biholomorphically equivalent

Definition.

Let $U, V \subset ℂ^{n}$ . If there exists a one-to-one and onto holomorphic mapping $ϕ : U \to V$ such that the inverse $ϕ^{- 1}$ 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 (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.

References

1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.