In mathematics, the Schwarz-Ahlfors-Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. It states that the Poincaré metric is distance-decreasing on harmonic functions.
The theorem states that every holomorphic automorphism of the unit disk U, or the upper half plane H, with distances defined by the Poincaré metric, is a contraction mapping. That is, every such analytic mapping will not increase the distance between points. Stated more precisely:
Theorem: (Schwarz-Ahlfors-Pick) For all holomorphic automorphisms
,
one has
for points
and Poincaré distance
- ρ.
For any tangent vector T, the hyperbolic length of the tangent vector does not increase:
