Hyperbolic metric

hyperbolic measure

A metric in a domain of the complex plane with at least three boundary points that is invariant under automorphisms of this domain.

The hyperbolic metric in the disc is defined by the line element

where is the line element of Euclidean length. The introduction of the hyperbolic metric in leads to a model of Lobachevskii geometry. In this model the role of straight lines is played by Euclidean circles orthogonal to and lying in ; the circle plays the role of the improper point. Fractional-linear transformations of onto itself serve as the motions in it. The hyperbolic length of a curve lying inside is defined by the formula

The hyperbolic distance between two points and of is

The set of points of whose hyperbolic distance from , , does not exceed a given number , , i.e. the hyperbolic disc in with hyperbolic centre at and hyperbolic radius , is a Euclidean disc with centre other than if .

The hyperbolic area of a domain lying in is defined by the formula

The quantities , and are invariant with respect to fractional-linear transformations of onto itself.

The hyperbolic metric in any domain of the -plane with at least three boundary points is defined as the pre-image of the hyperbolic metric in under the conformal mapping of onto ; its line element is defined by the formula

A domain with at most two boundary points can no longer be conformally mapped onto a disc. The quantity

is called the density of the hyperbolic metric of . The hyperbolic metric of a domain does not depend on the selection of the mapping function or of its branch, and is completely determined by . The hyperbolic length of a curve located in is found by the formula

The hyperbolic distance between two points and in a domain is

where is any function conformally mapping onto . A hyperbolic circle in is, as in the case of the disc , a set of points in whose hyperbolic distance from a given point of (the hyperbolic centre) does not exceed a given positive number (the hyperbolic radius). If the domain is multiply connected, a hyperbolic circle in is usually a multiply-connected domain. The hyperbolic area of a domain lying in is found by the formula

The quantities , and are invariant under conformal mappings of (one of the main properties of the hyperbolic metric in ).

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)

Comments

Generalizations to higher-dimensional domains (mainly strongly pseudo-convex domains) are, e.g., the Carathéodory metric, the Kobayashi metric and the Bergman metric (for the latter see Bergman kernel function).

Let be a domain, and . Denote by the set of holomorphic mappings , the unit ball in . Then the (infinitesimal version of the) Carathéodory metric is

and the (infinitesimal version of the) Kobayashi distance is

Instead of sometimes other domains (e.g. the unit polydisc) are taken. (See [a2], [a3].)

One correspondingly defines for these metrics distance and area.

References

[a1]	L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973)
[a2]	S. Lang, "Introduction to complex hyperbolic spaces" , Springer (1987)
[a3]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)