# Function Theory off the complexified unit circle:
Fréchet space structure and automorphisms

Michael Heins  M. Heins: Department of Mathematics, University of Würzburg,
Emil Fischer Strasse 40, 97074, Würzburg, Germany.
<EMAIL_ADDRESS>,  Annika Moucha§ A. Moucha:
Department of Mathematics, University of Würzburg, Emil Fischer Strasse 40,
97074, Würzburg, Germany<EMAIL_ADDRESS>and  Oliver Roth  O.
Roth: Department of Mathematics, University of Würzburg, Emil Fischer Strasse
40, 97074, Würzburg, Germany<EMAIL_ADDRESS>
###### Abstract.

Motivated by recent work on strict deformation quantization of the unit disk
and the Riemann sphere, we study the Fréchet space structure of the set of
holomorphic functions on the complement
$\Omega:=\\{(z,w)\in\hat{\mathbb{C}}^{2}\,:\,z\cdot w\not=1\\}$ of the
complexified unit circle ${\\{(z,w)\in\hat{\mathbb{C}}^{2}\,:\,z\cdot
w=1\\}}$. We also characterize the subgroup of all biholomorphic automorphisms
of $\Omega$ which leave the canonical Laplacian on $\Omega$ invariant.

${}^{\S}\,$Partially supported by the Alexander von Humboldt Stiftung

00footnotetext: 2020 Mathematics Subject Classification. Primary 30F45, 53A55,
46A35, 46A0400footnotetext: Key words and phrases. Schauder basis, Invariant
Laplacian, Conformal invariance August 28, 2024

##  1\. Introduction

We work on the Riemann sphere $\hat{\mathbb{C}}:=\mathbb{C}\cup\\{\infty\\}$.
The complement of the complexified unit circle is defined111By convention,
$z\cdot\infty=\infty\cdot z=\infty$ for all
$z\in\hat{\mathbb{C}}\setminus\\{0\\}$, while $0\cdot\infty=\infty\cdot 0=1$.
as the open subset

$\Omega:=\left\\{(z,w)\in\hat{\mathbb{C}}^{2}\,:\,z\cdot w\not=1\right\\}\,$

of $\hat{\mathbb{C}}^{2}$. The Fréchet space $\mathcal{H}(\Omega)$ of all
holomorphic functions $f:\Omega\to\mathbb{C}$, equipped with its natural
topology of locally uniform convergence, plays a peculiar and rather puzzling
key role in recent work [1, 3, 14, 22] on strict deformation quantization of
the unit disk $\mathbb{D}:={\\{z\in\mathbb{C}\,:\,|z|<1\\}}$ and the Riemann
sphere $\hat{\mathbb{C}}$. Moreover, it is shown in [7, 18], that the fine
structure of $\mathcal{H}(\Omega)$ also provides an efficient device to
investigate spectral properties of the invariant Laplacians on the disk and
the sphere (see e.g. Helgason [8], Rudin [20]) using methods from complex
analysis.

The purpose of the present paper is to analyze the structure of the Fréchet
space $\mathcal{H}(\Omega)$ and to identify the group of biholomorphic
automorphisms of $\Omega$ which leave the canonical Laplacian of $\Omega$
invariant. In order to place the results of this paper into a broader context,
we briefly indicate the role the Fréchet space $\mathcal{H}(\Omega)$ is
playing in strict deformation quantization as well as for the spectral theory
of the invariant Laplacian for the unit disk $\mathbb{D}$ and the sphere
$\hat{\mathbb{C}}$.

  * (i)

Roughly speaking, in the theory of strict deformation quantization, a star
product $\star_{\hbar}$ of Wick–type is an associative, non–commutative
product or “deformation” of the pointwise product of functions defined on some
complex manifold $M$ and depending on a complex parameter $\hbar$ s.t. the
product $\star_{\hbar}$ is compatible with the complex structure of $M$, see
e.g. [24] for a detailed survey. It turns out, see [1, 3, 14], that the vector
spaces

$\mathcal{A}(\hat{\mathbb{C}}):=\left\\{\hat{\mathbb{C}}\ni z\mapsto
f(z,-\overline{z})\,:\,f\in\mathcal{H}(\Omega)\right\\}$

and

$\mathcal{A}(\mathbb{D}):=\left\\{\mathbb{D}\ni z\mapsto
f(z,\overline{z})\,:\,f\in\mathcal{H}(\Omega)\right\\}\,,$

which are subspaces of the real analytic functions on $\hat{\mathbb{C}}$ resp.
$\mathbb{D}$, are of particular interest. In fact, they can be equipped with
star products $\star_{\hbar,\hat{\mathbb{C}}}$ and $\star_{\hbar,\mathbb{D}}$
such that $(\mathcal{A}(\hat{\mathbb{C}}),\star_{\hbar,\hat{\mathbb{C}}})$ and
$(\mathcal{A}(\mathbb{D}),\star_{\hbar,\mathbb{D}})$ are non–commutative
Fréchet algebras w.r.t. the subspace topology induced from the natural
topology of $\mathcal{H}(\Omega)$. In addition, these star products are
holomorphic w.r.t. $\hbar$, and they are invariant w.r.t. the action of the
groups $\text{Rot}(\hat{\mathbb{C}})$ and ${\sf Aut}(\mathbb{D})$ of
holomorphic isometries of $\hat{\mathbb{C}}$ resp. ${\mathbb{D}}$.

In [6] we go one step further and show that the star products
$\star_{\hbar,\mathbb{D}}$ and $\star_{\hbar,\hat{\mathbb{C}}}$ are simply the
restrictions of a naturally defined star product $\star_{\hbar}$ on the
“ambient” Fréchet space $\mathcal{H}(\Omega)$. In particular,
$(\mathcal{H}(\Omega),\star_{\hbar})$ becomes a Fréchet algebra when equipped
with its natural compact–open topology, and the dependence on the parameter
$\hbar$ is holomorphic. This gives a natural interpretation of the striking
similarities of $\mathcal{A}(\hat{\mathbb{C}})$ and $\mathcal{A}(\mathbb{D})$
observed in [3, 22]. Moreover, the star product $\star_{\hbar}$ on
$\mathcal{H}(\Omega)$ is fully invariant under the distinguished subgroup

(1.1) $\mathcal{M}:=\bigcup\limits_{\psi\in{\sf
Aut}(\hat{\mathbb{C}})}\left\\{(z,w)\mapsto\left(\psi(z),\frac{1}{\psi(1/w)}\right)\,,\,(z,w)\mapsto\left(\psi(w),\frac{1}{\psi(1/z)}\right)\right\\}\,$

of automorphisms of $\Omega$, which corresponds to the full Möbius group ${\sf
Aut}(\hat{\mathbb{C}})$ of all Möbius transformations.

  * (ii)

While the group ${\sf Aut}(\Omega)$ of all biholomorphic automorphisms is
exceedingly large, the distinguished subgroup $\mathcal{M}$ is characterized
as the group of exactly those automorphisms of $\Omega$ which leave the
natural Laplacian on $\Omega$ invariant, see Theorem 5.2 in Section 5 below.

  * (iii)

The complement of the complexified unit circle has several other equivalent
models such as the second configuration space
${\\{(z,w)\in\hat{\mathbb{C}}^{2}\,:\,w\not=z\\}}$ of the Riemann sphere
$\hat{\mathbb{C}}$ or the complex two–sphere
$\mathbb{S}^{2}_{\mathbb{C}}=\\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}\,:\,z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\\}$,
see Section 3 below. By the invariance properties of the star product
$\star_{\hbar}$ on $\Omega$, one therefore obtains a star product on
$\mathcal{H}(\mathbb{S}^{2}_{\mathbb{C}})$, which is invariant under the group
$\text{SO}(3,\mathbb{C})$ of all complex linear transformations preserving the
quadratic form $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}$. In the complex two–sphere
model, the restriction process of $\Omega$ to its “rotated diagonal”
$\\{(z,-\overline{z})\,:\,z\in\hat{\mathbb{C}}\\}\subseteq\Omega$ reveals
that, algebraically, the Fréchet algebra $\mathcal{A}(\hat{\mathbb{C}})$ is
the same as the set of spherical harmonics on the real two–sphere. This aspect
will be further discussed in the forthcoming paper [18] of the second–named
author where a coherent theory of complexified spherical harmonics is
developed.

  * (iv)

The star product $\star_{\hbar}$ on $\mathcal{H}(\Omega)$ has an explicit and
simple formula in terms of $\mathcal{M}$–invariant differential operators
acting on $\mathcal{H}(\Omega)$, see [6, Sec. 6]. These differential operators
are natural extensions to $\Omega$ of conformally invariant differential
operators acting on $\hat{\mathbb{C}}$ resp. $\mathbb{D}$ of Peschl–Minda
type, which have been continuously studied since the 1950s, see [12, 11, 19,
21, 23]. By restriction to $\hat{\mathbb{C}}$ and $\mathbb{D}$ one obtains
explicit formulas for the star products $\star_{\hbar,\hat{\mathbb{C}}}$ on
$\hat{\mathbb{C}}$ and $\star_{\hbar,\mathbb{D}}$ on $\mathbb{D}$ in terms of
those Peschl–Minda differential operators, see again [6, Sec. 6].

  * (v)

The Fréchet space $\mathcal{H}(\Omega)$ has an intriguing structure. It is the
direct sum of two complementary subspaces $\mathcal{H}_{-}(\Omega)$ and
$\mathcal{H}_{+}(\Omega)$ which we call the past and the future of
$\mathcal{H}(\Omega)$, see Theorem 4.2 below. These subspaces ultimately come
from composition operators induced by two biholomorphic maps from
$\mathbb{C}^{2}$ onto the open subsets

$\Omega_{+}:=\Omega\setminus\left\\{(z,\infty)\,:\,z\in\hat{\mathbb{C}}\right\\}\,,\qquad\Omega_{-}:=\Omega\setminus\left\\{(\infty,w)\,:\,w\in\hat{\mathbb{C}}\right\\}\,$

of $\Omega$, which are obtained from $\Omega$ by removing the horizontal resp.
vertical complex line through the point $(\infty,\infty)\in\Omega$, see Figure
1 below. In particular, the Fréchet spaces $\mathcal{H}(\Omega_{+})$ and
$\mathcal{H}(\Omega_{-})$ are both isomorphic to
$\mathcal{H}(\mathbb{C}^{2})$, the Fréchet space of all entire functions on
$\mathbb{C}^{2}$. Theorem 4.7 below shows that $\mathcal{H}(\Omega_{+})$ and
$\mathcal{H}(\Omega_{-})$ completely encode the Fréchet space structure of
$\mathcal{H}(\Omega)$. As a byproduct, we get (see Corollary 4.8) that the
functions

$f_{p,q}(z,w)=\frac{z^{p}w^{q}}{\left(1-zw\right)^{\max\\{p,q\\}}}\,,\quad
p,q\in\mathbb{N}_{0}\,,$

form a Schauder basis for $\mathcal{H}(\Omega)$.

  * (vi)

The Fréchet spaces $\mathcal{H}(\Omega_{+})$ and $\mathcal{H}(\Omega_{-})$
also play a crucial role for studying invariant differential operators on
$\Omega$ and its projections, the unit disk $\mathbb{D}$ and the Riemann
sphere $\hat{\mathbb{C}}$. In fact, it is shown in [6] that
$\mathcal{H}(\Omega_{+})$ and $\mathcal{H}(\Omega_{-})$ are identified as the
“natural” function spaces on which the Peschl–Minda type differential
operators (see (iv) above) operate. Moreover, the Fréchet spaces
$\mathcal{H}(\Omega_{+})$ and $\mathcal{H}(\Omega_{-})$ are utilised in [7] as
basic tools for a function–theoretic characterization of the “exceptional”
eigenspaces of the invariant Laplacian on the unit disk, which have been
studied by Rudin [20].

  * (vii)

In addition, the second–named author shows in [18] that these “exceptional”
eigenspaces span all of $\mathcal{H}(\Omega)$. In fact, much more is true:
there is a “natural” Schauder basis for $\mathcal{H}(\Omega)$ consisting of
eigenfunctions of the invariant Laplacian on $\Omega$. By restriction to
$\hat{\mathbb{C}}$ and $\mathbb{D}$ one obtains explicit Schauder bases of the
Fréchet algebras $\mathcal{A}(\hat{\mathbb{C}})$ and $\mathcal{A}(\mathbb{D})$
consisting precisely of the eigenfunctions of the invariant Laplacian on
$\hat{\mathbb{C}}$ resp. the “exceptional” eigenfunctions of Rudin [20] of the
invariant Laplacian on $\mathbb{D}$. This provides a geometrically pleasing
intrinsic description of the algebras $\mathcal{A}(\hat{\mathbb{C}})$ and
$\mathcal{A}(\mathbb{D})$.

