Schwarz–Ahlfors–Pick theorem explained

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

\rho ; let S be a Riemann surface endowed with a Hermitian metric \sigma whose Gaussian curvature is ≤ -1; let f:U → S be a holomorphic function. Then \sigma(f(z _{1),f(z} _{2))} \leq \rho(z _{1,z} _{2)} for all z _{1,z} _{2} \in U.

A generalization of this theorem was proved by Shing-Tung Yau in 1973.^[1]

Notes and References

1. From Schwarz to Pick to Ahlfors and Beyond. Robert. Osserman. Robert Osserman. Notices of the AMS. September 1999. 46. 8. 868–873.