Biholomorphically equivalent domains: Difference between revisions
(New page: ==Definition== Let <math>U,V \subset \mathbb{C}^n</math> be domains (open connected subset). We say that <math>U,V</math> are '''biholomorphically equivalent''' if there exists a [[h...) |
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Let <math>U,V \subset \mathbb{C}^n</math> be [[domain]]s (open connected subset). We say that <math>U,V</math> are '''biholomorphically equivalent''' if there exists a [[holomorphic function]] <math>\varphi:U \to V</math> with a holomorphic inverse <math>\varphi^{-1}:V \to U</math>. | Let <math>U,V \subset \mathbb{C}^n</math> be [[domain]]s (open connected subset). We say that <math>U,V</math> are '''biholomorphically equivalent''' if there exists a [[holomorphic function]] <math>\varphi:U \to V</math> with a holomorphic inverse <math>\varphi^{-1}:V \to U</math>. | ||
(By a holomorphic function from <math>U</math> to <math>V</math>, we mean a holomorphic function from <math>U</math> to <math>\C^n</math>, whose image lies completely inside <math>V</math>. In this case, we get a bijection from <math>U</math> to <math>V</math> that is holomorphic both ways. | (By a holomorphic function from <math>U</math> to <math>V</math>, we mean a holomorphic function from <math>U</math> to <math>\mathbb{C}^n</math>, whose image lies completely inside <math>V</math>. In this case, we get a bijection from <math>U</math> to <math>V</math> that is holomorphic both ways. | ||
Revision as of 18:26, 16 April 2008
Definition
Let be domains (open connected subset). We say that are biholomorphically equivalent if there exists a holomorphic function with a holomorphic inverse .
(By a holomorphic function from to , we mean a holomorphic function from to , whose image lies completely inside . In this case, we get a bijection from to that is holomorphic both ways.