In summary, the study of the set of holomorphic functions on the complement of
the complexified unit circle provides a unified framework for investigating
conformally invariant differential operators on the disk and the sphere within
their conjecturally natural habitat. In addition, our further investigations
in [6, 7, 18] offer a complex–analysis approach to Wick–type strict
deformation quantization and the spectral properties of the invariant
Laplacians on the unit disk, the Riemann sphere and the complex two–sphere as
well as its various interactions that might otherwise remain hidden.

The paper is organized in the following way. We start with two preliminary
sections. In Section 2 we briefly discuss the canonical Laplacian of $\Omega$
and its main properties. In Section 3 we construct two other very useful,
biholomorphically equivalent models of $\Omega$, the second configuration
space $\mathop{{\rm
Conf}_{2}(\hat{\mathbb{C}})}=\\{(z,w)\in\hat{\mathbb{C}}^{2}\,:\,z\not=w\\}$
of $\hat{\mathbb{C}}$ and the complex two–sphere
$\mathbb{S}_{\mathbb{C}}^{2}=\\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}\,:\,z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\\}$.
After these preparations, we turn to a discussion of the main results of this
paper. In Section 4 we describe in detail the fine structure of the Fréchet
space $\mathcal{H}(\Omega)$. Section 5 is concerned with the group ${\sf
Aut}(\Omega)$ of all biholomorphic automorphisms of $\Omega$. In particular,
we identify the Möbius subgroup $\mathcal{M}$ of ${\sf Aut}(\Omega)$ as the
set of those biholomorphisms of $\Omega$ which leave the Laplacian of $\Omega$
invariant. In Section 6 we introduce a canonical holomorphic Riemannian metric
on $\Omega$ and study its relation to the Möbius subgroup $\mathcal{M}$. The
proofs of the results of Section 4 are given in Section 7, while those of
Sections 5 and 6 are presented in Section 8.

We finally note that there is a natural higher dimensional analogue for the
complement of the complexified unit circle, namely the complement of the
complexified unit ball in $\mathbb{C}^{N}$,

$\Omega_{N}:=\left\\{(z_{1},\ldots,z_{N},w_{1},\ldots,w_{N})\in\hat{\mathbb{C}}^{2N}\,:\,z_{1}w_{1}+\ldots+z_{N}w_{N}\not=1\right\\}\,,$

and one can define a star product on the unit ball of $\mathbb{C}^{N}$ by
working on $\Omega_{N}$ instead of $\Omega=\Omega_{1}$, see [14].
Nevertheless, we have decided to present our work exclusively in the $N=1$
dimensional case. In fact, some of our results easily extend to the
$N$–dimensional situation, essentially without extra work except for adding
more variables, while others conjecturally do not. Perhaps “this is a feature,
not a bug” [10, p. xiv].

##  2\. The invariant Laplacian on $\Omega$

The set $\Omega$ is a complex manifold of complex dimension $2$. Its complex
structure is given by just two charts:

  * (i)

Standard chart

$\phi_{+}:\Omega\cap(\mathbb{C}\times\mathbb{C})\to\mathbb{C}^{2}\,,\qquad\phi_{+}(u,v)=(u,v)\,.$

  * (ii)

Flip chart

$\phi_{-}:\Omega\cap(\hat{\mathbb{C}}\setminus\\{0\\}\times\hat{\mathbb{C}}\setminus\\{0\\})\to\mathbb{C}^{2}\,,\qquad\phi_{-}(u,v):=(1/v,1/u)\,.$

In local coordinates the canonical Laplacian $\Delta_{zw}$ on $\Omega$ is
defined by

$\lx@glossaries@gls@link{main}{Delta_{zw}}{\leavevmode\Delta_{zw}}F(z,w)\coloneqq
4(1-zw)^{2}\mathop{}\\!\partial_{z}\mathop{}\\!\partial_{w}F(z,w)$

for $z,w\in\mathbb{C}$. Here, $\partial_{z}$ and $\partial_{w}$ denote the
complex Wirtinger derivatives for a function $F$ which is supposed to be twice
partially differentiable in the real sense and defined in an open neighborhood
$U\subseteq\Omega$ of $(z,w)$. We are mostly interested in the case when $F$
belongs to the set $\mathcal{H}(U)$ of holomorphic functions on $U$. It is not
difficult to check that $\Delta_{zw}$ is a well–defined continuous linear
operator acting on $\mathcal{H}(U)$ for every open subset $U$ of $\Omega$. The
canonical Laplacian $\Delta_{zw}$ on $\Omega$ can be viewed as the complex
counterpart of both the hyperbolic Laplacian $\Delta_{\mathbb{D}}$ on the unit
disk $\mathbb{D}$ defined by

$\Delta_{\mathbb{D}}f(z):=\left(1-|z|^{2}\right)^{2}\Delta f(z)\,,\quad
z=x+iy\in\mathbb{D}\,,$

as well as of the spherical Laplacian $\Delta_{\hat{\mathbb{C}}}$ on the
sphere $\hat{\mathbb{C}}$ defined in local coordinates by

$\Delta_{\hat{\mathbb{C}}}f(z):=\left(1+|z|^{2}\right)^{2}\Delta f(z)\,,\quad
z=x+iy\in\mathbb{C}\,.$

Here,

$\Delta:=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}$

is the standard Euclidean Laplacian. It is an elementary fact that the
hyperbolic Laplacian $\Delta_{\mathbb{D}}$ is invariant under the group ${\sf
Aut}(\mathbb{D})$ of all conformal automorphisms $T:\mathbb{D}\to\mathbb{D}$
of $\mathbb{D}$. This means that $\Delta_{\mathbb{D}}(f\circ
T)=\Delta_{\mathbb{D}}f\circ T$ for all $T\in{\sf Aut}(\mathbb{D})$ and all
$C^{2}$–functions $f:U\to\mathbb{D}$ defined on some open set
$U\subseteq\mathbb{D}$. In a similar way, the spherical Laplacian
$\Delta_{\hat{\mathbb{C}}}$ is invariant under the group ${\sf
Rot}(\hat{\mathbb{C}})$ of holomorphic rigid motions of $\hat{\mathbb{C}}$.
Both ${\sf Aut}(\mathbb{D})$ and ${\sf Rot}(\hat{\mathbb{C}})$ are subgroups
of the group ${\sf Aut}(\hat{\mathbb{C}})$ of all Möbius transformations, that
is, the biholomorphic self–maps $\psi:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$.
Clearly, every Möbius transformation $\psi\in{\sf Aut}(\hat{\mathbb{C}})$
gives rise to the following two biholomorphic self–maps of $\Omega$:

$(z,w)\mapsto\left(\psi(z),\frac{1}{\psi(1/w)}\right)\,,\quad(z,w)\mapsto\left(\psi(w),\frac{1}{\psi(1/z)}\right)\,,$

and the set of all such automorphisms of $\Omega$ is a subgroup of ${\sf
Aut}(\Omega)$, which we denote by $\mathcal{M}$. Slightly abusing language, we
call $\mathcal{M}$ the Möbius subgroup of ${\sf Aut}(\Omega)$ and its elements
Möbius automorphisms of $\Omega$. Just as ${\sf Aut}(\mathbb{D})$ and ${\sf
Rot}(\hat{\mathbb{C}})$ operate transitively on $\mathbb{D}$ resp.
$\hat{\mathbb{C}}$, the Möbius subgroup $\mathcal{M}$ acts transitively on
$\Omega$. Moreover, the canonical Laplacian $\Delta_{zw}$ is invariant under
every $T\in\mathcal{M}$:

$\Delta_{zw}\left(F\circ T\right)=\Delta_{zw}\left(F\right)\circ T$

for every $F\in\mathcal{H}(\Omega)$. For this reason we call $\Delta_{zw}$ the
invariant Laplacian on $\Omega$.

##  3\. Other models of $\Omega$

While for applications to deformation quantization and to the spectral
property of the invariant Laplacian on the unit disk, the set $\Omega$ seems
most suitable, some of the proofs become more transparent in the $\mathop{{\rm
Conf}_{2}(\hat{\mathbb{C}})}$–setting, and the connections to spherical
harmonics are best discussed in the $\mathbb{S}^{2}_{\mathbb{C}}$–model. We
therefore briefly discuss these two other biholomorphically equivalent models
for the manifold $\Omega$.

###  3.1. The second configuration space of the Riemann sphere

The mapping

$\mathcal{T}(z,w):=\left(z,1/w\right)$

defines a biholomorphic map from $\Omega$ onto

$G=\left\\{(z,w)\in\hat{\mathbb{C}}\,:\,z\not=w\right\\}\,,$

the so–called second configuration space of the sphere $\hat{\mathbb{C}}$, see
[2]. In the literature, the second configuration space of $\hat{\mathbb{C}}$
is often denoted by $\mathop{{\rm Conf}_{2}(\hat{\mathbb{C}})}$, but for ease
of notation we write $G$ instead of $\mathop{{\rm
Conf}_{2}(\hat{\mathbb{C}})}$. Clearly, $G$ is an open subset of
$\hat{\mathbb{C}}^{2}$ and pathconnected. It is known that $G=\mathop{{\rm
Conf}_{2}(\hat{\mathbb{C}})}$ is homotopy equivalent to $\hat{\mathbb{C}}$,
see [2, Example 2.4 (1)]. Thus its fundamental group $\pi_{1}(G)$ is trivial
and $G$ is simply connected. Moreover, it is easy to see that every entire
function $g:\mathbb{C}\to\mathbb{C}$ gives rise to the following biholomorphic
self–map of $G$:

$(z,w)\mapsto\left(z+g\left(\frac{1}{z-w}\right),w+g\left(\frac{1}{z-w}\right)\right)\,.$

In particular, the automorphism group ${\sf Aut}(G)$ of $G$ is
infinite–dimensional. By way of the biholomorphic map $\mathcal{T}:\Omega\to
G$, these properties of $G$ also hold for $\Omega$. Thus $\Omega$ is simply
connected and ${\sf Aut}(\Omega)$ is infinite–dimensional.

###  3.2. The complex two–sphere

Another useful model for $\Omega$ is the complex two–sphere

$\mathbb{S}_{\mathbb{C}}^{2}=\left\\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}\;\big{|}\;z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=1\right\\}\,.$

In fact, $\Omega$ is biholomorphically equivalent to
$\mathbb{S}_{\mathbb{C}}^{2}$ by way of the biholomorphic map
$S\colon\Omega\longrightarrow\mathbb{S}_{\mathbb{C}}^{2}$,

$S(z,w)=\begin{cases}\displaystyle\left(\frac{z-w}{1-zw},-i\frac{z+w}{1-zw},-\frac{1+zw}{1-zw}\right)&\phantom{if\quad}(z,w)\in\mathbb{C}^{2},\,zw\not=1,\\\\[11.38109pt]
\displaystyle\left(1/z,i/z,1\right)&\text{ if
\quad}z\in\hat{\mathbb{C}}\setminus\\{0\\},\,w=\infty,\\\\[11.38109pt]
\displaystyle\left(-1/w,i/w,1\right)&\phantom{if\quad}w\in\hat{\mathbb{C}}\setminus\\{0\\},\,z=\infty.\end{cases}$

The inverse map of $S\colon\Omega\longrightarrow\mathbb{S}_{\mathbb{C}}^{2}$
is given by $\pi\colon\mathbb{S}_{\mathbb{C}}^{2}\to\Omega$,

$\pi(z_{1},z_{2},z_{3})=\begin{cases}\displaystyle\left(\frac{z_{1}+iz_{2}}{1-z_{3}},-\frac{z_{1}-iz_{2}}{1-z_{3}}\right)&\phantom{if\quad}z_{3}\not=1,\\\\[11.38109pt]
\displaystyle\left(1/z_{1},\infty\right)&\text{ if
\quad}z_{3}=1,\,z_{2}=iz_{1},\\\\[11.38109pt]
\displaystyle\left(\infty,-1/z_{1}\right)&\phantom{if\quad}z_{3}=1,\,z_{2}=-iz_{1}.\end{cases}$

It is natural to call $\pi\colon\mathbb{S}_{\mathbb{C}}^{2}\to\Omega$ the
“complex stereographic projection”, since if we denote by

$\mathbb{S}_{\mathbb{R}}^{2}\coloneqq\mathbb{S}^{2}_{\mathbb{C}}\cap\mathbb{R}^{3}=\big{\\{}(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\;|\;x_{1}^{2}+x_{2}^{2}+x_{2}^{3}=1\big{\\}}$

the Euclidean two–sphere in $\mathbb{R}^{3}$, then

$\pi(x_{1},x_{2},x_{3})=\left(\frac{x_{1}+ix_{2}}{1-x_{3}},-\frac{x_{1}-ix_{2}}{1-x_{3}}\right)\quad\text{
for all }(x_{1},x_{2},x_{3})\in\mathbb{S}_{\mathbb{R}}^{2}\,$

and

$\mathbb{S}^{2}_{\mathbb{R}}\to\hat{\mathbb{C}}\,,\qquad(x_{1},x_{2},x_{3})\mapsto\frac{x_{1}+ix_{2}}{1-x_{3}}$

is the standard stereographic projection of $\mathbb{S}^{2}_{\mathbb{R}}$ onto
$\hat{\mathbb{C}}$. In particular, $\pi$ maps $\mathbb{S}_{\mathbb{R}}^{2}$
onto the “rotated diagonal”
$\\{(z,-\overline{z})\,:z\in\hat{\mathbb{C}}\\}\subseteq\Omega$. Furthermore,
if $H_{\mathbb{R}}^{2}$ denotes the image of the hyperboloid
$\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\,:\,-x_{1}^{2}-x_{2}^{2}+x_{3}^{2}=1\\}$
under the biholomorphic map $(z_{1},z_{2},z_{3})\mapsto(iz_{1},iz_{2},z_{3})$,
then $\pi$ maps the lower half $\\{(x_{1},x_{2},x_{3})\in
H_{\mathbb{R}}^{2}\,:\,x_{3}\leq-1\\}$ onto
$\\{(z,\overline{z})\\}\,:\,z\in\mathbb{D}\\}$ and the upper half
$\\{(x_{1},x_{2},x_{3})\in H_{\mathbb{R}}^{2}\,:\,x_{3}\geq 1\\}$ onto
$\\{(z,\overline{z})\\}\,:\,z\in\hat{\mathbb{C}}\setminus\overline{\mathbb{D}}\\}$.

