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 ${\displaystyle f:U\rightarrow S}$ 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