Schwarz–Ahlfors–Pick theorem について

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Schwarz–Ahlfors–Pick theorem ：ウィキペディア英語版

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

\rho

; let ''S'' be a Riemann surface endowed with a Hermitian metric

\sigma

whose Gaussian curvature is ≤ −1; let

f:U\rightarrow S

\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.
==References==

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