Schwarz–Ahlfors–Pick theorem

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 (Schwarz–Ahlfors–Pick). Let U be the unit disk with Poincaré metric {displaystyle rho } ; let S be a Riemann surface endowed with a Hermitian metric {displaystyle sigma } whose Gaussian curvature is ≤ −1; let {displaystyle f:Urightarrow S} be a holomorphic function. Then {displaystyle sigma (f(z_{1}),f(z_{2}))leq rho (z_{1},z_{2})} for all {displaystyle z_{1},z_{2}in U.} A generalization of this theorem was proved by Shing-Tung Yau in 1973.[1] References ^ Osserman, Robert (September 1999). "From Schwarz to Pick to Ahlfors and Beyond" (PDF). Notices of the AMS. 46 (8): 868–873. Categories: Hyperbolic geometryRiemann surfacesTheorems in complex analysisTheorems in differential geometry