biholomorphically equivalent


Let U,Vn. If there exists a one-to-one and onto holomorphic mapping ϕ:UV such that the inversePlanetmathPlanetmath ϕ-1 exists and is also holomorphic, then we say that U and V are biholomorphically equivalent or that they are biholomorphic. The mapping ϕ is called a biholomorphic mapping.

It is not an obvious fact, but if the source and target dimension are the same then every one-to-one holomorphic mapping is biholomorphic as a one-to-one holomorphic map has a nonvanishing jacobian.

When n=1 biholomorphic equivalence is often called conformal equivalence (, since in one complex dimension, the one-to-one holomorphic mappings are conformal mappingsMathworldPlanetmathPlanetmath.

Further if n=1 then there are plenty of conformal (biholomorhic) equivalences, since for example every simply connected domain ( other than the whole complex plane is conformally equivalent to the unit disc. On the other hand, when n>1 then the open unit ball and open unit polydisc are not biholomorphically equivalent. In fact there does not exist a proper ( holomorphic mapping from one to the other.


  • 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title biholomorphically equivalent
Canonical name BiholomorphicallyEquivalent
Date of creation 2013-03-22 14:29:47
Last modified on 2013-03-22 14:29:47
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 32H02
Synonym biholomorphic
Synonym biholomorphic equivalence
Defines biholomorphic mapping