Riemann mapping theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Any simply connected domain in $\mathbb {C}$ (i.e. any open, connected and simply connected subset of $\mathbb {C}$) that is not the whole of $\mathbb {C}$, is bihomolorphically equivalent to the open unit disk. A biholomorphic mapping from the simply connected domain to the open unit disk is termed a Riemann mapping.

Corollaries

Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to $\mathbb {C}$).

Facts used

• Maximum modulus principle
• Open mapping theorem

Proof

Proof outline

The standard proof of the Riemann mapping theorem has the following three steps:

1. Obtain a conformal mapping from the simply connected domain into the open unit disk (we do this by first obtaining a map into the upper half plane, and then using a fractional linear transformation to pass from the upper half plane to the open unit disk)
2. Maximize among all such mappings with respect to certain criteria. Prove that a maximal mapping must exist.
3. Prove that this maximal conformal mapping is also surjective, and hence, must be a biholomorphic mapping