathbb{D})=E_{\lambda_{0}}$. For $s\in G$ we define $\Lambda(s)$ by the following recipe.

Since each point in $G$ is in a flat geodesic and $\operatorname{Aut}G$ acts transitively on the flat geodesics, we may choose $m\in\operatorname{Aut}\mathbb{D}$ such that $\gamma_{m}^{-1}(s)\in F_{s_{0}}$ and hence there exists $z\in\mathbb{D}$ such that

[TABLE]

Let

[TABLE]

Certainly $\Lambda(s)\in\Omega$. To see that this recipe does define $\Lambda$ as a map from $G$ to $\Omega$, consider $z_{1},z_{2}\in\mathbb{D}$ and $m_{1},m_{2}\in\operatorname{Aut}\mathbb{D}$ such that

[TABLE]

We wish to show that

[TABLE]

Note first that equation (2.41) implies that if $m=m_{2}^{-1}\circ m_{1}$, then

[TABLE]

Since $g(z_{1}),g(z_{2})\in F_{s_{0}}$, Lemma 1.14 implies that $m\in\operatorname{Aut}_{z_{0}}\mathbb{D}$. Consequently, by Lemma 1.19

[TABLE]

which implies that

[TABLE]

By hypothesis, $(\Omega,D,\mathcal{E})$ is synchronous. According to Definition 2.23, it means (since $m\in\operatorname{Aut}_{z_{0}}\mathbb{D}$) that

[TABLE]

Hence

[TABLE]

Therefore equation (2.42) is true, and so $\Lambda(s)$ is unambiguously defined.

On taking $m=\mathrm{id}_{\mathbb{D}}$ in equation (2.40) we have

[TABLE]

for all $z\in\mathbb{D}$. In particular, $\Lambda(s_{0})=\lambda_{0}$.

Consider any $\upsilon\in\operatorname{Aut}\mathbb{D}$. Since

[TABLE]

by the definition (2.40) of $\Lambda$,

[TABLE]

Thus

[TABLE]

Now fix a general point $R(z)$ on the royal geodesic $\mathcal{R}$. If $m\in\operatorname{Aut}\mathbb{D}$ is such that $m(z_{0})=z$, then

[TABLE]

By equation (2.44) and the fact that $s_{0}=R(z_{0})$,

[TABLE]

Thus

[TABLE]

Now fix $z_{1}\in\mathbb{D}$ and choose $m$ such that $m(z_{0})=z_{1}$. Since $\gamma_{m}\circ R(z_{0})=R(z_{1})$, Lemma 1.14 implies that

[TABLE]

and since $\Theta(m)\circ d(z_{0})=d(z_{1})$, Condition (3) in Definition 2.21 implies that

[TABLE]

Therefore

[TABLE]

To summarize, we have shown that if $\mathcal{F}$ denotes the partition of $G$ in Lemma 1.12 and $\mathcal{E}$ denotes the partition of $\Omega$ in Definition 2.21, then $\Lambda$ induces a map $\Lambda^{\sim}:\mathcal{F}\to\mathcal{E}$ given by

[TABLE]

Furthermore, as the map $R(z)\mapsto d(z)$ from $\mathcal{R}$ to $D$ is a bijection, so also is $\Lambda^{\sim}$.

Consider any point $\lambda\in E_{d(z_{1})}$. Then $\Theta(m^{-1})(\lambda)\in E_{\lambda_{0}}$, and so $\Theta(m^{-1})(\lambda)=f(z)$ for some $z\in\mathbb{D}$. Hence $\lambda=\Theta(m)\circ f(z)$. By equations (2.39) and (2.40), $\lambda=\Lambda\circ\lambda_{m}\circ g(z)$. Thus $\lambda\in\Lambda(G)$, and so $\Lambda$ is surjective.

Suppose $s_{1},\,s_{2}\in G$ satisfy $\Lambda(s_{1})=\Lambda(s_{2})$. Since $\Lambda^{\sim}$ is a bijection, it follows that $s_{1},\,s_{2}$ lie in the same flat geodesic in $G$, say in $F_{R(z_{1})}$. Let $m\in\operatorname{Aut}\mathbb{D}$ be such that $m(z_{0})=z_{1}$. We have, for $j=1,2$,

[TABLE]

Hence $\gamma_{m}^{-1}(s_{j})=g(\zeta_{j})$ for some $\zeta_{1},\zeta_{2}\in\mathbb{D}$. By equation (2.40),

[TABLE]

Hence $\Theta(m)\circ f(\zeta_{1})=\Theta(m)\circ f(\zeta_{2})$, and therefore $\zeta_{1}=\zeta_{2}$. Thus

[TABLE]

We have shown that $\Lambda:G\to\Omega$ is bijective. Moreover, we can observe that

[TABLE]

There remains to prove that $\Lambda$ and $\Lambda^{-1}$ are holomorphic.

