biholomorphically equivalent

Definition.

Let $U,V\\subset{\\mathbb{C}}^{n}$ . If there exists a one-to-one and ontoholomorphic 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 arebiholomorphic. 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 (http://planetmath.org/ConformallyEquivalent), since in one complexdimension, 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 conformallyequivalent to the unit disc. On the other hand,when $n>1$ then the open unit ball and open unit polydiscare not biholomorphically equivalent. In fact there does not exista proper (http://planetmath.org/ProperMap) holomorphic mapping from one to the other.

References

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

单词	BiholomorphicallyEquivalent
释义	biholomorphically equivalent Definition. Let $U,V\\subset{\\mathbb{C}}^{n}$ . If there exists a one-to-one and ontoholomorphic 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 arebiholomorphic. 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 (http://planetmath.org/ConformallyEquivalent), since in one complexdimension, 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 conformallyequivalent to the unit disc. On the other hand,when $n>1$ then the open unit ball and open unit polydiscare not biholomorphically equivalent. In fact there does not exista proper (http://planetmath.org/ProperMap) holomorphic mapping from one to the other. References 1 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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