A Fatou-Bieberbach domain intersecting the plane in the unit disk (Q2846845)

A Fatou-Bieberbach domain \(\Omega\) is a proper subdomain \(\Omega \subsetneq \mathbb{C}^n\) which is biholomorphic to \(\mathbb{C}^n\). The existence of such domains for \(n \geq 2\) has been established by the works of Poincaré, Fatou and Bieberbach. A more recent treatment can be found in the article of \textit{J.-P. Rosay} and \textit{W. Rudin} [Trans. Am. Math. Soc. 310, No. 1, 47--86 (1988; Zbl 0708.58003)].NEWLINENEWLINEFatou-Bieberbach domains are of great importance in complex analysis, in particular because they can be used to construct proper holomorphic embeddings which is in general a difficult task. For example, a proper holomorphic embedding of the unit disc \(\mathbb{D}\) into \(\mathbb{C}^2\) can be constructed as follows: Take a Runge Fatou-Bieberbach domain \(\Omega \subset \mathbb{C}^2\). Then there exists an affine line \(L \subset \mathbb{C}^2\) such that \(L \cap \Omega \neq L\). Each connected component of \(L \cap \Omega\) is Runge as well and hence biholomorphic to \(\mathbb{D}\). This yields a proper holomorphic embedding of \(\mathbb{D}\) into \(\Omega \cong \mathbb{C}^2\).NEWLINENEWLINEIn [\textit{J.-P. Rosay} and \textit{W. Rudin}, Trans. Am. Math. Soc. 310, No. 1, 47--86 (1988; Zbl 0708.58003)] the question is raised whether there exists a Fatou-Bieberbach domain \(\Omega \subset \mathbb{C}^2\) such that \(\Omega \cap \left( \mathbb{C} \times \{0\} \right) = \mathbb{D} \times \{0\}\), i.e.,\ such that the intersection with a coordinate axis is the actual unit disc and not just biholomorphic to it. This question remains so far open, but two partial answers are known:NEWLINENEWLINE\textit{J. Globevnik} showed in [Math. Z. 229, No. 1, 91--106 (1998; Zbl 0919.32001)] that it is possible to choose \(\Omega\) such that this intersection is as close to the unit disc in \(\mathcal{C}^1\)-norm as desired.NEWLINENEWLINEIn the paper under review the author constructs \(\Omega\) such that its intersection with the first coordinate axis contains a component which is the unit disc. However, the existence of other connected components of this intersection is not excluded.NEWLINENEWLINEThe construction uses the method of exposing points and Andersén-Lempert theory (for an overview see Kaliman and Kutzschebauch: On the present state of the Andersén-Lempert theory). The crucial technical ingredient is Lemma 1.3 which combines Andersén-Lempert theory and the use of certain meromorphic shear maps, see also [\textit{E. F. Wold}, Int. J. Math. 17, No. 8, 963--974 (2006; Zbl 1109.32013)].