In mathematics, the SchwarzâAhlforsâPick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the PoincarĂ© half-plane model.
The SchwarzâPick 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 PoincarĂ© distance between points. The unit disk U with the PoincarĂ© metric has negative Gaussian curvature â1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces:
Theorem ( Schwarzâ Ahlforsâ Pick). Let U be the unit disk with PoincarĂ© metric ; let S be a Riemann surface endowed with a Hermitian metric whose Gaussian curvature is ≤ −1; let be a holomorphic function. Then
for all
A generalization of this theorem was proved by Shing-Tung Yau in 1973. [1]
In mathematics, the SchwarzâAhlforsâPick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the PoincarĂ© half-plane model.
The SchwarzâPick 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 PoincarĂ© distance between points. The unit disk U with the PoincarĂ© metric has negative Gaussian curvature â1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces:
Theorem ( Schwarzâ Ahlforsâ Pick). Let U be the unit disk with PoincarĂ© metric ; let S be a Riemann surface endowed with a Hermitian metric whose Gaussian curvature is ≤ −1; let be a holomorphic function. Then
for all
A generalization of this theorem was proved by Shing-Tung Yau in 1973. [1]