Envelopes of holomorphy of Hartogs and circular domains (Q1825998)

The classical problem of the univalence and description of the envelope of holomorphy of a domain in \({\mathbb{C}}^ n\) is investigated: the classes of Hartogs and circular domains are taken into consideration. In the Hartogs case preceding results are improved and completed. For instance, the problem is solved for domains such that the fibers of the projection onto the hyperplane of symmetry are connected: they include complete Hartogs domains as particular cases. These results are transferred to the circular case using the fact that a circular domain which does not intersect some hyperplane through the origin is biholomorphic to a Hartogs domain; the general case is then inferred from this and other considerations. An alternative interpretation of these results is given in terms of fiber bundles over the projective space \({\mathfrak P}^{n-1}({\mathbb{C}}).\) Suitable examples show the noneliminability of the assumptions of `connected sections' from any of our main statements.