- Fatou-Bieberbach domain
In
mathematics , a Fatou-Bieberbach domain comprises a proper subdomain of mathbb{C}^n which is biholomorphically equivalent to mathbb{C}^n; i.e. one calls an open Omega subset mathbb{C}^n ; (Omega eq mathbb{C}^n) a Fatou-Bieberbach domain if there exists abijective holomorphic function f:Omega ightarrow mathbb{C}^n and a holomorphicinverse function f^{-1}:mathbb{C}^n ightarrow Omega.History
As a consequence of the
Riemann mapping theorem , there are no Fatou-Bieberbach domains in the case of n = 1.Pierre Fatou andLudwig Bieberbach first explored such domains in higher dimensions in the 1920s, hence the name given to them later. Since the 1980s, Fatou-Bieberbach domains have again become the subject of mathematical research.References
* Fatou, Pierre: "Sur les fonctions méromorphs de deux variables. Sur certains fonctions uniformes de deux variables." "C.R." Paris 175 (1922)
* Bieberbach, Ludwig: "Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlichte volumtreue Abbildung des mathcal{R}_4 auf einen Teil seiner selbst vermitteln". Preussische Akademie der Wissenschaften. "Sitzungsberichte" (1933)
* Rosay, J.-P. and Rudin, W: "Holomorphic maps from mathbb{C}^n to mathbb{C}^n". "Trans. A.M.S." 310 (1988)
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