Schwarz–Ahlfors–Pick theorem

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 (SchwarzAhlforsPick). Let U be the unit disk with Poincaré metric ; let S be a Riemann surface endowed with a Hermitian metric whose Gaussian curvature is  1; let be a holomorphic function. Then

for all

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


  1. Osserman, Robert (September 1999). "From Schwarz to Pick to Ahlfors and Beyond" (PDF). Notices of the AMS. 46 (8): 868–873.
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