###### Remark 3.1

On the complex two–sphere $\mathbb{S}^{2}_{\mathbb{C}}$, the subgroup
$\mathcal{M}$ of ${\sf Aut}(\Omega)$ defined by (1.1) corresponds to

$\operatorname{SO}(3,\mathbb{C}):=\left\\{R\in\mathbb{C}^{3\times
3}\,:\,R^{T}R=I,\,\det R=1\right\\}\,.$

##  4\. The structure of the Fréchet space $\mathcal{H}(\Omega)$

In this section we describe the fine structure of the Fréchet space
$\mathcal{H}(\Omega)$. In particular, we shall see that the functions

(4.1) $f_{p,q}:\Omega\to\mathbb{C}\,,\qquad
f_{p,q}(z,w)=\frac{z^{p}w^{q}}{\left(1-zw\right)^{\max\\{p,q\\}}}\,,\quad
p,q\in\mathbb{N}_{0}\,,$

are a Schauder basis for $\mathcal{H}(\Omega)$. As immediate corollaries we
obtain that the functions ${z\mapsto f_{p,q}(z,\overline{z})}$ are a Schauder
basis for $\mathcal{A}(\mathbb{D})$ and the functions $z\mapsto
f_{p,q}(z,-\overline{z})$ are a Schauder basis for
$\mathcal{A}(\hat{\mathbb{C}})$. These corollaries have already been proven in
[14, Theorem 3.16] and [22, Proposition 6.4]. Our approach is much more
conceptional, and has various useful ramifications for studying invariant
differential operators and the spectral theory of the invariant Laplacian on
$\Omega$ for which we refer to [6, 7]. The basic tool is a natural
decomposition of $\mathcal{H}(\Omega)$ into a direct sum of two closed
subspaces $\mathcal{H}_{+}(\Omega)$ and $\mathcal{H}_{-}(\Omega)$, which are
defined as follows. We denote by $\mathop{\rm span}M$ the closure of all
finite linear combinations of a subset $M$ of a Fréchet space $X$.

######  Definition 4.1 (The past and the future in $\mathcal{H}(\Omega)$)

The closed subspace

$\mathcal{H}_{-}(\Omega):=\mathop{\rm span}\left\\{f_{p,q}\,:\,0\leq q\leq
p<\infty\right\\}$

is called the past in $\mathcal{H}(\Omega)$ and the closed subspace

$\mathcal{H}_{+}(\Omega):=\mathop{\rm span}\left\\{f_{p,q}\,:\,0\leq
p<q<\infty\right\\}$

is called the future in $\mathcal{H}(\Omega)$.

######  Theorem 4.2 (Canonical Decomposition of $\mathcal{H}(\Omega)$)

$\mathcal{H}(\Omega)$ is the topological direct sum of the past
$\mathcal{H}_{-}(\Omega)$ and the future $\mathcal{H}_{+}(\Omega)$,

(4.2)
$\mathcal{H}(\Omega)=\mathcal{H}_{-}(\Omega)\oplus\mathcal{H}_{+}(\Omega)\,.$

The decomposition (4.2) is intimately tied to the decomposition of the domain
$\Omega$,

$\Omega=\Omega_{+}\cup\Omega_{-}\cup\left\\{(\infty,\infty)\right\\}$

into the two subdomains

$\Omega_{+}:=\Omega\setminus\left\\{(z,\infty)\,:\,z\in\hat{\mathbb{C}}\right\\}\,,\qquad\Omega_{-}:=\Omega\setminus\left\\{(\infty,w)\,:\,w\in\hat{\mathbb{C}}\right\\}\,.$

Note that $\Omega_{-}\cap\Omega_{+}=\Omega\cap\mathbb{C}^{2}$ and

$\mathcal{H}(\Omega_{-})\cap\mathcal{H}(\Omega_{+})=\mathcal{H}(\Omega)\,.$

Moreover, it is an immediate consequence of (4.1) and Definition 4.1 that

$\mathcal{H}_{\pm}(\Omega)\subseteq\mathcal{H}(\Omega_{\pm})\,,$

and the inclusion is strict. The subdomains $\Omega_{+}$ and $\Omega_{-}$ of
$\Omega$ are visualized in Figure 1. The edges of each square represent points
near infinity. If the edge belongs to the domain, it is dashed. The blue dots
correspond to boundary points. Note that points on opposite edges and in
particular the four corners are identified.

Figure 1. Schematic picture of the domains $\Omega$ (left), $\Omega_{+}$
(center) and $\Omega_{-}$ (right) with points at infinity.

The following proposition provides the first step towards the proof of Theorem
4.2. It shows that $\Omega_{+}$ and $\Omega_{-}$ are simply connected
subdomains of $\Omega$ which can be mapped biholomorphically onto
$\mathbb{C}^{2}$ by means of the elementary maps

$\displaystyle\Psi_{+}:\Omega_{+}\to\mathbb{C}^{2}\,,\quad\Psi_{+}(z,w)=\left(\frac{z}{1-zw},w\right)\,,$
$\displaystyle\Psi_{-}:\Omega_{-}\to\mathbb{C}^{2}\,,\quad\Psi_{-}(z,w)=\left(z,\frac{w}{1-zw}\right)\,.$

In terms of the induced composition operators

(4.3) $\displaystyle\mathcal{C}_{\Psi_{+}}$
$\displaystyle:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega_{+})\,,\qquad\mathcal{C}_{\Psi_{+}}(F):=F\circ\Psi_{+}\,,$
(4.4) $\displaystyle\mathcal{C}_{\Psi_{-}}$
$\displaystyle:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega_{-})\,,\qquad\mathcal{C}_{\Psi_{-}}(F):=F\circ\Psi_{-}\,,$

this means that both Fréchet spaces, $\mathcal{H}(\Omega_{+})$ and
$\mathcal{H}(\Omega_{-})$, are isomorphic to $\mathcal{H}(\mathbb{C}^{2})$.

###### Proposition 4.3

The domains $\Omega_{+}$ and $\Omega_{-}$ are biholomorphically equivalent to
$\mathbb{C}^{2}$ and the composition operators (4.3) and (4.4) are continuous
and bijective.

The proof is by simple verification. The second ingredient for the proof of
Theorem 4.2 is the canonical decomposition of the Fréchet space
$\mathcal{H}(\mathbb{C}^{2})$ of all entire functions
$F:\mathbb{C}^{2}\to\mathbb{C}$,

$\mathcal{H}(\mathbb{C}^{2})=\mathcal{H}_{+}(\mathbb{C}^{2})\oplus\mathcal{H}_{-}(\mathbb{C}^{2})\,,$

where $\mathcal{H}_{+}(\mathbb{C}^{2})$ denotes the closed linear hull of the
monomials $z^{p}w^{q}$ with $p\geq q$, and $\mathcal{H}_{-}(\mathbb{C}^{2})$
is the closed linear hull of the monomials $z^{p}w^{q}$ with $p<q$. We call
$\mathcal{H}_{-}(\mathbb{C}^{2})$ the past and
$\mathcal{H}_{+}(\mathbb{C}^{2})$ the future in $\mathcal{H}(\mathbb{C}^{2})$.
Roughly speaking, we shall show that the direct sum of the restriction of the
operator $\mathcal{C}_{\Psi_{-}}$ to $\mathcal{H}_{-}(\mathbb{C}^{2})$ and the
restriction of the operator $\mathcal{C}_{\Psi_{+}}$ to
$\mathcal{H}_{+}(\mathbb{C}^{2})$ provides a continuous linear bijection from
$\mathcal{H}(\mathbb{C}^{2})$ onto $\mathcal{H}(\Omega)$. Along the way, we
will encounter several subtleties, which we carefully have to bypass. The
construction is in two steps. In the first step, we establish the following
Proposition, which provides the bridge from the past and the future in
$\mathcal{H}(\mathbb{C}^{2})$ to the past and the future in
$\mathcal{H}(\Omega)$.

###### Proposition 4.4

aa

  * (a)

For each $F^{+}\in\mathcal{H}_{+}(\mathbb{C}^{2})$ the function
$\mathcal{C}_{\Psi_{+}}(F^{+})\in\mathcal{H}(\Omega_{+})$ has a holomorphic
extension $\Phi_{+}(F^{+})\in\mathcal{H}_{+}(\Omega)$, and the induced linear
operator

$\Phi_{+}:\mathcal{H}_{+}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\Omega)$

is continuous and one-to-one.

  * (b)

For each $F^{-}\in\mathcal{H}_{-}(\mathbb{C}^{2})$ the function
$\mathcal{C}_{\Psi_{-}}(F^{-})\in\mathcal{H}(\Omega_{-})$ has a holomorphic
extension $\Phi_{-}(F^{-})\in\mathcal{H}_{-}(\Omega)$, and the induced linear
operator

$\Phi_{-}:\mathcal{H}_{-}(\mathbb{C}^{2})\to\mathcal{H}_{-}(\Omega)$

is continuous and one-to-one.

In particular, the linear operator
$\Phi:=\Phi_{+}\oplus\Phi_{-}:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega)$
is continuous and one-to-one. In the second step, it remains to prove that
$\Phi$ is onto in order to complete the following diagram:

${\mathcal{H}(\mathbb{C}^{2})}$${=}$${\mathcal{H}_{+}(\mathbb{C}^{2})}$${\oplus}$${\mathcal{H}_{-}(\mathbb{C}^{2})}$${\mathcal{H}(\Omega)}$${=}$${\mathcal{H}_{+}(\Omega)}$${\oplus}$${\mathcal{H}_{-}(\Omega)}$${\phantom{;;}\Phi_{\phantom{+}}}$${\phantom{;}=_{\phantom{a}}}$${\phantom{;;}\Phi_{+}}$${\hskip
4.26773pt\oplus_{\phantom{a}}}$${\phantom{;;}\Phi_{-}}$ Figure 2.
Decomposition of $\mathcal{H}(\Omega)$

For the proof that $\Phi:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega)$ is
onto, we show that the natural projections that map from
$\mathcal{H}(\mathbb{C}^{2})$ onto its closed complementary subspaces
$\mathcal{H}_{+}(\mathbb{C}^{2})$ and $\mathcal{H}_{-}(\mathbb{C}^{2})$ also
act as continuous projections from $\mathcal{H}(\Omega_{\pm})$ onto
$\mathcal{H}_{\pm}(\Omega)$. As we shall see, in combination with Proposition
4.3, this will imply that the linear operators
$\Phi_{\pm}:\mathcal{H}_{\pm}(\mathbb{C}^{2})\to\mathcal{H}_{\pm}(\Omega)$ in
Proposition 4.4 are surjective and also that
$\mathcal{H}(\Omega)=\mathcal{H}_{+}(\Omega)\oplus\mathcal{H}_{-}(\Omega)$.

