On the automorphism group of strongly pseudoconvex domains in almost complex manifolds
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 of any relatively compact \(D\) in an almost complex manifold. This is obtained by proving that the non-integrable model manifolds do not admit any relatively compact representation. Another result is a stability result of the automorphism group of such a \(D\) under a \(\mathcal{C}^{\infty}\)-small deformation of the almost complex structure (in analogy with a result of \textit{R. E. Greene} and \textit{S. G. Krantz} see [Math. Z. 190, 455--467 (1985; Zbl 0584.58014)]).
scientific article; zbMATH DE number 5541202
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