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# biholomorphically equivalent

###### Definition.

Let $U,V\subset{\mathbb{C}}^{n}$. If there exists a one-to-one and onto holomorphic mapping $\phi\colon U\to V$ such that the inverse $\phi^{{-1}}$ exists and is also holomorphic, then we say that $U$ and $V$ are biholomorphically equivalent or that they are biholomorphic. The mapping $\phi$ 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 mappings.

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.

# References

- 1 Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

## Mathematics Subject Classification

32H02*no label found*

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