We shall first show that $\Lambda$ is smooth as a mapping between real manifolds by giving a formula for $\Lambda$ which is clearly differentiable. The assumption that $z_{0}=0$, $g(z)=(0,z)$ and so $s_{0}=(0,0)$ loses no generality. It implies that $F_{s_{0}}=\{(0,z):z\in\mathbb{D}\}$.

Consider a point

[TABLE]

for some $\zeta,\eta\in\mathbb{D}$. To evaluate $\Lambda(s)$ we shall choose an automorphism $m$ of $\mathbb{D}$ satisfying $m^{\prime}(0)>0$ such that $\gamma_{m}^{-1}(s)\in F_{s_{0}}$. To see that this is possible take $m=B_{\alpha}$ for some $\alpha\in\mathbb{D}$. Then $m^{\prime}(0)>0$. We require $\gamma_{B_{\alpha}}^{-1}(s)\in F_{s_{0}}$, which is to say that

[TABLE]

Expressing this relation in terms of the components $s^{1},s^{2}$ of $s$, we must find $\alpha=\alpha(s)\in\mathbb{D}$ such that

[TABLE]

Compare this expression with that of the flat co-ordinates for $s$ given in equations (1.38) and (1.39):

[TABLE]

where

[TABLE]

One sees that it suffices to choose $\alpha(s)$ such that

[TABLE]

A suitable choice of $\alpha(s)$ is

[TABLE]

as may readily be checked. Clearly $\beta,\alpha\in\mathbb{D}$ and both $\beta$ and $\alpha$ are real-analytic functions of $s$. Moreover

[TABLE]

which is also real-analytic in $s$. By the definition of $\Lambda$,

[TABLE]

The map $s\mapsto B_{\alpha(s)}$ is real-analytic from $G$ to $\operatorname{Aut}\mathbb{D}$. Since the action of $\operatorname{Aut}\mathbb{D}$ on $\Omega$ is differentiable, by the regularity assumption on the royal manifold $(\Omega,D)$, we conclude that $\Lambda:G\to\Omega$ is differentiable.

Consider $s\in G$ and suppose that $s\in F_{R(z_{1})}$. Let $X=\Lambda^{\prime}(s)$ viewed as a real-linear mapping from $T_{s}G$ to $T_{\Lambda(s)}\Omega$.

Recall from Definition 1.13 that $s^{\flat}$ denotes the flat direction at $s$. Equation (2.46) implies that

[TABLE]

By equations (2.39) and (2.40), for all $z\in\mathbb{D}$ and $m\in\operatorname{Aut}\mathbb{D}$,

[TABLE]

In view of the definitions (1.6) and (2.10), this equation can be written

[TABLE]

On differentiating at $\mathrm{id}_{\mathbb{D}}$ we obtain

[TABLE]

For $z\neq z_{0}$, the point $g(z)\notin\mathcal{R}$, and therefore, by Proposition 2.28, $e^{\prime}_{g(z)}(\mathrm{id}_{\mathbb{D}})$ is invertible, and so

[TABLE]

By the hypothesis, $\operatorname{Aut}\Omega$ acts sharply on $\Omega$. By Lemma 2.38, it follows that $\Lambda^{\prime}\circ g(z)$ maps $g(z)^{\sharp}$ into $f(z)^{\sharp}$ and is complex-linear on $g(z)^{\sharp}$ whenever $g(z)\notin\mathcal{R}$.

