Proper holomorphic mappings in some class of unbounded domains
Elementary Reinhardt domains in \(\mathbb C^n\) are defined as the domains of the form \(D_\alpha = \{ z=(z_1,\dots,z_n)\in \mathbb C^n : |z_1|^{\alpha_1} \cdots |z_n|^{\alpha_n} < 1 \}\), where \(\alpha := (\alpha_1, \dots, \alpha_n) \in \mathbb R_+^n\) is said to be of rational type if there exists \(\lambda >0\) such that \(\lambda \alpha = (\alpha'_1, \dots, \alpha'_n)\) where the \(\alpha'_j\) are relatively prime natural numbers. Let \(F\) be a proper holomorphic mapping from \(D_\alpha\) to \(D_\beta\), two elementary Reinhardt domains. Then \(\alpha\) and \(\beta\) must have the same type (rational or irrational). If \(n=2\), the authors give an explicit form for \(F\) depending on a discrete set of parameters and a zero-free holomorphic function on the unit disk (the coordinates of \(F\) must be certain quotients of monomials multiplied or divided by this function). If \(n\) is arbitrary and \(F\) is furthermore biholomorphic, they prove that both domains have to be linearly biholomorphic. Those results improve on previous work of \textit{S. Shimizu} [Osaka J. Math. 28, No. 3, 609-621 (1991; Zbl 0786.32001); Kodai Math. J. 15, No. 3, 430-444 (1992; Zbl 0790.32001)]. The method of proof involves the pluricomplex Green function for several poles [\textit{P. Lelong}, J. Math. Pures Appl., IX. Sér. 68, No. 3, 319-347 (1989; Zbl 0633.32019)], which is invariant under proper holomorphic mappings. The authors compute this function for elementary Reinhardt domains using previous results for the Green function with one pole [\textit{M. Jarnicki} and \textit{P. Pflug}, `Invariant distances and metrics in complex analysis'. De Gruyter Expositions in Mathematics. 9. Berlin: Walter de Gruyter (1993; Zbl 0789.32001); and \textit{P. Pflug} and \textit{W. Zwonek}, Ann. Pol. Math. 69, No. 2, 175-196 (1998); erratum ibid. No. 3, 301 (1998; Zbl 0926.32015)]. This function is constant on the level sets of the function \(z^\alpha\) (which sends \(D_\alpha\) to the unit disk), so \(F\) must induce a map from the level sets of \(z^\alpha\) in \(D_\alpha\) and \(z^\beta\) in \(D_\beta\) which induces in fact a proper holomorphic map of the unit disk. Further arguments lead to both results.
scientific article; zbMATH DE number 1437241