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 ; let S be a Riemann surface endowed with a Hermitian metric whose Gaussian curvature is ≤ −1; let be a holomorphic function. Then
- Robert Osserman (September 1999). "From Schwarz to Pick to Ahlfors and Beyond" (PDF). Notices of the AMS.