{comment}

Next, observe that if $r\in\mathbb{R}$, $\alpha\in\mathbb{C}$, and $s\in G$, then

[TABLE]

In particular, if we let $s=f(z)$, then (2.43) implies that

[TABLE]

for all $z\in\mathbb{D}$, $r\in\mathbb{R}$, and $\alpha\in\mathbb{C}$. But $\operatorname{Aut}{\Omega}$ acts sharply on $\Omega$, so that

[TABLE]

for all $z\in\mathbb{D}$. Hence,

[TABLE]

which implies via equation (2.44) and Propositions (1.10) and (2.27) that

[TABLE]

Recalling that $X:T_{s}G\to T_{\Lambda(s)}\Omega$ is real-linear and that (by Proposition 1.15) $s^{\flat}$ and $s^{\sharp}$ are linearly independent, we infer from equation (2.47) that $X=\Lambda^{\prime}(s)$ is complex linear for all $s\in G\setminus\mathcal{R}$. Therefore $\Lambda$ is analytic on $G\setminus\mathcal{R}$. The restriction of $\Lambda$ to any co-ordinate plane $P_{\zeta}\stackrel{{\scriptstyle\rm def}}{{=}}\{s\in G:s^{1}=\zeta\}$, for $|\zeta|<2$, is analytic in $s^{2}$ except possibly at the sole point $(\zeta,\tfrac{1}{4}\zeta^{2})$ of $P_{\zeta}\cap\mathcal{R}$ and is continuous on $P_{\zeta}$. Hence $\Lambda|P_{\zeta}$ is analytic in $s^{2}$. Likewise the restriction of $\Lambda$ to any of the orthogonal co-ordinate planes is analytic in $s^{1}$. Thus $\Lambda$ is analytic on $G$. Every bijective holomorphic map between domains has a holomorphic inverse (for example, [20, Chapter 10, Exercise 37]). It follows easily that a bijective holomorphic map between a domain and a complex manifold has a holomorphic inverse. ∎

3. A characterization of $G$ via flat co-ordinates

Recall from Subsection 1.5 that $G$ is foliated by the sets

[TABLE]

for $\beta\in\mathbb{D}$ [4, Theorem 2.1]. Thus the map $\eta:\mathbb{D}^{2}\to G$ defined by the formula

[TABLE]

is a homeomorphism of $\mathbb{D}^{2}$ onto $G$.

We will call $\beta,z$ the flat co-ordinates for points of $G$.

In this section we shall use the variables $(\beta,z)$ for points in $\mathbb{D}^{2}$ and the variables $(s,p)$ for points in $G$, so that

[TABLE]

Flat co-ordinates provide another characterization of domains biholomorphic to $G$.

{comment}

If $f=f(s,p)$ is a differentiable function on $G$ and if $\xi=f\circ\eta$, then by the Chain Rule,

[TABLE]

We record this result as well as the results of the similar calculations of the other complex partial derivatives of $\xi$ in the following lemma.

The following lemma is a consequence of the Chain Rule.

Lemma 3.1**.**

If $f=f(s,p)$ is a differentiable function on $G$, $\eta$ is defined on $\mathbb{D}^{2}$ as in equation (3.1) and $\xi=f\circ\eta$, then the following relations hold.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Theorem 3.2**.**

If $\Omega$ is a domain in $\mathbb{C}^{2}$, then $\Omega$ is biholomorphic to $G$ if and only if there exists a differentiable homeomorphism $\Xi=(\xi_{1},\xi_{2})$ from $\mathbb{D}^{2}$ onto $\Omega$ satisfying

[TABLE]

and

[TABLE]

at all $(\beta,z)\in\mathbb{D}^{2}$.

Proof.

First assume that $F\in\Omega(G)$ is a biholomorphic map of $G$ onto $\Omega$ and let $\Xi=F\circ\eta$. Since $\eta$ is a smooth homeomorphism of $\mathbb{D}^{2}$ onto $G$, $\Xi$ is a smooth homeomorphism of $\mathbb{D}^{2}$ onto $\Omega$.

If we set $F=(f_{1},f_{2})$ and $\Xi=(\xi_{1},\xi_{2})$, then $f_{i}$ is holomorphic and $\xi_{i}=f_{i}\circ\eta$ for $i=1,2$. Hence, using equations (3.2) and (3.3), we see that

[TABLE]

which proves that equation (3.6) holds. Also, equation (3.5) implies that the relation (3.7) holds.

Now assume that $\Xi=(\xi_{1},\xi_{2})$ is a differentiable homeomorphism from $\mathbb{D}^{2}$ onto $\Omega$ satisfying equations (3.6) and (3.7). Define $F=(f_{1},f_{2})$ by $F=\Xi\circ\eta^{-1}$. Since $\eta$ is a differentiable homeomorphism of $\mathbb{D}^{2}$ onto $G$, it follows that $F$ is a differentiable homeomorphism of $G$ onto $\Omega$. There remains to show that $F$ is holomorphic.