Note that the bidisk $\mathbb{D}^{2}$ is contained in each of the domains
$\Omega$, $\Omega_{+}$ and $\Omega_{-}$, see again Figure 1. We denote by
$\pi_{+}:\mathcal{H}(\mathbb{D}^{2})\to\mathcal{H}_{+}(\mathbb{D}^{2})$ and
$\pi_{-}:\mathcal{H}(\mathbb{D}^{2})\to\mathcal{H}_{-}(\mathbb{D}^{2})$ the
natural projection maps defined by

(4.5)
$\pi_{+}\left(\sum\limits_{p,q=0}^{\infty}a_{p,q}u^{p}v^{q}\right):=\sum\limits_{p\geq
q}a_{p,q}u^{p}v^{q}\,,\qquad\pi_{-}\left(\sum\limits_{p,q=0}^{\infty}a_{p,q}u^{p}v^{q}\right):=\sum\limits_{p<q}a_{p,q}u^{p}v^{q}\,.$

Clearly, the restrictions of $\pi_{\pm}$ to the subspace
$\mathcal{H}(\mathbb{C}^{2})$ of $\mathcal{H}(\mathbb{D}^{2})$ are continuous
projections from $\mathcal{H}(\mathbb{C}^{2})$ onto
$\mathcal{H}_{\pm}(\mathbb{C}^{2})$. The following result shows that the
restrictions of $\pi_{\pm}$ to the subspaces $\mathcal{H}(\Omega_{\pm})$ of
$\mathcal{H}(\mathbb{D}^{2})$ are continuous projections from
$\mathcal{H}(\Omega_{\pm})$ onto $\mathcal{H}_{\pm}(\Omega)$ as well.

###### Proposition 4.5

aa

  * (a)

For each $f\in\mathcal{H}(\Omega_{+})$ the function
$\pi_{+}(f)\in\mathcal{H}(\mathbb{D}^{2})$ has a holomorphic extension to a
function in $\mathcal{H}_{+}(\Omega)$, and the induced linear operator

$\pi_{+}:\mathcal{H}(\Omega_{+})\to\mathcal{H}_{+}(\Omega)$

is continuous. Moreover,
$\pi_{+}:\mathcal{H}(\Omega_{+})\to\mathcal{H}(\Omega_{+})$ is a projection
onto $\mathcal{H}_{+}(\Omega)$, and the diagram in Figure 3 (a) commutes.

  * (b)

For each $f\in\mathcal{H}(\Omega_{-})$ the function
$\pi_{-}(f)\in\mathcal{H}(\mathbb{D}^{2})$ has a holomorphic extension to a
function in $\mathcal{H}_{-}(\Omega)$, and the induced linear operator

$\pi_{-}:\mathcal{H}(\Omega_{-})\to\mathcal{H}_{-}(\Omega)$

is continuous. Moreover,
$\pi_{-}:\mathcal{H}(\Omega_{-})\to\mathcal{H}(\Omega_{-})$ is a projection
onto $\mathcal{H}_{-}(\Omega)$, and the diagram in Figure 3 (b) commutes.

${\mathcal{H}(\mathbb{C}^{2})}$${\mathcal{H}(\Omega_{+})}$${\mathcal{H}_{+}(\mathbb{C}^{2})}$${\mathcal{H}_{+}(\Omega)}$${\mathcal{C}_{\Psi_{+}}}$${\cong}$${\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi_{+}}$${\pi_{+}}$${\Phi_{+}}$${\cong}$
(a)

${\mathcal{H}(\mathbb{C}^{2})}$${\mathcal{H}(\Omega_{-})}$${\mathcal{H}_{-}(\mathbb{C}^{2})}$${\mathcal{H}_{-}(\Omega)}$${\mathcal{C}_{\Psi_{-}}}$${\cong}$${\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi_{-}}$${\pi_{-}}$${\Phi_{-}}$${\cong}$
(b)

Figure 3. Commutation of composition operators and projections

Note that we are systematically abusing language by using the same symbols,
$\pi_{+}$ and $\pi_{-}$, for denoting the projections (4.5) which act on
different functions spaces depending on the context. This should cause no
serious confusion.

###### Remark 4.6

Proposition 4.5 implies that the set $\mathcal{H}_{+}(\Omega)$ is also closed
w.r.t. the Fréchet topology of the ambient space $\mathcal{H}(\Omega_{+})$.
Hence the map $\pi_{+}:\mathcal{H}(\Omega_{+})\to\mathcal{H}(\Omega_{+})$ is a
continuous projection onto the closed subspace $\mathcal{H}_{+}(\Omega)$ of
$\mathcal{H}(\Omega_{+})$. The same remark applies to $\pi_{-}$. However, for
the proof of Theorem 4.2 we need the full force of Proposition 4.5, that is,
the continuity of $\pi_{+}:\mathcal{H}(\Omega_{+})\to\mathcal{H}_{+}(\Omega)$
w.r.t. the canonical Fréchet topology of $\mathcal{H}_{+}(\Omega)$.

It follows at once from Proposition 4.5 that the operators
$\Phi_{+}:\mathcal{H}_{+}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\Omega)$ and
$\Phi_{-}:\mathcal{H}_{-}(\mathbb{C}^{2})\to\mathcal{H}_{-}(\Omega)$ in
Proposition 4.4 are not only continuous and one-to-one, but also surjective.
Their direct sum
$\mathcal{C}_{\Psi_{+}}\oplus\mathcal{C}_{\Psi_{-}}:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega)$
is a continuous bijection:

######  Theorem 4.7 (Canonical decomposition of $\mathcal{H}(\Omega)$, fine
structure)

The map

$\displaystyle\Phi\quad:$
$\displaystyle\quad\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega)\,,$
$\displaystyle\quad
F\mapsto\Phi(F):=\Phi_{+}\left(\pi_{+}(F)\right)+\Phi_{-}\left(\pi_{-}(F)\right)$

is a continuous linear bijection. Its inverse map is given by

$\displaystyle\Phi^{-1}\quad:$
$\displaystyle\quad\mathcal{H}(\Omega)\to\mathcal{H}(\mathbb{C}^{2})\,,$
$\displaystyle\quad
f\mapsto\Phi^{-1}(f):=\pi_{+}\left(f\circ\Psi_{+}^{-1}\right)+\pi_{-}\left(f\circ\Psi_{-}^{-1}\right)\,.$

In addition, the isomorphisms
$\Phi:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega)$ and
$\Phi^{-1}:\mathcal{H}(\Omega)\to\mathcal{H}(\mathbb{C}^{2})$ preserve the
past and the future,

$\Phi\left(\mathcal{H}_{-}(\mathbb{C}^{2})\right)=\mathcal{H}_{-}(\Omega)\,,\qquad\Phi\left(\mathcal{H}_{+}(\mathbb{C}^{2})\right)=\mathcal{H}_{+}(\Omega)\,.$

By construction, $\Phi$ maps each element $u^{p}v^{q}$ of the canonical
Schauder basis $(u^{p}v^{q})_{p,q}$ of $\mathcal{H}(\mathbb{C}^{2})$ to
$f_{p,q}\in\mathcal{H}(\Omega)$ defined in (4.1). Since
$\Phi:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}(\Omega)$ is a continuous
bijection, we finally arrive at the following result.

###### Corollary 4.8

$(f_{p,q})_{p,q\in\mathbb{N}_{0}}$ is a Schauder basis of
$\mathcal{H}(\Omega)$. In particular, each $f\in\mathcal{H}(\Omega)$ has a
unique representation as

$f(z,w)=\sum\limits_{p,q=0}^{\infty}a_{p,q}f_{p,q}(z,w)\,.$

This series converges absolutely and locally uniformly in $\Omega$, and the
Schauder coefficients $a_{p,q}$ of $f$ are given by

(4.6)
$a_{p,q}=\begin{cases}\displaystyle-\frac{1}{4\pi^{2}}\int\limits_{\partial\mathbb{D}}\int\limits_{\partial\mathbb{D}}f\left(z,\frac{w}{1+zw}\right)\frac{dzdw}{z^{p+1}w^{q+1}}&p<q\,,\\\\[5.69054pt]
\displaystyle-\frac{1}{4\pi^{2}}\int\limits_{\partial\mathbb{D}}\int\limits_{\partial\mathbb{D}}f\left(\frac{z}{1+zw},w\right)\frac{dzdw}{z^{p+1}w^{q+1}}&p\geq
q\end{cases}$

Clearly, $(f_{p,q})_{p<q}$ is a Schauder basis of $\mathcal{H}_{-}(\Omega)$
and $(f_{p,q})_{q\leq p}$ is a Schauder basis of $\mathcal{H}_{+}(\Omega)$.

######  Remark 4.9 (Terminology)

We briefly indicate why one might call $\mathcal{H}_{-}(\Omega)$ the past and
$\mathcal{H}_{+}(\Omega)$ the future in $\mathcal{H}(\Omega)$. As described in
the introduction, one motivation for our study of the Fréchet space
$\mathcal{H}(\Omega)$ comes from the fact that the restriction to the diagonal
yields the Fréchet algebra $\mathcal{A}(\mathbb{D})$ for which a continuous
star product has been constructed in [14]. In view of Theorem 4.2, the Fréchet
algebra $\mathcal{A}(\mathbb{D})$ decomposes in

$\mathcal{A}(\mathbb{D})=\mathcal{A}_{+}(\mathbb{D})\oplus\mathcal{A}_{-}(\mathbb{D})\,$

where each element $\phi$ in $\mathcal{A}_{+}(\mathbb{D})$ has the form

$\phi(re^{it})=\sum\limits_{p\geq
q}a_{p,q}\frac{r^{p+q}}{(1-r^{2})^{p}}e^{i(p-q)t}\,,\quad
z=re^{it}\in\mathbb{D}\,.$

Hence $t\mapsto\phi(re^{it})$ has no negatively indexed Fourier coefficients
and thus belongs to the Hardy space $H^{2}(\partial\mathbb{D})$, see e.g. [9].

##  5\. Automorphisms and the invariant Laplacian of $\Omega$

The complex manifold $\Omega$ has a rich and highly symmetric structure. In
particular, $\Omega$ is a Stein manifold with the holomorphic density
property, see [13, Sec. 3]. Roughly speaking, the density property implies by
way of Andersen–Lempert theory [4] that $\Omega$ has a very large automorphism
group ${\sf Aut}(\Omega)$. In particular, ${\sf Aut}(\Omega)$ is
infinite–dimensional. It is therefore of interest to characterize those
subgroups of ${\sf Aut}(\Omega)$, whose elements have meaningful geometric or
algebraic characteristic properties. For instance, since one can think of
$\Omega$ as a submanifold of the complex projective space
${\mathbb{C}}\operatorname{P}^{3}$, it is possible to speak about rational
automorphisms of $\Omega$, and these have been completely characterized in
[16]. In this section we give a characterization of those automorphisms of
$\Omega$, which leave the Laplacian $\Delta_{zw}$ on $\Omega$ invariant in the
following, very weak, sense.

###### Definition 5.1

Let $O\subseteq\Omega$ be a non–empty open set and let $T\colon O\to\Omega$ be
a holomorphic mapping. We say that the operator $\Delta_{zw}$ is $T$–invariant
if

(5.1) $\Delta_{zw}(F\circ T)=\Delta_{zw}(F)\circ T\quad\text{for every
$F\in\mathcal{H}(\Omega)$.}$

As it has been pointed out in Section 2, the operator $\Delta_{zw}$ is
$T$–invariant for every automorphism $T$ belonging to the Möbius subgroup
$\mathcal{M}$, see (1.1). Since ${\sf Aut}(\Omega)$ is much larger than
$\mathcal{M}$, one might ask if there exist other automorphisms $T$ for which
$\Delta_{zw}$ is $T$–invariant. The following result shows that this is not
the case: the Möbius automorphisms of $\Omega$ are the only locally defined
holomorphic maps on $\Omega$, for which the Laplacian $\Delta_{zw}$ is
invariant.

###### Theorem 5.2

Let $O$ be a subdomain of $\Omega$, and let $T\colon O\to\Omega$ be a
holomorphic mapping. Then the following conditions are equivalent:

  * (a)

$\Delta_{zw}$ is $T$–invariant.

  * (b)

$T\in\mathcal{M}$.

In short: The Möbius–invariant Laplacian is $T$–invariant if and only if $T$
is Möbius.

Note that in Theorem 5.2 we only assume $T:O\to\Omega$ is defined in some
subdomain $O$ of $\Omega$. It is to emphasize that the $T$–invariance
condition (5.1) is very weak, since we only assume this condition holds for
functions $F$ which are defined and holomorphic on all of $\Omega$ and not for
functions which are holomorphic merely on some open subset of $\Omega$, see
Remark 8.2 below. The crucial point is to make sure that $\mathcal{H}(\Omega)$
contains sufficiently many functions to ensure that condition (5.1) forces
$T\in\mathcal{M}$. In fact, condition (5.1) cannot be weakened any further,
since there is no open set in $\hat{\mathbb{C}}^{2}$ which is strictly larger
than $\Omega$ and carries nonconstant holomorphic functions:

###### Theorem 5.3

Let $F\in\mathcal{H}(\Omega)$. Suppose $\xi\in\partial\Omega$ and there is an
open neighborhood $U\subseteq\hat{\mathbb{C}}^{2}$ of $\xi$ such that $F$ is
bounded on $U$. Then $F$ is constant.

