Construction of envelopes of holomorphy for some classes of special domains
The authors construct explicitly the envelope of holomorphy over \(\mathbb{C}^ n\) of any domain \(\Omega \subset \mathbb{C}^ n\) which is either \(\alpha\)-circular for some nonzero multi-index \(\alpha=(\alpha_ 1,\dots,\alpha_ n) \in \mathbb{N}^ n\) (i.e., is such that \((w^{\alpha_ 1} x_ 1, \dots, w^{\alpha_ n} x_ n) \in \Omega\) for every \(x=(x_ 1, \dots, x_ n) \in \Omega\) and \(w \in \mathbb{S}^ 1 \subset \mathbb{C})\) or \(k\)-tabular for \(1 \leq k \leq n\) (i.e., is such that \(x+iy \in \Omega\) whenever \(x \in \Omega\) and \(y \in \mathbb{R}^ k \subset \mathbb{C}^ k \times \mathbb{C}^{n-k})\). (Hartogs, circular, respectively tube domains are particular cases of these.) If the group \(G=\mathbb{S}^ 1\), respectively \(G=\mathbb{R}^ k\), acts on \(\Omega\) in the obvious way, such envelope of holomorphy is built in terms of that of the categorical quotient \(\Omega \| G\) (a complex space of dimension at most \(n-\dim G)\) and of the largest plurisubharmonic minorants of functions defined on suitable \((n-\dim G)\)- dimensional spaces. As auxiliary results, conditions are proved for \(\Omega\) to be a domain of holomorphy, or to have a univalent envelope. Some examples illustrate the construction.
scientific article; zbMATH DE number 579772
scientific article