Since $\xi_{i}=f_{i}\circ\eta$, we have

[TABLE]

{comment}

[TABLE]

Thus

[TABLE]

on $G$. Since $|z|<1$ when $(s,p)\in G$ it follows that

[TABLE]

throughout $G$. Hence $f_{1},f_{2}$ are holomorphic on $G$. ∎

{comment}

Here is one interpretation of the $\beta$ co-ordinate. Consider a point

[TABLE]

with $\beta,z,\zeta,\eta\in\mathbb{D}$. Let $\alpha$ denote the hyperbolic mid-point of $\zeta$ and $\eta$. That is, $\alpha$ is the point mid-way between $\zeta$ and $\eta$ on the geodesic in the Poincaré disc through $\zeta$ and $\eta$. Then

[TABLE]

4. Asymmetry of domains

É. Cartan’s classification theorem [8]

is based on his theory of symmetric spaces,

in the sense of the first paragraph of the paper.

In $\mathbb{C}^{2}$ and $\mathbb{C}^{3}$ (but not $\mathbb{C}^{4}$) every bounded homogeneous domain is a symmetric space [8, 14]. In contrast, none of the ‘almost homogeneous’ domains that we consider is symmetric.

Let us say that a point $\lambda$ in a domain $\Omega$ is a point of symmetry of $\Omega$ if there exists a holomorphic self-map $\gamma$ of $\Omega$ such that $\gamma\circ\gamma={\mathrm{id}}_{\Omega}$ and $\lambda$ is an isolated fixed point of $\gamma$. Thus a domain is symmetric if every point of the domain is a point of symmetry.

From the fact that the automorphisms of the annulus $\mathbb{A}_{q}$

are the maps $\omega z$ and $\omega z^{-1}$ for $\omega\in\mathbb{T}$ (for example, [12, Theorem 6.2]), it is easy to see that the only points of symmetry in $\mathbb{A}_{q}$ are the points of the unit circle. Hence $\mathbb{A}_{q}$ is not a symmetric domain.

Proposition 4.1**.**

*Neither the symmetrized bidisc nor the tetrablock contains a point of symmetry. *

Proof.

We sketch the proof for the tetrablock; that for the symmetrized bidisc is similar but simpler. {comment} By Proposition 1.1, every isomorphism of $G$ has the form $\gamma_{m}$ for some $m\in\operatorname{Aut}\mathbb{D}$, as in equation (1.2). Since $m\mapsto\gamma_{m}$ is an isomorphism of groups, $\gamma_{m}$ is an involution if and only if $m\circ m=\mathrm{id}_{\mathbb{D}}$.

It is enough to show that the point $(0,p)$ is not a point of symmetry of $G$ for any $p\in\mathbb{D}$, since every orbit in $G$ contains a point of this form. Suppose a holomorphic involution $\gamma_{m}$ fixes $(0,p)$. Since the isomorphisms of $G$ permute the flat geodesics, $\gamma_{m}$ fixes the flat geodesic through $(0,p)$, which is the flat geodesic $F_{(0,0)}$ (as in Subsection 1.5). Hence $\gamma_{m}(0,0)$ lies in $F_{(0,0)}$. But also $\gamma_{m}$ fixes the royal variety, and therefore

[TABLE]

Thus $\gamma_{m}$ fixes the point $(0,0)$, which is to say that the involution $m$ fixes [math]. Hence $m(z)=\pm z$ and so either $\gamma_{m}=\mathrm{id}_{\mathbb{D}}$ or $\gamma_{m}(s^{1},s^{2})=(-s^{1},s^{2})$. The set of fixed points of $\gamma_{m}$ is therefore either $G$ or the whole of $F_{(0,0)}$, and so $(0,p)$ is not an isolated fixed point of $\gamma_{m}$. ∎

Proposition 4.2**.**

The tetrablock contains no points of symmetry.

Proof.

The proof follows the same pattern as for $G$.

Let $E$ denote the tetrablock defined in equation (0.5). Every orbit in $E$ contains a point of the form $(0,0,p)$ [26, Theorem 5.2], so it suffices to show that no such point is a point of symmetry. By [1, Theorem 2.2], the tetrablock is foliated by the ‘flat geodesics’

[TABLE]

where $|\beta_{1}|+|\beta_{2}|<1$. These geodesics are permuted by the automorphisms of $E$ [26, Theorem 5.1]. Moreover the ‘royal variety’ $\{x\in E:x^{1}x^{2}=x^{3}\}$ is invariant under all automorphisms of $E$ (see the proof of [26, Theorem 4.1]).