The proof of the non–trivial implication “(a) $\Longrightarrow$ (b)” of
Theorem 5.2 will be given in Section 8. It is rather long and is therefore
divided into several steps which might be of independent interest. We
therefore briefly outline the idea of the proof of Theorem 5.2. The situation
becomes somewhat simpler on the second configuration space $G$ of
$\hat{\mathbb{C}}$, see Section 3.1. By mapping $\Omega$ biholomorphically
onto $G$ as explained in Section 3.1, it is easy to check that Theorem 5.2 is
equivalent to

######  Theorem 5.4 (Invariance of the Laplacian on the second configuration
space)

Let $O$ be a subdomain of $G$ and $T=(T_{1},T_{2}):O\to G$ a holomorphic
mapping. Then the following conditions are equivalent:

  * (a)

(The Laplacian on $G$ is $T$–invariant)
For every $F\in\mathcal{H}(G)$ and all $(z,w)\in O$,

(5.2) $(z-w)^{2}\partial_{z}\partial_{w}\left(F\circ
T\right)(z,w)=\left(T_{1}(z,w)-T_{2}(z,w)\right)^{2}(\partial_{z}\partial_{w}F)(T(z,w))$

  * (b)

There exists ${\psi\in{\sf Aut}(\hat{\mathbb{C}})}$ such that
$T(z,w)=\left(\psi(z),\psi(w)\right)$ or
$T(z,w)=\left(\psi(w),\psi(z)\right)$.

The crucial part of the proof of Theorem 5.4 is to show that any holomorphic
mapping

$T:O\to G\,,\quad(z,w)\mapsto T(z,w)=\begin{pmatrix}T_{1}(z,w)\\\
T_{2}(z,w)\end{pmatrix}\,,$

for which the $T$–invariance property (5.2) holds has “separated variables” in
the sense that either $T_{1}$ depends only on $z$ and $T_{2}$ depends only on
$w$, or vice versa. By carefully choosing appropriate “test functions”
$F\in\mathcal{H}(G)$, we shall show as a first step that the difference
$T_{1}(z,w)-T_{2}(z,w)$ can be expressed as the difference $H_{1}(z)-H_{2}(w)$
of a holomorphic function $H_{1}$ depending only on $z$ and another
holomorphic function $H_{2}$ depending only on $w$. In addition, the
$T$–invariance property (a) in Theorem 5.4 then translates into an
ODE–condition for $H_{1}$ and $H_{2}$:

###### Proposition 5.5

Let $O\subseteq G$ be a non–empty open set and let $T\colon O\to G$ such that
the $T$–invariance condition (5.2) holds. Then there exist subdomains
$D_{1},D_{2}$ of $\mathbb{C}$ with $D_{1}\times D_{2}\subseteq O$ and
holomorphic functions $H_{1}\in\mathcal{H}(D_{1})$,
$H_{2}\in\mathcal{H}(D_{2})$ such that
$T_{1}(z,w)-T_{2}(z,w)=H_{1}(z)-H_{2}(w)$ and

(5.3)
$\left(\frac{H_{1}(z)-H_{2}(w)}{z-w}\right)^{2}=H_{1}^{\prime}(z)H_{2}^{\prime}(w)$

for all $(z,w)\in D_{1}\times D_{2}$.

In a next step, we show that equation (5.3) puts severe constraints on the
functions $H_{1}$ and $H_{2}$:

###### Proposition 5.6

Let $D_{1},D_{2}$ be open sets in $\mathbb{C}$. Suppose
$H_{1}\in\mathcal{H}(D_{1})$ and ${H_{2}\in\mathcal{H}(D_{2})}$ satisfy
condition (5.3) for all $(z,w)\in D_{1}\times D_{2}$. Then either
$H_{1},H_{2}\in{\sf Aut}(\hat{\mathbb{C}})$ or $H_{1},H_{2}$ are constant. In
both cases, $H_{1}=H_{2}$.

###### Remark 5.7

As we shall explain in Remark 8.1 (a), Proposition 5.6 can be seen as a
generalization of a result of Hawley and Schiffer [5]. Their result is the
special case of Proposition 5.6 when $D_{1}=D_{2}$ and $H_{1}=H_{2}$ is
assumed from the outset. Note however that for applying Proposition 5.6 in the
proof of Theorem 5.4, we need $D_{1}\times D_{2}\subseteq G$ and hence
$D_{1}\cap D_{2}=\emptyset$. In particular, the result of Hawley and Schiffer
does not suffice for proving Theorem 5.4.

Taking Propositions 5.5 and 5.6 together shows that for a holomorphic mapping
$T:O\to G$ with the $T$–invariance property (5.2) there is $H_{1}\in{\sf
Aut}(\hat{\mathbb{C}})$ such that

(5.4) $T_{1}(z,w)-T_{2}(z,w)=H_{1}(z)-H_{1}(w)\,\quad\text{ for all }(z,w)\in
O\,.$

The main trick now is to realize that the change of variables
$(z,w)\mapsto\left(H^{-1}_{1}(z),H^{-1}_{2}(w)\right)$ is permissible, which
simplifes condition (5.4) to

$T_{1}(z,w)-T_{2}(z,w)=z-w\,,$

or – equivalently – the $T$–invariance condition (5.2) becomes

(5.5) $\partial_{z}\partial_{w}(F\circ
T)(z,w)=\left(\partial_{z}\partial_{w}F\right)\left(T(z,w)\right)\,.$

Again choosing appropriate “test functions” $F\in\mathcal{H}(G)$, we finally
show that (5.5) implies $T_{1}$ is either independent of $z$ or independent of
$w$. From here, it is then immediate that $T_{1}$ and $T_{2}$ are Möbius.

###### Remark 5.8

If we restrict in Theorem 5.2 to the “diagonal”
$\\{(z,\overline{z})\,:\,z\in\mathbb{D}\\}\subset\Omega$ which we identify
with the unit disk $\mathbb{D}$ and to the “rotated diagonal”
$\\{(z,-\overline{z})\,:\,z\in\hat{\mathbb{C}}\\}\subset\Omega$ which we
identify with the Riemann sphere $\hat{\mathbb{C}}$, we get:

  * (i)

Let $O$ be a subdomain of $\mathbb{D}$, and $T:O\to\mathbb{D}$ holomorphic.
Then the following conditions are equivalent:

    * (a)

$\Delta_{\mathbb{D}}$ is $T$–invariant, that is,
$\Delta_{\mathbb{D}}\left(f\circ T\right)=\Delta_{\mathbb{D}}f\circ T$ for all
$f\in\mathcal{A}(\mathbb{D})$.

    * (b)

$T\in{\sf Aut}(\mathbb{D})$.

  * (ii)

Let $O$ be a subdomain of $\hat{\mathbb{C}}$, and $T:O\to\hat{\mathbb{C}}$
holomorphic. Then the following conditions are equivalent:

    * (a)

$\Delta_{\hat{\mathbb{C}}}$ is $T$–invariant, that is,
$\Delta_{\hat{\mathbb{C}}}\left(f\circ
T\right)=\Delta_{\hat{\mathbb{C}}}f\circ T$ for all
$f\in\mathcal{A}(\hat{\mathbb{C}})$.

    * (b)

$T\in{\sf Rot}(\hat{\mathbb{C}})$.

Similar to Theorem 5.2, in both cases $X=\mathbb{D}$ or $X=\hat{\mathbb{C}}$,
condition (a) is fairly weak in the sense that we assume $\Delta_{X}(f\circ
T)=\Delta_{X}f\circ T$ only for functions $f$ belonging to the set
$\mathcal{A}(X)$, which is a comparatively small subset of the set of all
real–analytic functions on $X$.

##  6\. Automorphisms and a canonical holomorphic Riemannian metric on
$\Omega$

Returning to Theorem 5.2 we now add a third equivalent condition in pure
differential geometric terms. For this purpose we define a holomorphic
analogue $g_{\Omega}$ of the hyperbolic and spherical Riemannian metric on
$\Omega$ in local coordinates by

(6.1) $g_{\Omega}(z,w)=\frac{1}{(1-zw)^{2}}dz\vee dw\,.$

It is easy to check that formula (6.1) defines a non-vanishing and holomorphic
section. In the literature, such a section is referred to as _holomorphic
Riemannian metric_ on $\Omega$, see e.g. [17]. It is not difficult to see that
the (complex–valued) Riemannian Laplacian corresponding to $g_{\Omega}$ is
exactly $\Delta_{zw}$.

###### Theorem 6.1

Let $O$ be a subdomain of $\Omega$, and let $T:O\to\Omega$ be a holomorphic
mapping. Then the following conditions are equivalent:

  1. (a)

$T^{*}g_{\Omega}=g_{\Omega}$ as two–forms.

  2. (b)

$T\in\mathcal{M}$.

Note that condition (a) of Theorem 5.2 is considerably weaker than condition
(a) of Theorem 6.1: the invariance of the two-form is a local condition,
whereas (5.1) merely poses a condition on _globally_ defined holomorphic
functions.

###### Remark 6.2

Let

$\lambda_{\mathbb{D}}(z):=\frac{1}{1-|z|^{2}}\qquad\text{ and
}\qquad\lambda_{\hat{\mathbb{C}}}(z)=\frac{1}{1+|z|^{2}}$

denote the density of the Poincaré metric on $\mathbb{D}$ and the density of
the spherical metric on $\hat{\mathbb{C}}$. If we restrict in Theorem 6.1 to
the “diagonal” $\\{(z,\overline{z})\,:\,z\in\mathbb{D}\\}\subset\Omega$ which
we identify with the unit disk $\mathbb{D}$ resp. to the “rotated diagonal”
$\\{(z,-\overline{z})\,:\,z\in\hat{\mathbb{C}}\\}\subset\Omega$ which we
identify with the Riemann sphere $\hat{\mathbb{C}}$, we get

  * (i)

Let $O$ be a subdomain of $\mathbb{D}$ and $T:O\to\mathbb{D}$ holomorphic.
Then the following conditions are equivalent:

    * (a)

$T^{*}\lambda_{\mathbb{D}}=\lambda_{\mathbb{D}}$.

    * (b)

$T\in{\sf Aut}(\mathbb{D})$.

  * (ii)

Let $O$ be a subdomain of $\hat{\mathbb{C}}$ and $T:O\to\hat{\mathbb{C}}$
holomorphic. Then the following conditions are equivalent:

    * (a)

$T^{*}\lambda_{\hat{\mathbb{C}}}=\lambda_{\hat{\mathbb{C}}}$.

    * (b)

$T\in{\sf Rot}(\hat{\mathbb{C}})$.

##  7\. Proofs of Propositions 4.4 and 4.5, Theorems 4.2 and 4.7, and
Corollary 4.8

### Proof of Proposition 4.4

We only prove part (a) since the proof of (b) merely requires some obvious
modifications. For every $F^{+}\in\mathcal{H}_{+}(\mathbb{C}^{2})$ there are
coefficients $a_{p,q}(F^{+})\in\mathbb{C}$ such that

(7.1) $F^{+}(u,v)=\sum\limits_{p\geq q}a_{p,q}(F^{+})u^{p}v^{q}\,,$

and the series converges uniformly and absolutely for $(u,v)$ in any compact
subset of $\mathbb{C}^{2}$. Since $\Psi_{+}:\Omega_{+}\to\mathbb{C}^{2}$ is
biholomorphic, it follows from Proposition 4.3 that

(7.2)
$\mathcal{C}_{\Psi_{+}}(F^{+})(z,w)=\left(F^{+}\circ\Psi_{+}\right)(z,w)=\sum\limits_{p\geq
q}a_{p,q}(F^{+})\frac{z^{p}w^{q}}{(1-zw)^{p}}\,$

for all $(z,w)\in\Omega_{+}$, and the convergence is absolute and locally
uniform in $\Omega_{+}$. Note that each individual term of the series in (7.2)
has the form $a_{p,q}(F^{+})f_{p,q}$ and is therefore a holomorphic function
on the larger domain $\Omega$. Our goal is to prove the following.

