Some new results on domains in complex space with non-compact automorphism group.
type: Item
instance of: scholarly article
title: Some new results on domains in complex space with non-compact automorphism group.
zbMATH Open document ID: 1035.32019
DOI: 10.1016/S0022-247X(03)00003-9
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(\Omega\subset\mathbb C^n\) be a bounded domain which is strictly pseudoconvex in a neighborhood of a given point \(p\in\partial \Omega \). Let \(G\) be the group of biholomorphic automorphisms of \(\Omega\) and assume that \(p\) is an orbit accumulation boundary point, i.e. \(p\) is contained in the closure of some \(G\)-orbit in \(\Omega \). By a theorem of \textit{B. Wong} [Invent. Math. 41, 253--257 (1977; Zbl 0385.32016)] and \textit{J.-P. Rosay} [Ann. Inst. Fourier 29, 91--97 (1979; Zbl 0402.32001)], \(\Omega \) and \(\mathbb B\) are biholomorphically equivalent. The authors give a new proof of this fact, based on the theorem of L. Bers that two pseudoconvex domains in \(\mathbb C^n\) are biholomorphically equivalent if their \(\mathbb C\)-algebras of holomorphic functions are isomorphic. In the above situation \(\Omega\) is globally pseudoconvex [see \textit{R. E. Greene} and \textit{S. G. Krantz}, Editoria Elettronica, Semin. Conf. 8, 108--135 (1991; Zbl 0997.32012)]. The method of establishing an isomorphism between function algebras can also be applied in the following situation: Without the above boundary condition on \(\Omega\) let \(p'\) be a smooth boundary point of a bounded symmetric domain \(E\) and \(X\) and \(Y\) the maximal analytic varieties in \(\partial \Omega\) and \(\partial E\) passing through \(p\) and \(p'\) respectively. If \(U\cap\overline\Omega\) and \(U'\cap\overline E\) are biholomorphically equivalent for suitable neighborhoods \(U\) of \(X\) and \(U'\) of Y, then \(\Omega\) and \(E\) are biholomorphically equivalent. It is also shown that every hyperbolic orbit boundary point of a bounded domain in \(\mathbb C^n\) with finite boundary type is a peak point, and that under some additional assumptions every parabolic orbit accumulation boundary point of a pseudoconvex domain in \(\mathbb C^n\) with \(C^\infty\)-boundary is of finite type.
Mathematics Subject Classification ID: 32T05
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32T15
Mathematics Subject Classification ID: 32T25
zbMATH DE Number: 1956871
zbMATH Keywords: boundary orbit accumulation point
zbMATH Keywords: peak function
zbMATH Keywords: boundary point of finite type
author: Kang-Tae Kim
author: Steven G. Krantz
reviewed by: Eberhard Oeljeklaus
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00003-9
OpenAlex ID: W2034086083
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description: scientific article; zbMATH DE number 1956871