Normal Families and the Riemann Mapping Theorem

The Riemann mapping theorem was stated but not proved: why should a conformal map onto the disc exist? The missing ingredient is compactness in a space of holomorphic functions — the theory of normal families. This page develops Montel's theorem and its companions and uses them to prove the mapping theorem by an extremal argument. It builds on schwarz-lemma.md, harmonic-functions.md, and conformal-mapping.md.

Normal families

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

Normality is compactness for the topology of local uniform convergence — the setting in which limits of holomorphic functions stay holomorphic (a Morera/Weierstrass fact). It is the tool for extracting a solution to an extremal problem as a limit of near-solutions.

Montel's theorem

The criterion for normality is a uniform bound, via the holomorphic strengthening of Arzelà–Ascoli:

Arzelà–Ascoli. A family of continuous functions is precompact (in local uniform convergence) iff it is locally bounded and equicontinuous.

For holomorphic functions the Cauchy estimates turn a bound on $f$ into a bound on $f^{\prime }$, so local boundedness alone forces equicontinuity:

Montel's theorem. A locally uniformly bounded family of holomorphic functions on $U$ is normal.

This is the workhorse. A far stronger version — a family omitting two fixed values is normal — lies deeper and yields the great Picard theorem (picard.md).

Vitali–Porter and Hurwitz

Two companions sharpen how the limits behave:

Vitali–Porter theorem. A locally bounded sequence of holomorphic functions that converges pointwise on a set with a limit point converges locally uniformly on all of $U$ (the identity theorem made quantitative).

Hurwitz's theorem. If $f_n\to f$ locally uniformly and none of the $f_n$ vanishes on $U$, then either $f\equiv 0$ or $f$ has no zeros. In particular a locally uniform limit of injective holomorphic functions is injective or constant.

Hurwitz (via the argument principle) is what keeps the extremal map below injective in the limit.

Proof of the Riemann mapping theorem

Let $U⊊\mathbb{C}$ be simply connected and $z_0\in U$. Consider $$\mathcal{F}=\{f:U\to \mathbb{D}\text{holomorphic, injective},f(z_0)=0,f^{\prime }(z_0)>0\}.$$

1. $\mathcal{F}$ is non-empty. Simple connectivity provides a holomorphic branch of a square root or logarithm that, composed with a Möbius map, lands $U$ inside the disc — a normalized injection exists.
2. Extremal problem. $\mathcal{F}$ is locally bounded (values in $\mathbb{D}$), so normal by Montel. The functional $f\mapsto f^{\prime }(z_0)$ is bounded above on $\mathcal{F}$; take a sequence approaching the supremum and, by normality, a locally uniform limit $F$. By Hurwitz $F$ is injective (not constant, since $F^{\prime }(z_0)>0$), and $F:U\to \mathbb{D}$.
3. $F$ is onto. If $F$ missed a point $w\in \mathbb{D}$, a Blaschke-and-square-root construction (schwarz-lemma.md) would produce a member of $\mathcal{F}$ with a larger derivative at $z_0$ — contradicting maximality.

Riemann mapping theorem. Every simply connected $U⊊\mathbb{C}$ is conformally equivalent to $\mathbb{D}$, by a map unique once $F(z_0)=0$ and $F^{\prime }(z_0)>0$ are fixed.

The maximal derivative singles out the map; normality supplies its existence.

Boundary behaviour

The interior map extends to the boundary when the boundary is reasonable:

Carathéodory's theorem. If $\partial U$ is a Jordan curve, the Riemann map extends to a homeomorphism of the closures $\overline{U}\to \overline{\mathbb{D}}$.

This makes the theorem usable for the Dirichlet problem (transport Poisson's formula through $F$) and for explicit conformal maps in physics. The global generalization to arbitrary simply connected Riemann surfaces is the uniformization theorem.

References

• Ahlfors, Complex Analysis, Ch. 5 §5 (normal families) and Ch. 6 §1 (RMT).
• Stein & Shakarchi, Complex Analysis, Ch. 8 (Montel, the mapping theorem).
• Conway, Functions of One Complex Variable, Ch. VII §§2–4.