Riemann mapping theorem

Statement

If $D\subsetneq\mathbb C$ is a nonempty simply connected domain, then there is a biholomorphism

f:D\longrightarrow\mathbb D, \qquad \mathbb D=\{z\in\mathbb C:|z|<1\}.

Normalization and uniqueness

Given $a\in D$ , there is a unique such map satisfying

f(a)=0,\qquad f'(a)>0.

Without normalization, two Riemann maps differ by a holomorphic automorphism of the disc.

Excluded cases and scope

The properness hypothesis excludes $D=\mathbb C$ , which cannot be biholomorphic to the disc by Liouville's theorem. Simple connectivity is essential: an annulus is not biholomorphic to a disc. The theorem classifies domains as Riemann surfaces, not their boundary regularity; extending $f$ continuously or smoothly to the boundary needs additional hypotheses.

Proof architecture

Standard proofs use a normal family of injective holomorphic maps and an extremal derivative argument. The result is a striking bridge from topology to conformal geometry.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter VI, §3.