###### Proposition 7.1

Every point $(z_{0},w_{0})\in\Omega$ has a neighborhood $U\subseteq\Omega$
with the following properties:

  * (i)

for each $F^{+}\in\mathcal{H}_{+}(\mathbb{C}^{2})$ of the form (7.1) the
series in (7.2) converges absolutely and uniformly in $U$ to some
$f^{+}\in\mathcal{H}(U)$;

  * (ii)

if $(F^{+}_{n})\subseteq\mathcal{H}_{+}(\mathbb{C}^{2})$ converges locally
uniformly in $\mathbb{C}^{2}$ to $F^{+}\in\mathcal{H}_{+}(\mathbb{C}^{2})$,
then $(f^{+}_{n})$ converges uniformly in $U$ to $f^{+}$.

Hence for each $F^{+}\in\mathcal{H}_{+}(\mathbb{C}^{2})$ the holomorphic
function $\mathcal{C}_{\Psi_{+}}(F^{+})\in\mathcal{H}(\Omega_{+})$ has a
holomorphic extension $\Phi_{+}(F^{+})\in\mathcal{H}_{+}(\Omega)$ given by the
series in (7.2), and this series converges locally uniformly in $\Omega$.
Moreover, the mapping
$\Phi_{+}:\mathcal{H}_{+}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\Omega)$ is
continuous and linear, and is explicitly given by

(7.3) $\displaystyle\Phi_{+}\left(\sum\limits_{p\geq
q}a_{p,q}u^{p}v^{q}\right)$ $\displaystyle:=\sum\limits_{p\geq
q}a_{p,q}f_{p,q}\,.$

Clearly, $\Phi_{+}:\mathcal{H}_{+}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\Omega)$
is one–to–one. Hence we have reduced Proposition 4.4 to Proposition 7.1.

###### Proof of Proposition 7.1.

Throughout the proof, we write

${\lVert f\rVert}_{K}\coloneqq\max_{(z,w)\in K}\lvert f(z,w)\rvert$

for $K\subseteq\Omega$ compact and continuous functions $f\colon
K\to\mathbb{C}$. Fix a compact set $K\subseteq\Omega$ and consider the compact
subsets $K_{1}\coloneqq\\{(z,w)\in K\colon\lvert w\rvert\leq 1\\}$ as well as
$K_{2}\coloneqq\\{(z,w)\in K\colon\lvert w\rvert\geq 1\\}$. Here, we define
$\lvert\infty\rvert\coloneqq\infty$. By continuity of the functions

$f_{1,0}(z,w)=\frac{z}{1-zw}\quad\text{and}\quad f_{1,1}(z,w)=\frac{zw}{1-zw}$

we have

$M\coloneqq\max\left\\{1,\,{\lVert f_{1,0}\rVert}_{K_{1}},\,{\lVert
f_{1,1}\rVert}_{K_{2}}\right\\}<\infty\,.$

Let $B\coloneqq\\{(z,w)\in\mathbb{C}^{2}\,\colon\,\lvert z\rvert\leq
2M,\,\lvert w\rvert\leq 2M\\}$. As $F^{+}\in\mathcal{H}_{+}(\mathbb{C}^{2})$,
the Cauchy estimates show

$\big{\lvert}a_{p,q}(F^{+})\big{\rvert}\leq\frac{{\lVert
F^{+}\rVert}_{B}}{(2M)^{p+q}}\,,\qquad p\geq q.$

Using the submultiplicativity of the norms $\lVert\,\cdot\,\rVert_{K_{1}}$ and
$\lVert\,\cdot\,\rVert_{K_{2}}$, we estimate

$\displaystyle\sum_{p\geq q}\lvert a_{p,q}(F^{+})\rvert\cdot{\lVert
f_{p,q}\rVert}_{K_{1}}\leq\sum_{p\geq q}\lvert
a_{p,q}(F^{+})\rvert\cdot{\lVert f_{1,0}\rVert}^{p}_{K_{1}}\cdot{\lVert
w^{q}\rVert}_{K_{1}}\leq\sum_{p\geq q}\frac{\lVert
F^{+}\rVert_{B}}{(2M)^{p+q}}M^{p}\leq\frac{8}{3}{\lVert F^{+}\rVert}_{B}$

as well as

$\displaystyle\sum_{p\geq q}\lvert a_{p,q}(F^{+})\rvert\cdot{\lVert
f_{p,q}\rVert}_{K_{2}}\leq\sum_{p\geq q}\lvert
a_{p,q}(F^{+})\rvert\cdot{\lVert f_{1,1}\rVert}^{p}_{K_{2}}\cdot{\lVert
w^{q-p}\rVert}_{K_{2}}\leq\sum_{p\geq q}\frac{{\lVert
F^{+}\rVert}_{B}}{(2M)^{p+q}}M^{p}\leq\frac{8}{3}{\lVert F^{+}\rVert}_{B}\,.$

As $K=K_{1}\cup K_{2}$, this implies

(7.4) $\sum_{p\geq q}\lvert a_{p,q}(F^{+})\rvert\cdot{\lVert
f_{p,q}\rVert}_{K}\leq\frac{8}{3}{\lVert F^{+}\rVert}_{B}<\infty\,.$

Thus, the series (7.2) converges uniformly and absolutely on $K$. Varying $K$,
we see that the series (7.2) converges uniformly on compact subsets of
$\Omega$ to a limit $f^{+}\in\mathcal{H}(\Omega)$. Finally, note that every
$(z_{0},w_{0})\in\Omega$ has a compact neighbourhood $K$ in $\Omega$. This
completes the proof of the first statement by setting $U\coloneqq K$. Let now
$(F^{+}_{n})\subseteq\mathcal{H}_{+}(\mathbb{C}^{2})$ converge locally
uniformly in $\mathbb{C}^{2}$ to $F^{+}\in\mathcal{H}_{+}(\mathbb{C}^{2})$. By
linearity of the coefficient functionals $a_{p,q}$, the estimate (7.4) implies

$\big{\lVert}f^{+}-f_{n}^{+}\big{\rVert}_{U}\leq\sum_{p\geq q}\lvert
a_{p,q}(F^{+}-F^{+}_{n})\rvert\cdot{\lVert f_{p,q}\rVert}_{U}\leq 4{\lVert
F^{+}-F^{+}_{n}\rVert}_{B}\,,\qquad n\in\mathbb{N}.$

As $B\subseteq\mathbb{C}^{2}$ is compact, $(f^{+}_{n})$ therefore converges
uniformly to $f^{+}$ on $U$. ∎

### Proof of Proposition 4.5

The proof of Proposition 4.5 is based on Proposition 4.4 and the following
simple observation.

###### Proposition 7.2

Let $p,q\in\mathbb{N}_{0}$. Then

$\displaystyle\pi_{+}\left(\frac{z^{p}w^{q}}{(1-zw)^{p}}\right)$
$\displaystyle=\begin{cases}\displaystyle\frac{z^{p}w^{q}}{(1-zw)^{p}}&\text{
if }p\geq q\\\ 0&\text{ if }p<q\end{cases}$
$\displaystyle\pi_{-}\left(\frac{z^{p}w^{q}}{(1-zw)^{p}}\right)$
$\displaystyle=\begin{cases}0&\text{ if }p\geq q\\\
\displaystyle\frac{z^{p}w^{q}}{(1-zw)^{p}}&\text{ if }p<q\end{cases}$

###### Proof.

This follows from

$\frac{z^{p}w^{q}}{(1-zw)^{p}}=\begin{cases}\displaystyle\frac{(zw)^{q}}{(1-zw)^{p}}z^{p-q}\in\mathcal{H}_{+}(\mathbb{D}^{2})&\text{
if }p\geq q\\\\[11.38109pt]
\displaystyle\frac{(zw)^{p}}{(1-zw)^{p}}w^{q-p}\in\mathcal{H}_{-}(\mathbb{D}^{2})&\text{
if }p<q\,.\qed\end{cases}$

###### Proof of Proposition 4.5.

It suffices to prove Part (a). Let $f\in\mathcal{H}(\Omega_{+})$. Then, by
Proposition 4.3, $F:=f\circ\Psi_{+}^{-1}\in\mathcal{H}(\mathbb{C}^{2})$, so

$F(u,v)=\sum\limits_{p,q=0}^{\infty}a_{p,q}u^{p}v^{q}\,,\quad(u,v)\in\mathbb{C}^{2}\,,$

or, equivalently,

$f(z,w)=\sum\limits_{p,q=0}^{\infty}a_{p,q}\frac{z^{p}w^{q}}{(1-zw)^{p}}\,,\qquad(z,w)\in\Omega_{+}\,.$

This series converges in particular absolutely and locally uniformly in
$\mathbb{D}^{2}$, so

$\pi_{+}(f)(z,w)=\sum\limits_{p,q=0}^{\infty}a_{p,q}\pi_{+}\left(\frac{z^{p}w^{q}}{(1-zw)^{p}}\right)=\sum\limits_{p\geq
q}a_{p,q}f_{p,q}(z,w)\,,\qquad(z,w)\in\mathbb{D}^{2}\,$

by Proposition 7.2. Recall that by slight abuse of notation, we denote both
projections $\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\mathbb{C}^{2})$
and $\mathcal{H}(\Omega_{+})\to\mathcal{H}_{+}(\Omega)$ by the same symbol
$\pi_{+}$. In view of (7.3), this implies

(7.5)
$\pi_{+}(f)=\Phi_{+}(\pi_{+}(F))=\left(\mathcal{C}_{\Psi_{+}}\circ\pi_{+}\circ\mathcal{C}_{\Psi_{+}}^{-1}\right)f\,.$

Note that the right–hand side is a composition of the three continuous
operators

$\mathcal{C}^{-1}_{\Psi_{+}}:\mathcal{H}(\Omega_{+})\to\mathcal{H}(\mathbb{C}^{2})\,,\quad\pi_{+}:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\mathbb{C}^{2})\,,\quad\mathcal{C}_{\Psi_{+}}:\mathcal{H}_{+}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\Omega)\,,$

see Proposition 4.3 and Proposition 4.4. Thus
$\pi_{+}:\mathcal{H}(\Omega_{+})\to\mathcal{H}_{+}(\Omega)$ is continuous and
(7.5) also shows that the diagram in Figure 3 (a) commutes. Using (7.5) twice
and the fact that
$\pi_{+}:\mathcal{H}(\mathbb{C}^{2})\to\mathcal{H}_{+}(\mathbb{C}^{2}$) is a
projection implies

$\pi_{+}(\pi_{+}(f))=\pi_{+}\circ(\mathcal{C}_{\Psi_{+}}\circ\pi_{+}\circ\mathcal{C}_{\Psi_{+}}^{-1})f=(\mathcal{C}_{\Psi_{+}}\circ\pi_{+}\circ\pi_{+}\circ\mathcal{C}_{\Psi_{+}}^{-1})f=(\mathcal{C}_{\Psi_{+}}\circ\pi_{+}\circ\mathcal{C}_{\Psi_{+}}^{-1})f=\pi_{+}(f)$

for all $f\in\mathcal{H}(\Omega_{+})$, so
$\pi_{+}:\mathcal{H}(\Omega_{+})\to\mathcal{H}_{+}(\Omega)$ is a projection as
well. It remains to prove that $\pi_{+}(f)=f$ for all
$f\in\mathcal{H}_{+}(\Omega)$. However, recalling that
$\mathcal{H}_{+}(\Omega)$ is the closed linear span of the functions $f_{p,q}$
with $p\geq q$ in $\mathcal{H}(\Omega)$, this follows at once from Proposition
7.2 since $\pi_{+}(f_{p,q})=f_{p,q}$ whenever $p\geq q$. ∎

### Proof of Theorem 4.2 and Theorem 4.7

Proposition 4.3 and Proposition 4.4 imply that the direct sum
$\Phi:=\Phi_{+}\oplus\Phi_{-}:\mathcal{H}_{+}(\mathbb{C}^{2})\oplus\mathcal{H}_{-}(\mathbb{C}^{2})\to\mathcal{H}(\Omega)$
is linear, continuous and injective. To prove its surjectivity, let
$f\in\mathcal{H}(\Omega)$. Then

$F^{+}:=\pi_{+}(f\circ\Psi_{+}^{-1})\in\mathcal{H}_{+}(\mathbb{C}^{2})\quad\text{and}\quad
F^{-}:=\pi_{-}(f\circ\Psi_{-}^{-1})\in\mathcal{H}_{-}(\mathbb{C}^{2})\,.$

Therefore, $f^{+}:=\Phi_{+}(F^{+})\in\mathcal{H}_{+}(\Omega)$ and
$f^{-}:=\Phi_{-}(F^{-})\in\mathcal{H}_{-}(\Omega)$, and we deduce

