Schwarz–Ahlfors–Pick theorem
Template:Short description In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model.
The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane H to itself, will not increase the Poincaré distance between points. The unit disk U with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces:
Theorem (Schwarz–Ahlfors–Pick). Let U be the unit disk with Poincaré metric <math>\rho</math>; let S be a Riemann surface endowed with a Hermitian metric <math>\sigma</math> whose Gaussian curvature is ≤ −1; let <math>f:U\rightarrow S</math> be a holomorphic function. Then
- <math>\sigma(f(z_1),f(z_2)) \leq \rho(z_1,z_2)</math>
for all <math>z_1,z_2 \in U.</math>
A generalization of this theorem was proved by Shing-Tung Yau in 1973.<ref>Template:Cite journal</ref>