Holomorphic equivalence and proper mapping of bounded Reinhardt domains not containing the origin
Bounded Reinhardt domains \(D\subset {\mathbb{C}}^ n\) satisfying the following condition (*) are considered in the paper: (*) There is an integer \(k=k(D)\), \(0\leq k\leq n\), such that \(D\cap \{z_ j=0\}=\emptyset\) for \(j=1,...,k\) and \(\bar D\cap \{z_ j=0\}=\emptyset\) for \(j=k+1,...,n.\) Theorem 1. If \(D_ 1\) and \(D_ 2\) are holomorphically equivalent bounded Reinhardt domains in \({\mathbb{C}}^ n\) satisfying (*), then \(k(D_ 1)=k(D_ 2)\), and there is a biholomorphic map \(F: D_ 1\to D_ 2\) of the form \(F(z',z'')=(c_ 1z^{''\beta_ 1}z_{\sigma (1)},...,c_ kz^{''\beta_ k}z_{\sigma (k)},\quad c_{k+1}z^{''\alpha_{k+1}},...,c_ nz^{''\alpha_ n})\) where \(k=k(D_ 1)=k(D_ 2)\), \(z''=(z_{k+1},...,z_ n)\), \(\beta_ j\in {\mathbb{Z}}^{n-k}\), \((\alpha_{k+1},...,\alpha_ n)^ t\in GL(n- k,{\mathbb{Z}})\), \(\sigma\) is a permutation of \(\{\) 1,...,k\(\}\) and \(c_ 1,...,c_ n\) are positive constants. For domains containing the origin this was proved by \textit{T. Sunada} [Math. Ann. 235, 111-128 (1978; Zbl 0357.32001)]. In the case when \(k(D_ 1)=k(D_ 2)=0\) Theorem 1 may be viewed as a special case of a \textit{E. Bedford}'s result [Pac. J. Math. 87, 271-281 (1980; Zbl 0449.32024)]. Theorem 2. If \(F: D_ 1\to D_ 2\) is a proper holomorphic map of bounded Reinhardt domains in \({\mathbb{C}}^ n\) and if \(D_ 1\) satisfies (*), then F extends holomorphically to a neighborhood of \(\bar D{}_ 1.\) For complete domains this result was proved by \textit{S. Bell} [Trans. Am. Math. Soc. 270, 685-691 (1982; Zbl 0482.32007)].
holomorphically equivalent bounded Reinhardt domains
scientific article; zbMATH DE number 3969068