# biholomorphically equivalent

###### Definition.

Let $U,V\subset {\u2102}^{n}$. 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.

## References

- 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 |