$\displaystyle\Phi(F^{+}+F^{-})$
$\displaystyle=\Phi_{+}(F^{+})+\Phi_{-}(F^{-})=\Phi_{+}\left(\pi_{+}(f\circ\Psi_{+}^{-1})\right)+\Phi_{-}\left(\pi_{-}(f\circ\Psi_{-}^{-1})\right)$
$\displaystyle=\pi_{+}\left(\mathcal{C}_{\Psi_{+}}(f\circ\Psi_{+}^{-1})\right)+\pi_{-}\left(\mathcal{C}_{\Psi_{-}}(f\circ\Psi_{-}^{-1})\right)$

in view of the commutation relations of Figure 3. Thus

$\Phi(F^{+}+F^{-})=\pi_{+}(f)+\pi_{-}(f)=f\,,$

and $\Phi$ is surjective. Note that

$F^{+}+F^{-}=\pi_{+}(f\circ\Psi_{+}^{-1})+\pi_{-}(f\circ\Psi_{-}^{-1})=\Phi^{-1}(f)\,.$

The proof of Theorem 4.7 (and Theorem 4.2) is complete. $\square$

### Proof of Corollary 4.8

Let $f\in\mathcal{H}(\Omega)$. By Theorem 4.7,
$F:=\Phi^{-1}(f)\in\mathcal{H}(\mathbb{C}^{2})$, so

$F(u,v)=\sum\limits_{p,q=0}^{\infty}a_{p,q}u^{p}v^{q}\,$

with

(7.6) $a_{p,q}=\frac{1}{(2\pi
i)^{2}}\int\limits_{\partial\mathbb{D}}\int\limits_{\partial\mathbb{D}}\frac{F(z,w)}{z^{p+1}w^{q+1}}\,dzdw\,.$

We can write

$F(u,v)=\pi_{+}(F)(u,v)+\pi_{-}(F)(u,v)=\sum\limits_{p\geq
q}a_{p,q}u^{p}v^{q}+\sum\limits_{p<q}a_{p,q}u^{p}v^{q}\,.$

Hence, again by Theorem 4.7,

$\displaystyle f(z,w)$
$\displaystyle=\Phi(F)(z,w)=\Phi_{+}(\pi_{+}(F))(z,w)+\Phi_{-}(\pi_{-}(F))(z,w)$
$\displaystyle=\sum\limits_{p\geq
q}a_{p,q}f_{p,q}(z,w)+\sum\limits_{p<q}a_{p,q}f_{p,q}(z,w)\,.$

In the last step, we have used identity (7.3) for $\Phi_{+}$ and the
corresponding identity for $\Phi_{-}$. Since both series converge absolutely
and locally uniformly in $\Omega$, we can rearrange the order of summation.
The explicit formula for the Schauder coefficients $a_{p,q}$ in terms of $f$
follows immediately from (7.6). Indeed, if $p\geq q$, then $a_{p,q}$ is the
corresponding Taylor coefficient of $\pi_{+}(F)$ and

$\pi_{+}(F)=\pi_{+}\left(f\circ\Psi_{+}^{-1}\right)=\pi_{+}(f)\circ\Psi_{+}^{-1}\,,$

in view of Figure 3(a). On the other hand,
$\pi_{+}(f)\circ\Psi_{+}^{-1}\in\mathcal{H}_{+}(\mathbb{C}^{2})$, so the
Cauchy integral formula shows that

$a_{p,q}=\frac{1}{(2\pi
i)^{2}}\int\limits_{\partial\mathbb{D}}\int\limits_{\partial\mathbb{D}}\frac{\pi_{+}(F)(z,w)}{z^{p+1}w^{q+1}}\,dzdw=\frac{1}{(2\pi
i)^{2}}\int\limits_{\partial\mathbb{D}}\int\limits_{\partial\mathbb{D}}\frac{(f\circ\Psi_{+}^{-1})(z,w)}{z^{p+1}w^{q+1}}\,dzdw$

Inserting the explicit expression for $\Psi_{+}^{-1}$ proves (4.6) for $p\geq
q$. The proof for $p<q$ is identical. $\square$

##  8\. Proofs of Propositions 5.5 and 5.6, Theorems 5.2 and 5.3, and Theorem
6.1

### Proof of Proposition 5.5

Let $T=(T_{1},T_{2})\colon O\longrightarrow G$ be a holomorphic mapping
defined on some non–empty open subset $O$ of $G$ such that

(8.1) $(z-w)^{2}\partial_{z}\partial_{w}\left(F\circ
T\right)(z,w)=\left(T_{1}(z,w)-T_{2}(z,w)\right)^{2}(\partial_{z}\partial_{w}F)(T(z,w))$

for all $F\in\mathcal{H}(G)$ and all $(z,w)\in O$. We shall make use of
condition (8.1) by inserting several carefully chosen specific functions
${F\in\mathcal{H}(G)}$.

First, we insert

$F\in\mathcal{H}(G)\,,\quad F(z,w)=\frac{1}{z-w}$

into (8.1). By an elementary, but lengthy computation we find that condition
(8.1) for this choice of $F$ is equivalent to

(8.2)
$\begin{split}2\left(\frac{T_{1}(z,w)-T_{2}(z,w)}{z-w}\right)^{2}&=\left(T_{1}-T_{2}\right)\left(\partial_{z}\partial_{w}T_{1}-\partial_{z}\partial_{w}T_{2}\right)\\\
&\qquad\qquad-2\left(\partial_{w}T_{1}-\partial_{w}T_{2}\right)\left(\partial_{z}T_{1}-\partial_{z}T_{2}\right)\end{split}$

If we take

$F\in\mathcal{H}(G)\,,\quad F(z,w)=\frac{1}{(z-w)^{2}}$

in (8.1), then another computation implies that condition (8.1) for this
choice of $F$ is equivalent to

(8.3)
$\begin{split}3\left(\frac{T_{1}(z,w)-T_{2}(z,w)}{z-w}\right)^{2}&=\left(T_{1}-T_{2}\right)\left(\partial_{z}\partial_{w}T_{1}-\partial_{z}\partial_{w}T_{2}\right)\\\
&\qquad\qquad-3\left(\partial_{w}T_{1}-\partial_{w}T_{2}\right)\left(\partial_{z}T_{1}-\partial_{z}T_{2}\right)\end{split}$

Together, the two conditions (8.2) and (8.3) clearly imply

$\left(T_{1}-T_{2}\right)\left(\partial_{z}\partial_{w}T_{1}-\partial_{z}\partial_{w}T_{2}\right)=0\,.$

Since $T=(T_{1},T_{2})$ maps the open set $O\subseteq G$ into $G$, we get
$T_{1}(z,w)\not=T_{2}(z,w)$ for all $(z,w)\in O$ by definition of $G$, and
hence

$\partial_{z}\partial_{w}(T_{1}-T_{2})=0\quad\text{on }O\,.$

Consequently, all terms with mixed derivatives in the Taylor expansion of
$T_{1}-T_{2}$ about each $(z_{0},w_{0})\in O$ vanish. This way, we obtain
auxiliary functions $H_{1}\in\mathcal{H}(D_{1})$, $H_{2}\in\mathcal{H}(D_{2})$
for certain domains $D_{1},D_{2}\subseteq\mathbb{C}$ with $D_{1}\times
D_{2}\subseteq O$ such that

$T_{1}(z,w)-T_{2}(z,w)=H_{1}(z)-H_{2}(w),\qquad(z,w)\in D_{1}\times D_{2}\,.$

Note that we can choose $D_{1},D_{2}\subseteq\mathbb{C}$ since $O$ and
$G\cap\mathbb{C}^{2}$ always have a nonempty intersection. Inserting this
expression into (8.2), it is easily checked that condition (5.3) holds for
$H_{1}$ and $H_{2}$, and the proof of Proposition 5.5 is complete. $\square$

### Proof of Proposition 5.6

If $H_{2}$ is constant on $D_{2}$, then (5.3) implies $H_{1}$ is constant on
$D_{2}$ and $H_{1}=H_{2}$. If $H_{2}$ is not constant on $D_{2}$, then (5.3)
implies $H_{1}$ is not constant on $D_{1}$ either and in fact $H_{1}$ has
vanishing Schwarzian derivative away from possible zeros of $H_{1}^{\prime}$,
that is,

$\operatorname{S}_{H_{1}}(z):=\bigg{(}\frac{H_{1}^{\prime\prime}(z)}{H_{1}^{\prime}(z)}\bigg{)}^{\prime}-\frac{1}{2}\bigg{(}\frac{H_{1}^{\prime\prime}(z)}{H_{1}^{\prime}(z)}\bigg{)}^{2}=0\,.$

This follows by fixing $w\in D_{2}$ and differentiating (5.3) twice w.r.t.
$z\in D_{1}$. It is a classical fact (see [15]) that
$\operatorname{S}_{H_{1}}=0$ is equivalent to $H_{1}\in{\sf
Aut}(\hat{\mathbb{C}})$. Switching the roles of $H_{1}$ and $H_{2}$, we see
that either $H_{1}$, $H_{2}$ are both constant or $H_{1},H_{2}\in{\sf
Aut}(\hat{\mathbb{C}})$. Finally, if $H_{1}\in{\sf Aut}(\hat{\mathbb{C}})$ and
$H_{2}\in{\sf Aut}(\hat{\mathbb{C}})$, then the right–hand side of (5.3) is
never zero. Hence the left–hand side is never zero either, and thus
$H_{1}(z)\neq H_{2}(w)$ for all $z,w\in\mathbb{C}$ such that $z\not=w$. This
is clearly only possible if $H_{1}=H_{2}$. $\square$

###### Remark 8.1

aa

  1. (a)

As noted earlier in Remark 5.7, Proposition 5.6 is an extension of a result of
Hawley and Schiffer [5], who considered the special case $D_{1}=D_{2}$. We
therefore briefly discuss this case a bit further in order to put Proposition
5.6 into a somewhat broader context. Note that if $D_{1}=D_{2}$ or merely
$D_{1}\cap D_{2}\not=\emptyset$, it follows immediately from (5.3) that
$H_{1}\equiv H_{2}$. Let $H:=H_{1}$. Excluding the trivial case that $H$ is
constant, condition (5.3) implies $H$ is univalent in $D_{1}$. Therefore, the
function

$\hat{\operatorname{S}}_{H}:G_{1}\to\mathbb{C}\,,\quad\hat{\operatorname{S}}_{H}(z,w):=\begin{cases}\displaystyle\frac{H^{\prime}(z)H^{\prime}(w)}{\left(H(z)-H(w)\right)^{2}}-\frac{1}{(z-w)^{2}}\;&\text{for}\;z\neq
w\\\\[11.38109pt] \operatorname{S}_{H}(z)\;&\text{for}\;z=w\\\ \end{cases}$

is well–defined and it is not difficult to show that
$\hat{\operatorname{S}}_{H}$ is holomorphic on $D_{1}\times D_{2}$. A
discussion of this fact and its implications can be found in [5]. The
difference between the point of view of Hawley–Schiffer [5] and the one we
pursue in this paper is now apparent: While [5] extends the Schwarzian
derivative $\operatorname{S}_{H}$, a function of one variable, to a
holomorphic function $\hat{\operatorname{S}}_{H}$ of two variables $(z,w)$ in
a neighborhood of the “diagonal” $(z,z)$, our approach to Theorem 5.4 is based
on Proposition 5.6 for the opposite case $D_{1}\cap D_{2}=\emptyset$, so we
exclude the diagonal.

  2. (b)

Fixing $w\in D_{2}$, equation (5.3) is a Riccati differential equation for the
unknown function $H_{1}:D_{1}\to\mathbb{C}$. It is an elementary fact that
each Riccati equation can be transformed in a canonical way into normal form
$u^{\prime}=A(z)+u^{2}$, see (see [15, p. 165]). For the specific Riccati
equation (5.3) the canonical transformation is given by

$H_{1}(z)=H_{2}^{\prime}(w)(z-w)^{2}u(z)+H_{2}(w)+\frac{H_{2}^{\prime}(w)}{2}(z-w)$

and turns the Riccati equation (5.3) into

$u^{\prime}=u^{2}\,,$

whose set of solutions is given by
$\\{(z+c)^{-1}\,:\,c\in\mathbb{C}\\}\cup\\{0\\}$. Elementary considerations
then again lead to the conclusion that either $H_{1}$ is constant or belongs
to ${\sf Aut}(\hat{\mathbb{C}})$. This provides a slightly different approach
to Proposition 5.6.

### Proof of Theorem 5.4 “(a) $\Longrightarrow$ (b)”

Let $T=(T_{1},T_{2})\colon O\longrightarrow G$ be a holomorphic mapping
defined on some subdomain $O$ of $G$ such that condition (5.2) holds for all
${F\in\mathcal{H}(G)}$ and all $(z,w)\in O$.

