Schwarz-Pick and the Hyperbolic Metric

The lemma that only worked at the centre

Guide 1 of this rung handed you the Schwarz lemma: if f is a holomorphic map of the unit disk into itself, and f(0) = 0, then |f(z)| <= |z| everywhere and |f'(0)| <= 1. It is a stunning amount of control from almost no hypotheses — but look closely and you will see a crutch. Everything is anchored at the single point 0. The conclusion compares |f(z)| to |z|, which is the Euclidean distance from 0; the derivative bound is taken at 0. Move the action anywhere else and the lemma falls silent.

That normalization f(0) = 0 always felt like a convenience rather than a law of nature, and it is. The fix is exactly the family of maps you spent guide 2 building: the disk automorphisms, the holomorphic bijections of the disk onto itself. Recall their shape — each is a Blaschke factor dressed with a rotation, phi_a(z) = (a - z)/(1 - a-bar z) for a point a inside the disk, possibly composed with a rotation z -> e^(i theta) z. The single property we need right now is that phi_a swaps a and 0: phi_a(a) = 0 and phi_a(0) = a.

Sliding the centre to where you need it

Here is the whole idea in one breath. Take any holomorphic f from the disk to the disk, and pick two points z_1 and z_2 inside. Set w_1 = f(z_1) and w_2 = f(z_2). Build a new map g by pre-composing with the automorphism that drags z_1 to 0, and post-composing with the automorphism that drags w_1 back from 0. Then g sends 0 to 0, it is still a holomorphic self-map of the disk, and so the plain Schwarz lemma applies to g without any apology.

1. Let phi_(z_1) be the automorphism with phi_(z_1)(z_1) = 0, and phi_(w_1) the one with phi_(w_1)(w_1) = 0.
2. Form g = phi_(w_1) o f o (phi_(z_1))-inverse. Each piece is a holomorphic self-map of the disk, so g is too.
3. Check g(0) = 0: feeding 0 into (phi_(z_1))-inverse gives z_1, then f gives w_1, then phi_(w_1) sends w_1 back to 0.
4. Apply the plain Schwarz lemma to g, then unwind the automorphisms back to f, z_1, z_2 — and read off the conclusion.

When you carry out that unwinding, the right combination to track is not |w_1 - w_2| over |z_1 - z_2| but the automorphism-flavoured ratio. Define the pseudo-hyperbolic distance d(z_1, z_2) = |(z_1 - z_2)/(1 - z_1-bar z_2)| — note the Blaschke shape lurking in it, the same denominator that made automorphisms work. Schwarz applied to g says precisely d(w_1, w_2) <= d(z_1, z_2). That is the Schwarz-Pick lemma: every holomorphic self-map of the disk shrinks (or preserves) this distance between any two points, not just distances measured from the centre.

From points to a ruler: the infinitesimal form

There is a second face of the lemma that is even more telling. Let z_2 slide toward z_1, so the two points merge. The pseudo-hyperbolic distance between them becomes infinitesimal, and dividing out by |z_1 - z_2| in the limit picks up the derivative. The result is a clean inequality at every single point: |f'(z)| / (1 - |f(z)|^2) <= 1 / (1 - |z|^2). Read it as a statement about how a tiny tangent vector is stretched: f never magnifies length when length is measured with the weight 1/(1 - |z|^2) instead of the ordinary Euclidean one.

Euclidean length of a step dz at z:   |dz|

hyperbolic (Poincare) length:         2 |dz| / (1 - |z|^2)

Schwarz-Pick, infinitesimal form:

      |f'(z)|              1
   --------------  <=  -----------
    1 - |f(z)|^2        1 - |z|^2

equality everywhere  <=>  f is a disk automorphism

That weight is the Poincare metric (also called the hyperbolic metric) on the disk: at the point z you measure a step dz not as |dz| but as 2|dz|/(1 - |z|^2). The factor 2 is a conventional normalization chosen to make the curvature come out to exactly -1; some books drop it, and nothing essential changes. The key feature is the blow-up of the weight as |z| approaches 1: near the rim of the disk, lengths are scaled up without bound. The boundary is infinitely far away. You can never reach it by travelling a finite hyperbolic distance.

When equality holds, and what isometries are

The Schwarz lemma had a sharp equality clause — |f(z)| = |z| at one interior point forces f to be a rotation — and Schwarz-Pick inherits an even more beautiful one. Equality d(w_1, w_2) = d(z_1, z_2) for a single pair of distinct points, or equality in the infinitesimal inequality at a single point, forces f to be a disk automorphism. In other words: among all holomorphic self-maps of the disk, the ones that preserve the hyperbolic ruler exactly, rather than shrinking it, are precisely the elements of the automorphism group. Everyone else is a genuine contraction.

This is what finally explains why guide 2's automorphisms deserved so much attention. They are not just an algebraic curiosity; they are the rigid motions of a geometry. An isometry is a map that preserves distance, and we have just discovered that the isometries of the disk equipped with the Poincare metric are exactly the disk automorphisms (plus the reflection z -> z-bar, if you allow orientation-reversing ones). The Blaschke factors slide points around, the rotations spin them, but none of them changes any hyperbolic distance — they are the rigid rotations and translations of a non-Euclidean world.

A whole geometry, and where it leads next

Step back and feel what just happened. We started with a one-point lemma about maps fixing the centre. By sliding automorphisms in front and behind, we promoted it to a statement at every pair of points, then to an infinitesimal statement about a weighted ruler, and that ruler turned out to define a complete, homogeneous, negatively curved geometry on the disk — the same disk you have mapped to and from a dozen times. The Schwarz lemma was never really about the centre. It was the first whisper of the fact that holomorphic self-maps of the disk are contractions of a hidden metric.

Two doors now open. Guide 4 walks all the way through the first: taking the Poincare disk seriously as a model of the hyperbolic plane — its geodesics are the diameters and the circular arcs meeting the boundary at right angles, its triangles have angle sums below pi, and Euclid's parallel postulate fails in the most generous possible way, with infinitely many parallels through a point. Everything you proved here about automorphisms-as-isometries is the engine that makes that model run.

The second door is more surprising. The whole spirit of this rung — a maximum-principle argument squeezing a function against the boundary, then transplanted by a clever conformal change of region — is exactly what powers the final guide. There you will meet the Phragmen-Lindelof principle and the three-lines theorem, where the same instinct (bound a holomorphic function on the edges of a strip, conclude control inside) carries you off the disk and onto unbounded regions. The disk taught you the grammar; guide 5 speaks it in a new dialect.