Biholomorphism

Def: Let $U, V \subseteq C$ be regions. We say that $U$ and $V$ are biholomorphic or biholomorphically equivalent, if there's a bijective analytic function $f : U \to V$ . We write it as $U \approx V$ . If we know the biholomorphism $f$ , then we can write it as $U \approx_{f} V$

Obs: The $\approx$ is an equivalence relation.

$D := B_{1} (0)$ , for the rest of the note.

We can see that $C ≉ D$ since if there was a a function $f : C \to D$ , analytic then $f$ would be constant.

We see that if $f : C \to C$ is analytic and injective then $f (z) = a z + b$ , then $f$ is surjective. Meaning that if $U \approx C$ then $U = C$ .

The goal from Riemann was to characterize all the $U \subset C$ such that $U \approx D$

Def: Let $U \subseteq C$ be a region. We say that $U$ satisfies the Riemann condition ( $U \in RC$ as a shorthand), if for every $f \in H (U)$ such that $Z (f) = \emptyset$ then $\exists g \in H (U)$ , that $g ² = f$

We see that $D \approx D$ , and we want to find all the biholomorphisms. We can use Schwarz Lemma to find them all.

Let $a \in D$ , and we define

φ_{a} (z) = \frac{z - a}{1 - \overset{―}{a} z}

is a Möbius transformation. with $φ_{a} (a) = 0$ , and $φ_{a} [\partial D] = \partial D$ . Since we can see that $φ_{a}$ has a pole at $1 / \overset{―}{a}$ , and $| 1 / \overset{―}{a} | > 1$ , we see that $φ_{a}$ is analytic on $B_{| 1 / \overset{―}{a} |} (0)$ which contains $D$ . Thus $φ_{a} : D \to D$ is a biholomorphism.

Additionally, $φ_{a}^{- 1} = φ_{- a}$

We need to look at special properties of $φ_{a}$ . $$\varphi_a'(z) = \frac{(1-\overline az) +(z-a)\overline a}{(1-\overline a z)}$$
and we see that$$\varphi_a'(a) = \frac{1}{1-|a|²}, \qquad\quad \varphi_a'(0) =1-|a|²$$
Lemma: Let $f : D \to D$ be analytic, and $a \in D$ , we call $α = f (a)$ , then $$|f'(\alpha)| \le \frac{1-|\alpha|²}{1-|a|^2}$$Furthermore, if $| f^{'} (α) | = \frac{1 - | α | ²}{1 - | a |^{2}}$ , then there's $c \in \partial D$ such that $$f(z) = \varphi_{-a}(c\varphi_a(z))$$in particular, $f$ is injective.

Th: $D \approx_{f} D$ , and $a \in D$ , $f (a) = 0$ , then there's $c \in \partial D$ such that $f = c φ_{a}$

Riemann Theorem

Let $U \subset C$ be a region. If $U$ satisfies the Riemann condition, for every $a \in U$ , there exists a unique biholomorphism $f : U \to D$ , such $f (a) = 0$ and $f^{'} (a) > 0$ .

The proof has 3 main parts:

Existence of biholomorphism(the hardest), requires the use of Normal families
Given that biholomorphism construct one that satisfies the additional constraints
Uniqueness of the biholomorphism