Local characterization of holomorphic automorphisms of Siegel domains (Q1062190)

Extending previous work, the authors investigate the Bergman-Shilov boundary of an arbitrary ''nondegenerate'' Siegel domain and prove the following basic theorem: Let D and D' be nondegenerate Siegel domains in \({\mathbb{C}}^{n+m}\) and M and M' their Bergman-Shilov boundary respectively. Let \(\phi\) : \(S\to S'\) be a homeomorphism of connected open subsets \(S\subset M\) and S'\(\subset M'\), satisfying (componentwise in the sense of distribution theory) the tangential Cauchy-Riemann equations on S. Then the map \(\phi\) can be extended to a biholomorphic map \(\phi\) : \(D\to D'\). In section 1 simple facts about the Shilov boundary are proved. In section 2 deep results (e.g. of Naruki and Alexander) are used to investigate the ''analyticity of a CR-homeomorphisms of the skeleton of a Siegel domain''. Section 3 contains the proof of the basic theorem and the last section indicates ''additional results''.