On the upper-semicontinuity of automorphism groups of model domains in almost complex manifolds
type: Item
instance of: scholarly article
title: On the upper-semicontinuity of automorphism groups of model domains in almost complex manifolds
zbMATH Open document ID: 1322.32018
author: Jisoo Byun
author: Kang-Hyurk Lee
author: Sunhong Lee
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-08-26T00:00:00Z
reviewed by: Swagata Sarkar
Mathematics Subject Classification ID: 32Q60
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 6334524
zbMATH Keywords: almost complex manifolds
zbMATH Keywords: pseudoconvex domains
zbMATH Keywords: automorphism groups
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.07.004
OpenAlex ID: W1985914102
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DOI: 10.1016/J.JMAA.2014.07.004
review text: This work deals with the upper semicontinuity of automorphism groups of almost complex manifolds. A typical result for the upper semicontinuity of holomorphic automorphism groups is the Greene-Krantz Theorem [\textit{R. E. Greene} and \textit{S. G. Krantz}, Math. Ann. 261, 425--446 (1982; Zbl 0531.32016); Math. Z. 190, 455--467 (1985; Zbl 0584.58014)], which deals with the automorphism groups in case of deformation of complex structures on bounded pseudo convex domains. It says that if \(\Omega\) is a \(\mathbb C^{\infty} \) bounded pseudoconvex domain in \(\mathbb C^{n} \) with the standard complex structure \( J_{\mathrm{st}} \), and if \(( J_{\nu} ) \) is a sequence of complex structures on \(\bar {\Omega} \) which converge to \(J_{\mathrm{st}} \) in the \( \mathbb C^{\infty} \) topology of \(\bar {\Omega} \), then \(\text{Aut} ( \Omega, J_{\nu}) \) is isomorphic to a subgroup of \(\text{Aut} (\Omega , J_{\mathrm{st}}) \), for sufficiently large \(\nu \).NEWLINENEWLINEFor general (even unbounded) strongly pseudo convex domains in \( \mathbb C^{n} \), a local version of [\textit{J.-P. Rosay}, Ann. Inst. Fourier 29, No. 4, 91--97 (1979; Zbl 0402.32001)] (see also [\textit{H. Gaussier} et al., Complex Variables, Theory Appl. 47, No. 9, 761--768 (2002; Zbl 1044.32019)]) which characterizes the unit ball, \(\mathbb B^{n}\), in \(\mathbb C^{n} \) is used. However, this characterization fails in case of almost complex manifolds [\textit{H. Gaussier} and \textit{A. Sukhov}, Math. Z. 254, No. 3, 567--589 (2006; Zbl 1107.32009)]. There are many strongly pseudoconvex homogeneous domains in almost complex manifolds, which are biholomorphically distinct from each other. Such a domain is called a model domain in the paper under review.NEWLINENEWLINELet \( J_{\mathrm{st}}\) denote the standard complex structure on \(\mathbb C^{n} \). More precisely, a model domain is a pair \( ( \mathbb H^{n+1} , J_{ \mathfrak B } ) \), where \(\mathbb H^{n+1} \) is the Siegel half plane in \(\…
description: scientific article; zbMATH DE number 6334524