domäninvarianta

Domäninvarianta (domain invariant) is a mathematical concept referring to quantities, properties, or constructions that are attached to a domain and remain unchanged under a specified class of domain mappings, most commonly biholomorphic or conformal maps. The idea is to capture features of the domain that do not depend on the particular presentation of the domain but only on its intrinsic structure within the chosen category.

In complex analysis and several complex variables, domain invariants are used to classify domains up to equivalence. Under biholomorphic (holomorphic and bijective with holomorphic inverse) maps, certain objects grow out of the domain in a way that preserves their essential character. Examples of such invariants include the automorphism group of the domain and metric structures like the Bergman metric and Bergman kernel, as well as the Carathéodory and Kobayashi (pseudo)metrics. These invariants provide tools to distinguish domains that are not biholomorphically equivalent and to study their geometric and function-theoretic properties.

Concrete instances help illustrate the concept. For example, the unit ball in complex n-space has a highly symmetric automorphism group and a explicit Bergman kernel, making its domain invariants particularly tractable. Different domains, such as the unit disk, unit polydisk, and annuli, exhibit distinct invariants, reflecting their differing geometric and function-theoretic structures. While invariants are powerful for classification, they are not always complete: a collection of invariants may separate many domains but may not uniquely determine the domain up to the chosen equivalence.

See also: conformal invariants, biholomorphic invariants, Bergman kernel, Kobayashi metric, Carathéodory metric.