Statement
Theorem. Every nonempty simply connected proper domain \(U\subsetneq\mathbb C\) is biholomorphic to the unit disc. After choosing \(a\in U\), there is a unique map \(f:U\to\mathbb D\) with \(f(a)=0\) and \(f'(a)>0\).
Why proper matters
Remark. The whole plane is simply connected but not biholomorphic to \(\mathbb D\), because a bounded entire biholomorphic image would contradict Liouville's theorem. The theorem is exactly about proper simply connected plane domains.
Extremal construction
Proof. Normalize maps \(f:U\to\mathbb D\) with \(f(a)=0\), and maximize \(|f'(a)|\). Normal families give a locally uniform limit of a maximizing sequence. One then proves the extremal map is injective and onto; otherwise a square-root or automorphism modification improves the derivative.
Uniqueness
Proposition. If two normalized maps satisfy \(f(a)=g(a)=0\) and positive real derivative at \(a\), then \(g\circ f^{-1}\) is an automorphism of \(\mathbb D\) fixing \(0\), hence a rotation. The derivative normalization forces the rotation to be the identity.
Boundary behavior
Remark. The theorem is an interior conformal classification. Extending the map continuously or smoothly to the boundary requires additional hypotheses such as Jordan boundary or smooth boundary.
Consequences
Example. Simply connected domains such as half-planes, strips, and slit planes are all conformally equivalent to the disc, although explicit maps may require exponentials, square roots, or more delicate constructions.
Statement and normalization
Theorem (Riemann mapping). Every nonempty simply connected proper domain \(\Omega\subsetneq\mathbb C\) is biholomorphic to the unit disc. Given \(z_0\in\Omega\), there is a unique such map \(f\) satisfying \[ f(z_0)=0,\qquad f'(z_0)>0. \]
Properness is necessary because every bounded entire function is constant, so \(\mathbb C\) cannot be biholomorphic to \(\mathbb D\). Simple connectedness is preserved by homeomorphisms.
Extremal construction
Let \(\mathcal F\) be the family of injective holomorphic maps \(f:\Omega\to\mathbb D\) with \(f(z_0)=0\) and \(f'(z_0)>0\). It is nonempty: choose \(a\notin\Omega\), take a holomorphic square root of \(z-a\), use its separation from one reflected value, and compose with a Möbius map into the disc.
Theorem (extremal map is onto). A map \(f\in\mathcal F\) maximizing \(f'(z_0)\) is surjective onto \(\mathbb D\).
Proof. Montel's theorem gives a locally uniformly convergent maximizing subsequence. Hurwitz preserves injectivity for the nonconstant limit, so an extremal exists. If \(a\in\mathbb D\setminus f(\Omega)\), then \(\phi_a\circ f\) is nonvanishing. Since \(\Omega\) is simply connected it has a holomorphic square root \(g\). Compose \(g\) with a disc automorphism sending \(g(z_0)\) to zero and normalize rotation. The resulting map lies in \(\mathcal F\) and Schwarz's lemma applied to the inverse construction shows its derivative at \(z_0\) is larger, contradicting extremality. Hence no point is omitted.
Uniqueness
Proof of normalized uniqueness. If \(f,g\) are normalized maps, then \(g\circ f^{-1}\) is a disc automorphism fixing zero. It is a rotation by Schwarz's lemma. Positivity of the derivatives forces that rotation to be the identity.
Boundary extension
The theorem itself concerns interiors. Carathéodory's theorem says a Riemann map extends to a homeomorphism of closures when the boundary is a Jordan curve. More generally, continuous extension is governed by local connectivity of the boundary, and prime ends provide the correct boundary object for arbitrary simply connected domains. Smooth boundaries yield stronger differentiability of the extension.
Consequences
Pulling the disc's Poisson kernel, Green function, and hyperbolic metric back to \(\Omega\) solves boundary and metric problems conformally. The map is rarely explicit; its force is classification and existence rather than a computational formula.