The Schwarz Lemma, Disc Automorphisms, and the Hyperbolic Metric

A holomorphic map of the unit disc to itself that fixes the origin cannot expand — the Schwarz lemma — and this one inequality organizes the whole conformal geometry of the disc: its automorphism group, the invariant Poincaré (hyperbolic) metric, and the fact that holomorphic maps are metric contractions. This is geometric function theory in miniature, and it identifies the disc's symmetries with the isometries of the hyperbolic plane. It builds on complex-plane.md and the conformal picture.

The Schwarz lemma

Write $\mathbb{D}=\{|z|<1\}$.

Schwarz lemma. If $f:\mathbb{D}\to \mathbb{D}$ is holomorphic with $f(0)=0$, then $|f(z)|\le |z|$ for all $z$ and $|f^{\prime }(0)|\le 1$. If equality holds at a single interior point (or in the derivative bound), then $f(z)=e^{i\alpha }z$ is a rotation.

The proof is a one-line maximum principle applied to $f(z)/z$ (holomorphic after removing the zero at $0$). The force of the lemma is its rigidity: a self-map of the disc that so much as preserves the infinitesimal scale at the centre must be a rigid rotation.

Automorphisms of the disc

Which holomorphic bijections map $\mathbb{D}$ onto itself? The Blaschke factors $${\phi }_a(z)=\frac{z-a}{1-\bar{a}z},a\in \mathbb{D},$$ each send $\mathbb{D}\to \mathbb{D}$ bijectively with ${\phi }_a(a)=0$. Composing with a rotation gives them all:

Theorem (automorphism group of the disc). Every conformal automorphism of $\mathbb{D}$ has the form $f(z)=e^{i\alpha }\frac{z-a}{1-\bar{a}z}$. Thus $\mathrm{Aut}(\mathbb{D})\cong PSU(1,1)\cong PSL(2,\mathbb{R})$, a three-parameter group acting transitively on the disc.

The proof is pure Schwarz: pre-compose an arbitrary automorphism with a Blaschke factor to fix $0$, apply the lemma to it and its inverse, and conclude it is a rotation. These are the Möbius transformations preserving the unit circle; transferring by the Cayley map $z\mapsto \frac{z-i}{z+i}$ gives $\mathrm{Aut}$ of the upper half-plane as the real Möbius group $PSL(2,\mathbb{R})$, and the automorphisms of the sphere are all of $PSL(2,\mathbb{C})$.

The Schwarz–Pick lemma

Dropping the normalization $f(0)=0$ upgrades the estimate to an invariant one:

Schwarz–Pick lemma. If $f:\mathbb{D}\to \mathbb{D}$ is holomorphic, then for all $z,w$ $$\left|\frac{f(z)-f(w)}{1-\overline{f(w)}f(z)}\right|>\le \left|\frac{z-w}{1-\bar{w}z}\right|,>>\frac{|f^{\prime }(z)|}{1-|f(z){|}^2}\le \frac{1}{1-|z{|}^2},$$ with equality (anywhere) iff $f$ is an automorphism.

The quantity being contracted is Möbius-invariant — a hint that the right geometry on the disc is not Euclidean.

The Poincaré metric

Define the Poincaré (hyperbolic) metric on $\mathbb{D}$ by the line element $$ds=\frac{2|dz|}{1-|z{|}^2}.$$ The Schwarz–Pick lemma is exactly the statement that every holomorphic self-map of the disc is distance-non-increasing for this metric, and the automorphisms are its isometries. The metric has constant curvature $-1$; its geodesics are the diameters and the circular arcs meeting $\partial \mathbb{D}$ at right angles.

Identification with hyperbolic geometry. $(\mathbb{D},ds)$ is the Poincaré disc model of the hyperbolic plane; $\mathrm{Aut}(\mathbb{D})=PSL(2,\mathbb{R})$ is its orientation-preserving isometry group. Complex analysis and hyperbolic geometry meet here: holomorphic self-maps are the hyperbolic contractions, automorphisms the rigid motions.

This viewpoint powers fixed-point theorems (a self-map with an interior fixed point is a hyperbolic isometry or a strict contraction), underlies the hyperbolic metric on any hyperbolic Riemann surface via the uniformization theorem, and gives the cleanest route to Montel's theorem and the great Picard theorem (picard.md).

References

• Ahlfors, Complex Analysis, Ch. 6 §1 (Schwarz's lemma, automorphisms).
• Needham, Visual Complex Analysis, Ch. 3 (hyperbolic geometry of the disc).
• Beardon, The Geometry of Discrete Groups (the Poincaré metric and $PSL(2,\mathbb{R})$).