On the automorphism group of strongly pseudoconvex domains in almost complex manifolds
type: Item
instance of: scholarly article
title: On the automorphism group of strongly pseudoconvex domains in almost complex manifolds
zbMATH Open document ID: 1168.32021
author: Jisoo Byun
author: Hervé Gaussier
author: Kang-Hyurk Lee
publication date: 2009-04-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10393
review text: Let \(D\) denote a connected open subset of \(\mathbb{C}^n\), with boundary \(\partial D\). B. Wong proved: Assume that \(D\) is bounded, strongly pseudoconvex, and \(\partial D\) is smooth. If \(\Aut(D)\) is non-compact, then \(D\) is biholomorphic to the unit ball. Using that for any bounded \(D\) with non-compact automorphism group there exists an orbit accumulating at some boundary point (H. Cartan), J.-P. Rosay improved Wong's result -- by localizing the boundary assumptions -- and obtained: Assume that \(D\) is bounded. If there exists \(\zeta \in \partial D\), at which \(\partial D\) is smooth and strongly pseudoconvex, and such that there exists \(z\in D\) whose orbit under \(\Aut(D)\) accumulates at \(\zeta\), then \(D\) is biholomorphic to the unit ball. Strengthening Rosay's theorem, A. M. Efimov dropped the assumption that \(D\) be bounded. Strengthening further, Gaussier-Kim-Krantz took \(D\) in any manifold. (see \textit{B. Wong} [Invent. Math. 41, 253--257 (1977; Zbl 0385.32016)]; \textit{J.-P. Rosay} [Ann. Inst. Fourier 29, 91--97 (1979; Zbl 0402.32001)]; \textit{A. M. Efimov} [Sb. Math. 186, No.~7, 967--976 (1995); translation from Mat. Sb. 186, No.~7, 41--50 (1995; Zbl 0865.32020)]; \textit{H. Gaussier}, \textit{K.-T. Kim} and \textit{S. G. Krantz} [Complex Variables, Theory Appl. 47, No.~9, 761--768 (2002; Zbl 1044.32019)].) Now let's consider the analogous questions in the more general setting of \(D\) contained in an almost complex manifold. Under the assumptions of the Gaussier-Kim-Krantz theorem, \(D\) must be either the unit ball or some explicit non-integrable deformation of it called a ``model manifold'' (see \textit{H. Gaussier} and \textit{A. Sukhov} [Math. Z. 254, No.~3, 567--589 (2006; Zbl 1107.32009)] and \textit{K.-H. Lee} [Mich. Math. J. 54, No.~1, 179--206 (2006; Zbl 1145.32014); J. Reine Angew. Math. 623, 123--160 (2008; Zbl 1160.32024)]). In the article under review, the authors extend Rosay's theorem as stated above to the case o…
reviewed by: Dan Zaffran
Mathematics Subject Classification ID: 32Q60
Mathematics Subject Classification ID: 32T15
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 5541202
zbMATH Keywords: almost complex manifolds
zbMATH Keywords: strongly pseudoconvex domains
zbMATH Keywords: non-integrable deformations
zbMATH Keywords: automorphism groups
MaRDI profile type: Publication
OpenAlex ID: W2295186732
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DOI: 10.5802/AIF.2431
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5541202