biholomorphically equivalent

Definition.

Let $U,V\subset {ℂ}^n$. If there exists a one-to-one and onto holomorphic mapping $\varphi :U\to V$ such that the inverse ${\varphi }^{-1}$ exists and is also holomorphic, then we say that $U$ and $V$ are biholomorphically equivalent or that they are biholomorphic. The mapping $\varphi$ 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 (http://planetmath.org/ConformallyEquivalent), since in one complex dimension, the one-to-one holomorphic mappings are conformal mappings.

Further if $n=1$ then there are plenty of conformal (biholomorhic) equivalences, since for example every simply connected domain (http://planetmath.org/Domain2) 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 (http://planetmath.org/ProperMap) holomorphic mapping from one to the other.

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