Biholomorphically equivalent domains - Revision history

Vipul: 3 revisions

2008-05-18T19:10:00Z

3 revisions

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Vipul at 20:56, 16 April 2008

2008-04-16T20:56:56Z

← Older revision		Revision as of 20:56, 16 April 2008
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	(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.		(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.

			When <math>n=1</math>, biholomorphically equivalent domains are also termed '''conformally equivalent'''.

			==Relation with other equivalence relations==

			===Weaker equivalence relations===

			* [[Diffeomorphic domains]]
			* [[Homeomorphic domains]]
			* [[Homotopy-equivalent domains]]

			==Facts==

			Any two simply connected open subsets of <math>\mathbb{C}</math> are biholomorphically equivalent. This is a consequence of the [[Riemann mapping theorem]].

Vipul at 18:26, 16 April 2008

2008-04-16T18:26:13Z

← Older revision		Revision as of 18:26, 16 April 2008
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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.

Vipul: New page: ==Definition== Let $U,V \subset \mathbb{C}^n$ be domains (open connected subset). We say that $U,V$ are '''biholomorphically equivalent''' if there exists a [[h...

2008-04-16T18:25:20Z

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...

New page

==Definition==

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.

Biholomorphically equivalent domains - Revision history

Vipul: 3 revisions

Vipul at 20:56, 16 April 2008

Vipul at 18:26, 16 April 2008

Vipul: New page: ==Definition== Let U,V \subset \mathbb{C}^n be domains (open connected subset). We say that U,V are '''biholomorphically equivalent''' if there exists a [[h...

Vipul: New page: ==Definition== Let $U,V \subset \mathbb{C}^n$ be domains (open connected subset). We say that $U,V$ are '''biholomorphically equivalent''' if there exists a [[h...