Riemann Mapping Theorem

The Riemann mapping theorem is one of the deepest results in complex analysis: every simply connected domain (other than $\mathbb{C}$ itself) is conformally equivalent to the unit disk.

Statement

Theorem7.3Riemann mapping theorem

Let $D \subsetneq \mathbb{C}$ be a simply connected domain with at least two boundary points. Then there exists a biholomorphic (conformal and bijective) map $f: D \to \mathbb{D}$ . Moreover, for any $z_0 \in D$ , there is a unique such map with $f(z_0) = 0$ and $f'(z_0) > 0$ .

RemarkSignificance

The Riemann mapping theorem asserts that the conformal geometry of simply connected planar domains is trivial: they are all the same (conformally speaking). The theorem is non-constructive — it guarantees existence but does not provide an explicit formula. Finding explicit conformal maps is the subject of Schwarz-Christoffel theory and other constructive methods.

Proof Strategy (Outline)

Definition7.4Normal family

A family $\mathcal{F}$ of holomorphic functions on a domain $D$ is normal if every sequence in $\mathcal{F}$ has a subsequence converging uniformly on compact subsets of $D$ .

Theorem7.4Montel's theorem

A uniformly bounded family of holomorphic functions on a domain is normal. That is, if $\{f_n\} \subset \text{Hol}(D)$ with $|f_n(z)| \leq M$ for all $n$ and $z \in D$ , then there exists a subsequence converging uniformly on compact subsets.

RemarkProof outline of Riemann mapping theorem

Define the family $\mathcal{F} = \{f: D \to \mathbb{D} \mid f \text{ injective}, f(z_0) = 0\}$ and show $\mathcal{F} \neq \emptyset$ .
Maximize $|f'(z_0)|$ : Let $\lambda = \sup\{|f'(z_0)| : f \in \mathcal{F}\}$ . By Montel's theorem, there exists $f_n \in \mathcal{F}$ with $|f_n'(z_0)| \to \lambda$ and a convergent subsequence $f_{n_k} \to f$ .
Show $f \in \mathcal{F}$ : By Hurwitz's theorem, the limit $f$ is injective (since all $f_n$ are injective and $f$ is non-constant).
Show $f$ is surjective: If $w_0 \in \mathbb{D} \setminus f(D)$ , construct $g \in \mathcal{F}$ with $|g'(z_0)| > |f'(z_0)| = \lambda$ , a contradiction.
Uniqueness: If $f_1, f_2$ are two solutions with $f_j(z_0) = 0$ , $f_j'(z_0) > 0$ , then $f_2 \circ f_1^{-1}$ is an automorphism of $\mathbb{D}$ fixing $0$ with positive derivative, hence the identity.

Consequences

ExampleExplicit Riemann maps

While the general theorem is non-constructive, explicit maps are known for many domains:

Half-plane $\mathbb{H}$ : $f(z) = \frac{z-i}{z+i}$ (Cayley map).
Vertical strip $\{-\pi/2 < \text{Re}(z) < \pi/2\}$ : $f(z) = \sin z$ maps to $\mathbb{C} \setminus ((-\infty,-1] \cup [1,\infty))$ .
Sector $\{0 < \arg z < \alpha\}$ : $f(z) = z^{\pi/\alpha}$ followed by the Cayley map.
Disk minus a slit: $f$ involves elliptic functions.

RemarkBoundary behavior

The Riemann map extends continuously to the boundary when $\partial D$ is a Jordan curve (Caratheodory's theorem). More precisely, $f$ extends to a homeomorphism $\overline{D} \to \overline{\mathbb{D}}$ . For smooth boundaries, the boundary extension inherits regularity.

RemarkHigher dimensions

The Riemann mapping theorem fails in higher dimensions. In $\mathbb{C}^n$ ( $n \geq 2$ ), the unit ball and the unit polydisk are not biholomorphically equivalent (Poincare, 1907). Conformal geometry in higher dimensions is far richer.