Domains with prescribed automorphism group
Let \(G\) be a connected compact Lie group and \(H\subset G\) a connected closed subgroup. Assume that \(G\) is realized as the full automorphism group Aut\(D\) of a strictly pseudoconvex bounded domain \(D\) in \(\mathbb C^N\) with \(\mathcal{C}^\infty\) boundary. The semicontinuity theorem of \textit{R. E. Greene} and \textit{S. G. Krantz} for automorphism groups [Math. Ann. 261, 425--446 (1982; Zbl 0531.32016)] states that perturbations of \(D\) within a sufficiently small neighborhood \(U\) of \(\overline D\) yield domains \(\tilde D\) diffeomorphic to \(D\) whose automorphism groups are subgroups of \(G\). The main result of the present paper says that if \(N\geq 5(1+\dim G)\) then for every \(\epsilon >0\) exists an \(\epsilon\)-close \(\mathcal{C}^\infty\) perturbation \(\tilde D\) of \(D\) such that Aut\(\tilde D=H\). The proof, which is based on a method of \textit{R. Saerens} and \textit{W. Zame} [Trans. Am. Math. Soc. 301, 413--429 (1987; Zbl 0621.32025)] for the construction of strictly pseudoconvex bounded domains with prescribed automorphism groups, uses concepts of complex differential geometry, scalar CR-invariants on strictly pseudoconvex hypersurfaces and multi-jet bundles. In an appendix of the paper the author proves the following: If the above domain \(D\) is the result of the Saerens-Zame construction, then there exists without any further assumption on \(N\) a strictly pseudoconvex domain \(D'\) with \(\mathcal{C}^{\infty}\) boundary such that \(D'\) is \(\mathcal{C}^{\infty}\)-close to \(D\) and Aut\(D' =H\). For the construction of strictly pseudoconvex bounded domains with prescribed automorphism group see also \textit{E. Bedford} and \textit{J. Dadok} [Comment. Math. Helv. 62, 561--572 (1987; Zbl 0647.32027)]
scientific article; zbMATH DE number 5624085
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