mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Schwarz–Ahlfors–Pick theorem is an extension of the

Schwarz lemma In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm , \partial f , ^2 of a holomorphic map f:(X,g_X) \to (Y,g_Y) between Hermitian manifo ...

for

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

, such as the

Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with co ...

. The Schwarz–Pick lemma states that every

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

from the

unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...

''U'' to itself, or from the

upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

''H'' to itself, will not increase the Poincaré distance between points. The unit disk ''U'' with the Poincaré metric has negative

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

−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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

endowed with a

Hermitian metric In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as ...

\sigma

whose

is ≤ −1; let

f:U\rightarrow S

be a

. Then :

\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 Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

in 1973.

References

Hyperbolic geometry Riemann surfaces Theorems in complex analysis Theorems in differential geometry {{Riemannian-geometry-stub