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 $ E = \{ {z } : {| z | < 1 } \} $ is defined by the line element

$$ d \sigma _ {z} = \ \frac{| dz | }{1 - | z | ^ {2} } , $$

where $ | dz | $ is the line element of Euclidean length. The introduction of the hyperbolic metric in $ E $ leads to a model of Lobachevskii geometry. In this model the role of straight lines is played by Euclidean circles orthogonal to $ | z | = 1 $ and lying in $ E $; the circle $ | z | = 1 $ plays the role of the improper point. Fractional-linear transformations of $ E $ onto itself serve as the motions in it. The hyperbolic length of a curve $ L $ lying inside $ E $ is defined by the formula

$$ \mu _ {E} ( L) = \ \int\limits _ { L } \frac{| dz | }{1 - | z | ^ {2} } . $$

The hyperbolic distance between two points $ z _ {1} $ and $ z _ {2} $ of $ E $ is

$$ r _ {E} ( z _ {1} , z _ {2} ) = \ { \frac{1}{2} } \mathop{\rm ln} \ \frac{| 1 - z _ {1} \overline{ {z _ {2} }}\; | + | z _ {1} - z _ {2} | }{| 1 - z _ {1} \overline{ {z _ {2} }}\; | - | z _ {1} - z _ {2} | } . $$

The set of points of $ E $ whose hyperbolic distance from $ z _ {0} $, $ z _ {0} \in E $, does not exceed a given number $ R $, $ R > 0 $, i.e. the hyperbolic disc in $ E $ with hyperbolic centre at $ z _ {0} $ and hyperbolic radius $ R $, is a Euclidean disc with centre other than $ z _ {0} $ if $ z _ {0} \neq 0 $.

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

$$ \Delta _ {E} ( B) = \ {\int\limits \int\limits } _ { B } \frac{dx dy }{( 1 - | z | ^ {2} ) ^ {2} } ,\ \ z = x + iy. $$

The quantities $ \mu _ {E} ( L) $, $ r _ {E} ( z _ {1} , z _ {2} ) $ and $ \Delta _ {E} ( B) $ are invariant with respect to fractional-linear transformations of $ E $ onto itself.

The hyperbolic metric in any domain $ D $ of the $ z $- plane with at least three boundary points is defined as the pre-image of the hyperbolic metric in $ E $ under the conformal mapping $ \zeta = \zeta ( z) $ of $ D $ onto $ E $; its line element is defined by the formula

$$ d \sigma _ {z} = \ \frac{| \zeta ^ \prime ( z) | | dz | }{1 - | \zeta ( z) | ^ {2} } . $$

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

$$ \rho _ {D} ( z) = \ \frac{| \zeta ^ \prime ( z) | }{1 - | \zeta ( z) | ^ {2} } $$

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

$$ \mu _ {D} ( L) = \ \int\limits _ { L } \rho _ {D} ( z) | dz | . $$

The hyperbolic distance between two points $ z _ {1} $ and $ z _ {2} $ in a domain $ D $ is

$$ r _ {D} ( z _ {1} , z _ {2} ) = { \frac{1}{2} } \mathop{\rm ln} \ \frac{| 1 - \zeta ( z _ {1} ) \overline{ {\zeta ( z _ {2} ) }}\; | + | \zeta ( z _ {1} ) - \zeta ( z _ {2} ) | }{| 1 - \zeta ( z _ {1} ) \overline{ {\zeta ( z _ {2} ) }}\; | - | \zeta ( z _ {1} ) - \zeta ( z _ {2} ) | } , $$

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

$$ \Delta _ {D} ( B) = \ {\int\limits \int\limits } _ { B } \rho _ {D} ^ {2} ( z) dx dy. $$

The quantities $ \mu _ {D} ( L) $, $ r _ {D} ( z _ {1} , z _ {2} ) $ and $ \Delta _ {D} ( B) $ are invariant under conformal mappings of $ D $( one of the main properties of the hyperbolic metric in $ D $).

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 $ \Omega \subset \mathbf C ^ {n} $ be a domain, $ z \in \Omega $ and $ \xi \in \mathbf C ^ {n} $. Denote by $ B ( \Omega ) $ the set of holomorphic mappings $ f : \Omega \rightarrow B $, $ B $ the unit ball in $ \mathbf C ^ {n} $. Then the (infinitesimal version of the) Carathéodory metric is

$$ F _ {C} ( z, \xi ) = \ \sup _ {\begin{array}{c} f \in B ( \Omega ) \\ f( z) = 0 \end{array} } \ \left | \sum _ { j= } 1 ^ { n } \frac{\partial f }{\partial z _ {j} } ( z) \cdot \xi _ {j} \right | , $$

and the (infinitesimal version of the) Kobayashi distance is

$$ F _ {K} ( z, \xi ) = $$

$$ = \ \inf \{ \alpha : \alpha > 0 \textrm{ and there is a holomorphic mapping } \ $$

$$ {} f : B \rightarrow \Omega \textrm{ with } f( z)= 0, ( f ^ { \prime } ( 0) ) ( 1 , 0 \dots 0) = \xi / \alpha \} . $$

Instead of $ B $ 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)