Automorphisms of certain domains in the Riemann sphere

Let ${\mathbb{C}}$ the set of complex numbers (complex plane) and $\widehat{\mathbb{C}}$ the Riemann sphere, the compactification of $\mathbb{C}$ by adding the point $\infty$. We will use the following notation for certain subsets of the complex plane:

$$\mathrm{upper half plane,}\hspace{12pt} {\mathbb{H}} = \{z \in {\mathbb{C}}; \mathrm{Im}(z) > 0\}$$

$$\mathrm{lower half plane,}\hspace{12pt} {\mathbb{L}} = \{z \in {\mathbb{C}}; \mathrm{Im}(z) < 0\}$$

$$\mathrm{unit disc,}\hspace{12pt} {\mathbb{D}} = \{z \in {\mathbb{C}};\vert z\vert < 1\}$$

A holomorphic mapping with a (necessarily holomorphic) inverse is called biholomorphic. By an automorphism of an open set we understand a biholomorphic mapping of the set onto itself.

The following results, describing the group of automorphisms of some domains in the Riemann sphere are classical. But first we remark that when we talk of mappings on $\widehat{\mathbb{C}}$, or with values in the Riemann sphere, we do not make any difference regarding the point of $\infty$, and thus we talk of holomorphic functions to include meromorphic functions as well (think of holomorphic functions on Riemann surfaces).

Prop 1.1   The group of automorphisms of the Riemann sphere consists of the group of Möbius transformations; that is, mappings of the form $z \mapsto \frac{az+b}{cz+d}$, where $a$, $b$, $c$ and $d$ are complex numbers satisfying $ad -bc \neq 0$.

Clearly we can multiply all coefficients of a Möbius transformation by a non-zero number and the mapping does not change, so we can assume, if needed, that $ad -bc = 1$. Then the group of Möbius transformations is then identified with the quotient of $\mathrm{GL}(2,{\mathbb{C}})$ by the multiples of the identity, or with the quotient of $\mathrm{SL}(2,{\mathbb{C}})$ by $\pm Id$. We denote these quotients by $\mathrm{PGL}(2,{\mathbb{C}})$ and $\mathrm{PSL}(2,{\mathbb{C}})$, respectively. We say that a matrix $(\begin{smallmatrix}a & b c & d\end{smallmatrix})$ represents a Möbius transformation $M$ if $M$ is given by $M(z) = (az+b)/(cz+d)$.

Prop 1.2   The group of holomorphic automorphisms of the complex plane consists of the Möbius transformations of the form $z \mapsto az+b$, where $a\neq 0$.

Prop 1.3   The group of holomorphic automorphisms of the upper half plane consists of the Möbius transformations of the form $z \mapsto \frac{az+b}{cz+d}$, with real coefficients satisfying $ad -bc > 0$, or equivalently $ad -bc = 1$.

Prop 1.4   The group of holomorphic automorphisms of the unit disc consists of the Möbius transformations of the form $z \mapsto \lambda\frac{z-w}{1-\overline{w}z}$, where $\vert\lambda\vert = 1$ and $\vert w\vert < 1$ are complex numbers.

A more general statement is the following result of B. Maskit [32], which we will not use but quote here for general reference.

Theorem 1.5 (Maskit)   Let $D$ be a domain (connected, open set) of the plane and $G$ the group of automorphisms of $D$. Then there exists a one-to-one holomorphic mapping $f:D\to D'$ onto a domain $D'$ so that every element of $f G f^{-1}$ is a Möbius transformation.