biholomorphically

Biholomorphically is an adverb used to describe a correspondence or mapping that is biholomorphic. In complex analysis, a map f between open subsets of the complex plane (or more generally between complex manifolds) is biholomorphic if it is a holomorphic bijection with a holomorphic inverse. When such a map exists, the two domains are said to be biholomorphically equivalent.

A biholomorphic map is, in particular, a conformal isomorphism: it preserves angles and the complex structure. In one complex variable, this requires f to be holomorphic with a holomorphic inverse, which by the holomorphic inverse function theorem implies f has everywhere nonzero derivative. The inverse map is also holomorphic, so the global condition ensures a full, structure-preserving equivalence between the domains.

Common examples and consequences include Möbius transformations, which are biholomorphic automorphisms of the extended complex plane, and standard conformal equivalences such as the Cayley transform that bijectively maps the unit disk to a half-plane. In several variables or on more general complex manifolds, biholomorphisms play the role of isomorphisms in the category of complex manifolds; they preserve many analytic and geometric properties, while not necessarily preserving global geometric quantities like size.

Biholomorphic equivalence provides a fundamental notion of sameness for domains in complex analysis: two domains are considered the same up to biholomorphic equivalence if there exists a biholomorphism between them. This concept underlies classifications and the study of conformal types, with results such as the Riemann mapping theorem illustrating the power of biholomorphic inferences in one complex variable.