Consider a holomorphic involution $\gamma$ of $E$ that fixes $(0,0,p)$. Then $\gamma$ fixes the flat geodesic containing $(0,0,p)$, which is $C_{00}$. Hence $\gamma$ fixes the only common point of $C_{00}$ and the royal variety, which is easily seen to be $(0,0,0)$. It is shown in [26, Proof of Theorem 4.1, foot of page 766] that an automorphism $\gamma$ of $E$ fixes $(0,0,0)$ if and only if either

[TABLE]

or

[TABLE]

for some $\omega_{1},\ \omega_{2}\in\mathbb{T}$.

In the case that $\gamma$ is of the form (4.1), since $\gamma$ is an involution, we have $\omega_{1}^{2}=\omega_{2}^{2}=1$. Thus the four involutions of this form that fix $(0,0,p)$ have fixed points as in the following table.

[TABLE]

where $z$ ranges over $\mathbb{D}$. Hence in the case (4.1), $(0,0,p)$ is not an isolated fixed point of $\gamma$.

In case (4.2),

[TABLE]

and so the involutory property of $\gamma$ corresponds to the condition $\omega_{1}\omega_{2}=1$. Hence $\gamma(x)=(\bar{\omega}x^{2},\omega x^{1},x^{3})$ for some $\omega\in\mathbb{T}$. Then the fixed points of $\gamma$ are the points $(x^{1},\omega x^{1},x^{3})$ in $E$. Hence $(x^{1},\omega x^{1},p)$ is a fixed point of $\gamma$ for all $x^{1}$ in a neighborhood of [math], and so $(0,0,p)$ is not an isolated fixed point of $\gamma$. ∎

Index

• $(d,\Theta)$ Definition 2.4

• $(\Omega,D,\mathcal{E})$ Definition 2.21

• $(\Omega,D)$ Definition 2.1

• analytic disc

• properly embedded Introduction

• annulus §4, Introduction

• $\mathbb{A}_{q}$ §4, Introduction

• automorphism Introduction

• $\operatorname{Aut}_{\lambda}$ §1.6

• $B_{\alpha}$ Introduction

• Cartan

• É. §4, Introduction

• H. §1, §2.1

• Cartan’s theorem §1.1, Introduction

• concomitant pair Definition 2.4

• consistent Definition 2.7

• $\mathbb{D}$ Introduction

• domain Introduction

• bounded symmetric homogeneous §4, Introduction

• $e_{s}$ §1.1

• evaluation map §1.1

• Figure 1 §2.6

• flat

• direction Definition 1.13, Definition 2.21

• fibration §1.5, Definition 2.21

• geodesic §1.5

• flat co-ordinates §3

• $F_{s}$ item (1)

• $F^{\beta}$ §1.5

• $G$ Introduction

• geodesic

• complex Introduction

• flat Introduction

• homogeneous Introduction

• irreducible §1

• Lempert Introduction

• Lie

• algebra §1.3

• group §1, §1.1

• $\operatorname{\mathrm{Lie}(\mathrm{Aut}~{}\mathbb{D})}$ §1.3

• main theorem §2.6

• orbit Introduction

• $\operatorname{{\rm Orb}}_{U}(\lambda)$ §1.3

• $P(\mu)$ §2.5

• Poincaré parameter §2.5, Introduction

• royal

• disc Definition 2.1, Introduction

• manifold Definition 2.1

• flatly fibered Definition 2.21

• regular Definition 2.14

• variety Introduction

• $s^{\flat}$ Definition 1.13

• $s^{\sharp}$ Definition 1.9

• sharp

• action §2.5, Introduction

• direction Definition 1.9, §2.4, Introduction

• sharply Definition 2.35

• stabilizer §1.6

• symmetric §4, Introduction

• symmetrized bidisc Introduction

• synchronous Definition 2.23

• synchrony §1.6, §1.6, Introduction

• $\mathbb{T}$ §1.6

• taut §2

• tetrablock Proposition 4.1, Introduction

• $T_{\lambda}\Omega$ Introduction

• $U_{\Omega}(K,O)$ §1

• $\mathcal{V}(s)$ §1.3

• $v_{r,\alpha}$ Lemma 1.7

• $\gamma_{m}$ §1.1

• $\lambda^{\flat}$ Definition 2.21

• $\lambda^{\sharp}$ §2.4, Introduction

• $\rho_{\eta}$ §1.6

• $\Omega(U)$ Introduction