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 a bijective holomorphic function $f:Omega\; ightarrow\; mathbb\{C\}^n$ and a holomorphic inverse 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 and Ludwig 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)