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biholomorphically equivalent
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(Definition)
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Definition 1 Let
 . If there exists a one-to-one and onto holomorphic mapping
 such that the inverse  exists and is also holomorphic, then we say that  and  are biholomorphically equivalent or that they are biholomorphic. The mapping  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  biholomorphic equivalence is often called conformal equivalence, since in one complex dimension, the one-to-one holomorphic mappings are conformal mappings.
Further if  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  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.
- 1
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"biholomorphically equivalent" is owned by jirka. [ full author list (2) ]
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(view preamble | get metadata)
| Other names: |
biholomorphic, biholomorphic equivalence |
| Also defines: |
biholomorphic mapping |
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Cross-references: polydisc, unit, open, unit disc, conformally equivalent, complex plane, simply connected, equivalences, conformal, conformal mappings, complex, Jacobian, dimension, source, obvious, inverse, mapping, holomorphic, onto, one-to-one
There are 8 references to this entry.
This is version 4 of biholomorphically equivalent, born on 2004-07-27, modified 2005-11-03.
Object id is 6032, canonical name is BiholomorphicallyEquivalent.
Accessed 4684 times total.
Classification:
| AMS MSC: | 32H02 (Several complex variables and analytic spaces :: Holomorphic mappings and correspondences :: Holomorphic mappings, embeddings and related questions) |
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Pending Errata and Addenda
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