The Riemann Mapping Theorem — Formal Statement and Proof Outline

The Riemann mapping theorem establishes a conformal equivalence between any proper simply connected domain and the unit disk, providing the most fundamental uniformization result in one complex variable.

Formal Statement

Theorem7.5Riemann mapping theorem (precise)

Let $D \subsetneq \mathbb{C}$ be a simply connected domain. Then there exists a biholomorphic map $\varphi: D \to \mathbb{D}$ . For any fixed $z_0 \in D$ , the map is unique subject to $\varphi(z_0) = 0$ and $\varphi'(z_0) > 0$ .

Proof

Existence of an injective map into $\mathbb{D}$ .

Since $D \neq \mathbb{C}$ , there exists $a \notin D$ . Since $D$ is simply connected, $z - a$ has a holomorphic square root on $D$ : there exists $h: D \to \mathbb{C}$ holomorphic with $h(z)^2 = z - a$ . Since $h$ is injective (if $h(z_1) = h(z_2)$ then $z_1 - a = z_2 - a$ ), and $h(D) \cap (-h(D)) = \emptyset$ (otherwise $h(z_1) = -h(z_2)$ implies $z_1 = z_2$ and $h(z_1) = 0$ , contradicting $a \notin D$ ).

The set $h(D)$ is open, so pick $w_0 \in h(D)$ and $\varepsilon > 0$ with $D(w_0, \varepsilon) \subset h(D)$ . Then $D(-w_0, \varepsilon) \cap h(D) = \emptyset$ (since $h(D) \cap (-h(D)) = \emptyset$ ). The map $g(z) = \frac{1}{h(z) + w_0}$ is bounded on $D$ (since $|h(z) + w_0| \geq \varepsilon/2$ away from $-w_0$ ), and after rescaling, we get an injective holomorphic $\psi: D \to \mathbb{D}$ .

Maximization argument.

Define $\mathcal{F} = \{f: D \to \mathbb{D} \mid f \text{ holomorphic, injective, } f(z_0) = 0\}$ . We showed $\mathcal{F} \neq \emptyset$ . Let $\lambda = \sup_{f \in \mathcal{F}} |f'(z_0)|$ . By the Schwarz lemma applied to the inverse, $\lambda < \infty$ .

Choose $f_n \in \mathcal{F}$ with $|f_n'(z_0)| \to \lambda$ . By Montel's theorem (the family is bounded by $1$ ), extract a subsequence $f_{n_k} \to \varphi$ uniformly on compacts.

By Hurwitz's theorem, $\varphi$ is either injective or constant. Since $|\varphi'(z_0)| = \lambda > 0$ , $\varphi$ is injective.

Surjectivity.

Suppose $w_0 \in \mathbb{D} \setminus \varphi(D)$ . Consider $\psi = \varphi_{w_0} \circ \varphi$ where $\varphi_{w_0}(w) = (w-w_0)/(1-\bar{w}_0 w)$ . Then $\psi: D \to \mathbb{D} \setminus \{0\}$ is injective with $\psi(z_0) = -w_0/(1-|w_0|^2) \cdot |w_0|^2/(-w_0) \neq 0$ .

Taking a square root $\sigma(z) = \sqrt{\psi(z)}$ (possible since $D$ is simply connected and $\psi \neq 0$ ), then normalizing to get $\tilde{\sigma} \in \mathcal{F}$ with $|\tilde{\sigma}'(z_0)| > |\varphi'(z_0)| = \lambda$ . This contradicts the maximality of $\lambda$ .

Therefore $\varphi$ is surjective, and $\varphi: D \to \mathbb{D}$ is biholomorphic. Normalizing by a rotation ensures $\varphi'(z_0) > 0$ , and uniqueness follows from the Schwarz lemma. $\blacksquare$

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Applications

ExampleSolving Dirichlet problems

The Riemann mapping theorem reduces the Dirichlet problem on any simply connected domain to the Dirichlet problem on the disk (which is solved by the Poisson integral formula). If $u$ is harmonic on $D$ with boundary values $g$ , and $\varphi: D \to \mathbb{D}$ is the Riemann map, then $u = (g \circ \varphi^{-1}) \circ P$ where $P$ is the Poisson kernel.

RemarkCaratheodory's extension theorem

If $\partial D$ is a Jordan curve, then $\varphi$ extends to a homeomorphism $\overline{D} \to \overline{\mathbb{D}}$ . This boundary correspondence is crucial for applications and connects the Riemann mapping theorem to potential theory.