(i) By Proposition 5.5 and Proposition 5.6 there exists a function
$H:=H_{1}:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$, which is either constant or
belongs to ${\sf Aut}(\hat{\mathbb{C}})$, such that

(8.4) $\left(\frac{H(z)-H(w)}{z-w}\right)^{2}=H^{\prime}(z)H^{\prime}(w)$

and

(8.5) $T_{1}(z,w)-T_{2}(z,w)=H(w)-H(z)\,$

for all $(z,w)\in O$. Since $T$ maps into $G$, we have
$T_{1}(z,w)\not=T_{2}(z,w)$ for all $(z,w)\in O$, so the case $H$ is constant
cannot occur, and therefore $H\in{\sf Aut}(\hat{\mathbb{C}})$.

(ii) We claim, without loss of generality, that we may assume

(8.6) $T_{1}(z,w)-T_{2}(z,w)=w-z\quad\text{ for all }(z,w)\in O\,.$

To prove this claim, consider

$S(z,w):=\left(H^{-1}(z),H^{-1}(w)\right)\,,\qquad(z,w)\in G\,.$

Note that $S$ is a biholomorphic self–map of $G$ and $\tilde{O}:=S^{-1}(O)$ is
a subdomain of $G$. The holomorphic mapping

$\tilde{T}:=T\circ S:\tilde{O}\to G\,$

then satisfies

$\tilde{T}_{1}(z,w)-\tilde{T}_{2}(z,w)=T_{1}\left(S(z,w)\right)-T_{2}\left(S(z,w)\right)=H\left(H^{-1}(w)\right)-H\left(H^{-1}(z)\right)\,$

in view of (8.5), and hence

(8.7) $\tilde{T}_{1}(z,w)-\tilde{T}_{2}(z,w)=w-z\quad\text{ for all
}(z,w)\in\tilde{O}\,.$

To finish the proof of Theorem 5.4 it therefore suffices to show that the
Laplacian on $G$ is $\tilde{T}$–invariant, that is, condition (5.2) holds with
$T$ replaced by $\tilde{T}$. To check this, observe that

(8.8)
$\begin{split}(z-w)^{2}\partial_{z}\partial_{w}\left(F\circ\tilde{T}\right)(z,w)&=(z-w)^{2}\partial_{z}\partial_{w}\left(\left(F\circ
T\right)\circ S\right)(z,w)\\\
&=(z-w)^{2}\left(\partial_{z}\partial_{w}\left(F\circ
T\right)(S(z,w))\right)\left(H^{-1}\right)^{\prime}(z)\left(H^{-1}\right)^{\prime}(w)\end{split}$

by the chain rule and the particular form of
$S(z,w)=\left(H^{-1}(z),H^{-1}(w)\right)$. Since $T$ satisfies condition
(5.2), we have

(8.9) $\begin{split}\partial_{z}\partial_{w}\left(F\circ
T\right)(S(z,w))&=\left(\frac{H^{-1}(z)-H^{-1}(w)}{H^{-1}(z)-H^{-1}(w)}\right)^{2}\partial_{z}\partial_{w}\left(F\circ
T\right)(S(z,w))\\\
&=\left(\frac{\tilde{T}_{1}(z,w)-\tilde{T_{2}}(z,w)}{H^{-1}(z)-H^{-1}(w)}\right)^{2}\left(\partial_{z}\partial_{w}F\right)\left(\tilde{T}(z,w)\right)\\\
&=\left(\frac{z-w}{H^{-1}(z)-H^{-1}(w)}\right)^{2}\left(\partial_{z}\partial_{w}F\right)\left(\tilde{T}(z,w)\right)\,\end{split}$

where we have taken (8.7) into account in the last identity. Being a Möbius
transformation, $H^{-1}$ has the property

$\left(\frac{z-w}{H^{-1}(z)-H^{-1}(w)}\right)^{2}=\frac{1}{\left(H^{-1}\right)^{\prime}(z)\left(H^{-1}\right)^{\prime}(w)}\,.$

This is exactly condition (8.4), but for the inverse $H^{-1}$ of $H$ instead
of $H$. Therefore, combining (8.8) and (8.9), leads to

$\displaystyle(z-w)^{2}\partial_{z}\partial_{w}\left(F\circ\tilde{T}\right)(z,w)$
$\displaystyle=(z-w)^{2}\left(\partial_{z}\partial_{w}F\right)\left(\tilde{T}(z,w)\right)$
$\displaystyle=\left(\tilde{T}_{1}(z,w)-\tilde{T}_{2}(z,w)\right)^{2}\left(\partial_{z}\partial_{w}F\right)\left(\tilde{T}(z,w)\right)$

again making use of (8.7) in the last step. Consequently, (5.2) holds with $T$
replaced by $\tilde{T}$.

(iii) In view of (ii) it remains to prove Theorem 5.4 under the additional
assumption that $T$ satisfies (8.6).

We therefore return to (5.2), where we choose

$F\in\mathcal{H}(G)\,,\quad F(z,w)=\frac{z}{(z-w)^{2}}\,.$

It is easily seen that equation (5.2) then takes the form

(8.10)
$\frac{2\left(\partial_{w}T_{1}(z,w)-\partial_{z}T_{1}(z,w)-1\right)}{w-z}+\partial_{z}\partial_{w}T_{1}(z,w)=0\,,$

With

$F\in\mathcal{H}(G)\,,\quad F(z,w)=\frac{z^{2}}{(z-w)^{2}}\,,$

equation (5.2) has the form

$2\partial_{w}T_{1}(z,w)\partial_{z}T_{1}(z,w)+2T_{1}(z,w)\left(\frac{2\left(\partial_{w}T_{1}(z,w)-\partial_{z}T_{1}(z,w)-1\right)}{w-z}+\partial_{z}\partial_{w}T_{1}(z,w)\right)=0\,.$

Taken together with (8.10), this implies

$(\partial_{w}T_{1}(z,w))(\partial_{z}T_{1}(z,w))=0\,,$

and either $T_{1}$ is independent of $z$ or independent of $w$. Going back to
(8.10), we see that either $T_{1}(z,w)=w+\gamma$ or $T_{1}(z,w)=z+\gamma$ for
some $\gamma\in\mathbb{C}$. In a similar way, we can deduce that either
$T_{2}(z,w)=w+\gamma^{\prime}$ or $T_{2}(z,w)=z+\gamma^{\prime}$ for some
$\gamma^{\prime}\in\mathbb{C}$. Finally, since $T_{1}(z,w)-T_{2}(z,w)=w-z$, we
see that

$\text{ either}\quad T(z,w)=(z+\gamma,w+\gamma)\quad\text{ or }\quad
T(z,w)=(w+\gamma,z+\gamma)$

for some $\gamma\in\mathbb{C}$. The proof is complete. $\square$

###### Remark 8.2

The preceding proof would become much simpler under the much stronger
assumption that (5.2) holds for all $F$ holomorphic merely in a neighborhood
of the origin in $\mathbb{C}^{2}$, since then one could choose the monomials
$z^{n}w^{m}$, $n,m=0,1,\ldots$ as “test functions” instead of functions
holomorphic on all of $\Omega$ resp. on $G=\mathop{{\rm
Conf}_{2}(\hat{\mathbb{C}})}$.

### Proof of Theorem 5.3

Let $\xi=(z_{0},w_{0})\in\partial\Omega$ and choose $r>0$ such that
$K_{r}(z_{0})\times K_{r}(w_{0})\subseteq U$. For fixed $w_{1}\in
K_{r}(w_{0})$ consider the holomorphic function

$F_{w_{1}}:\hat{\mathbb{C}}\setminus\\{1/w_{1}\\}\to\mathbb{C}\,,\quad
z\mapsto F(z,w_{1})\,.$

By assumption, $F_{w_{1}}$ is bounded on
$\hat{\mathbb{C}}\setminus\\{1/w_{1}\\}$, and thus by Liouville’s theorem
there is a constant $c(w_{1})\in\mathbb{C}$ such that $F_{w_{1}}(z)=c(w_{1})$
for all $z\in\hat{\mathbb{C}}\setminus\\{1/w_{1}\\}$. A similar argument shows
that for each $z_{1}\in K_{r}(z_{0})$ there is a constant $\tilde{c}(z_{1})$
such that $F_{z_{1}}(w):=F(z_{1},w)=\tilde{c}(z_{1})$ for all
$w\in\hat{\mathbb{C}}\setminus\\{1/z_{1}\\}$. Notice that for
$(z_{1},w_{1})\in U\cap\Omega$, we have
$c(w_{1})=F(z_{1},w_{1})=\tilde{c}(z_{1})$. This clearly implies that $F$ is
constant in $U\cap\Omega$, and hence on $\Omega$. $\square$

### Proof of Theorem 6.1

We only prove the non–trivial implication “(a) $\Longrightarrow$ (b)”. Note
again that $O$ and $\mathbb{C}^{2}\cap\Omega$ always have a nonempty
intersection, i.e. we may work in the standard chart and assume
$O\subseteq\mathbb{C}^{2}$. Let $T(z,w)=(T_{1}(z,w),T_{2}(z,w))$. Suppressing
the arguments, we compute

$\displaystyle T^{*}g_{\Omega}\big{|}_{(z,w)}$
$\displaystyle=\big{(}1-T_{1}T_{2}\big{)}^{-2}d(T_{1})\vee d(T_{2})$
$\displaystyle=\big{(}1-T_{1}T_{2}\big{)}^{-2}\big{(}\partial_{z}T_{1}dz+\partial_{w}T_{1}dw\big{)}\vee\big{(}\partial_{z}T_{2}dz+\partial_{w}T_{2}dw\big{)}$
$\displaystyle=\big{(}1-T_{1}T_{2}\big{)}^{-2}\Big{(}\big{(}(\partial_{z}T_{1})(\partial_{w}T_{2})+(\partial_{w}T_{1})(\partial_{z}T_{2})\big{)}dz\vee
dw$
$\displaystyle+(\partial_{z}T_{1})(\partial_{w}T_{1})dz^{2}+(\partial_{z}T_{2})(\partial_{w}T_{2})dw^{2}\Big{)}.$

Comparing coefficients with (6.1) yields first that
$(\partial_{z}T_{1})(\partial_{w}T_{1})=0=(\partial_{z}T_{2})(\partial_{w}T_{2})$.
That is, each component of $T$ only depends on $z$ or $w$. If $T$ depended on
$z$ only, then $T^{*}dw=0$, which is absurd in view of
$T^{*}g_{\Omega}=g_{\Omega}$. Consequently, we may assume
$T_{1}(z,w)=T_{1}(z)$ and ${T_{2}(z,w)=T_{2}(w)}$. Making this more precise,
we fix $(z_{0},w_{0})\in O$ and define the holomorphic auxiliary functions

$\displaystyle H_{1}\colon
O_{1}=\big{\\{}z\in\mathbb{C}\;\big{|}\;(z,w_{0})\in
O\big{\\}}\longrightarrow\hat{\mathbb{C}},\quad H_{1}(z)=T_{1}(z,w_{0})$ and
$\displaystyle H_{2}\colon
O_{2}=\big{\\{}w\in\mathbb{C}\;\big{|}\;(z_{0},w)\in
O\big{\\}}\longrightarrow\hat{\mathbb{C}},\quad H_{2}(z)=T_{2}(z_{0},w).$

As $O$ is open, the same is true for its nonempty projections $O_{1}$ and
$O_{2}$. By construction, we have
$H_{1}^{\prime}(z)=\partial_{z}T_{1}(z,w_{0})$ and
$H_{2}^{\prime}(w)=\partial_{w}T_{2}(z_{0},w)$ for every $(z,w)\in O_{1}\times
O_{2}$. Comparing the remaining coefficients this way yields

$H_{1}^{\prime}(z)H_{2}^{\prime}(w)=\frac{\big{(}1-H_{1}(z)H_{2}(w)\big{)}^{2}}{(1-zw)^{2}},\qquad(z,w)\in
O_{1}\times O_{2}.$

This is the $\Omega$–version of (5.3), hence Proposition 5.6 now implies
$H_{1},H_{2}\in{\sf Aut}(\hat{\mathbb{C}})$ and ${H_{1}(z)=1/H_{2}(1/z)}$ for
all $z\in\hat{\mathbb{C}}$, as both functions $H_{1},H_{2}$ being constant
would imply $T^{*}g_{\Omega}=0$. By construction, this means

$\big{(}T_{1}(z,w),T_{2}(z,w)\big{)}=\big{(}H_{1}(z),H_{2}(w)\big{)}=\big{(}H_{1}(z),1/H_{1}(1/w)\big{)}$

for all $(z,w)\in\Omega$, that is, $T\in\mathcal{M}$, see again (1.1).
$\square$

## Acknowledgements

The authors thank Daniela Kraus for useful discussions about Schwarzian
derivatives and for suggesting the alternative approach for proving
Proposition 5.6 which is outlined in Remark 8.1(b).

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