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
ρ
{\displaystyle \rho }
; let S be a Riemann surface endowed with a Hermitian metric
σ
{\displaystyle \sigma }
whose Gaussian curvature is ≤ −1; let
f
:
U
→
S
{\displaystyle f:U\rightarrow S}
be a holomorphic function. Then
σ
(
f
(
z
1
)
,
f
(
z
2
)
)
≤
ρ
(
z
1
,
z
2
)
{\displaystyle \sigma (f(z_{1}),f(z_{2}))\leq \rho (z_{1},z_{2})}
for all
z
1
,
z
2
∈
U
.
{\displaystyle z_{1},z_{2}\in U.}
A generalization of this theorem was proved by Shing-Tung Yau in 1973.