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The Poincare distance has the formula $2 artanh ∣ φ_{z} (w) ∣$ and is disc-automorphism invariant

Statement

For $z, w \in D$ , the Poincare distance on the unit disc satisfies

$d_{D} (z, w) = 2 artanh ∣ φ_{z} (w) ∣,$

where $φ_{z}$ is the Blaschke factor carrying $z$ to $0$ . Moreover every disc automorphism preserves this distance.

Facts & Assumptions

Given: The Poincare metric and distance on $D$ .

[F1]

The Poincare length and distance are defined by integrating $2 ∣ d z ∣ / (1 - ∣ z ∣^{2})$ along piecewise $C^{1}$ curves (The Poincare metric and distance on the unit disc).

[F2]

Every Blaschke factor is an automorphism of $D$ (Blaschke factors are automorphisms of the disc).

[F3]

Every automorphism of $D$ is a rotated Blaschke factor (Every automorphism of the disc is a rotated Blaschke factor).

Proof

technique · direct

1.1F1F2givenalgebra

For a Blaschke factor $φ_{a}$ , a direct differentiation gives $φ_{a}^{'} (z) = - (1 - ∣ a ∣^{2}) / (1 - \overline{a} z)^{2}$ and $1 - ∣ φ_{a} (z) ∣^{2} = (1 - ∣ a ∣^{2}) (1 - ∣ z ∣^{2}) / ∣ 1 - \overline{a} z ∣^{2}$ . Therefore $2 ∣ φ_{a}^{'} (z) ∣ / (1 - ∣ φ_{a} (z) ∣^{2}) = 2 / (1 - ∣ z ∣^{2})$ , so $φ_{a}$ preserves Poincare length of every piecewise $C^{1}$ curve.

2.1F1F2F3step 1.1algebra

Since length is preserved under $φ_{a}$ , taking infima in [F1] gives $d_{D} (φ_{a} (z), φ_{a} (w)) = d_{D} (z, w)$ . By [F3], every disc automorphism is a composition of a Blaschke factor and a rotation, and rotations satisfy the same identity, so every disc automorphism preserves $d_{D}$ .

3.1F1step 2.1algebra

By step 2.1, $d_{D} (z, w) = d_{D} (0, φ_{z} (w))$ . Write $r = ∣ φ_{z} (w) ∣$ . If $r = 0$ , then $φ_{z} (w) = 0$ , so $z = w$ and both sides are $0 = 2 artanh 0$ . If $r > 0$ , the radial segment $γ (t) = t φ_{z} (w) / r$ from $0$ to $φ_{z} (w)$ has Poincare length $\int_{0}^{r} 2 d t / (1 - t^{2}) = 2 artanh r$ , so $d_{D} (0, φ_{z} (w)) \leq 2 artanh r$ .

4.1F1step 3.1algebra

For any piecewise $C^{1}$ curve $γ$ from $0$ to $φ_{z} (w)$ , one has $∣ (∣ γ ∣)^{'} ∣ \leq ∣ γ^{'} ∣$ , hence $ℓ_{D} (γ) \geq \int 2 ∣ (∣ γ ∣)^{'} ∣ / (1 - ∣ γ ∣^{2}) d t \geq \int_{0}^{r} 2 d s / (1 - s^{2}) = 2 artanh r$ . Taking the infimum over all such curves gives the reverse inequality.

5.1step 2.1step 3.1step 4.1∎

Combining steps 3.1 and 4.1 yields $d_{D} (z, w) = 2 artanh ∣ φ_{z} (w) ∣$ , and step 2.1 gives automorphism invariance.

Depends on

Used by

Poincaré metric on a hyperbolic Riemann surface Definition
Hyperbolic distances and geodesics in disc and half-plane Example
Deck transformations preserve the hyperbolic metric Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiri Lebl, Guide to Cultivating Complex Analysis, §3.5 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §2 (standard reference, not scraped)

The Poincare distance has the formula 2artanh⁡∣φz(w)∣2artanh∣φz​(w)∣ and is disc-automorphism invariant