On the compactness of the automorphism group of a domain. (English) Zbl 1086.32020

Byun, Jisoo; Gaussier, Hervé

C. R., Math., Acad. Sci. Paris 341, No. 9, 545-548 (2005).

Let \(\Omega\) be a smooth, bounded domain in \({\mathbb C}^{n}\). One would like to relate the automorphism group \(\text{ Aut}(\Omega)\) to the geometry of \(\partial\Omega\) with one underlying motivation in this regard being a resolution of the conjecture of R. E. Greene and S. G. Krantz [Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987–88, Math. Notes 38, 389–410 (1993; Zbl 0779.32017)] that an orbit accumulation point is of finite type.
One approach is to consider the sets \(P_{m}(M)\) of all points in \(M\) of type \(m\), with \(m\) either a positive integer or infinity, where \(M\subset{\mathbb C}^{n}\) is a hypersurface. For \(\Omega\subset{\mathbb C}^{2}\) M. Landucci [Ill. J. Math. 48, 875–885 (2004; Zbl 1065.32016)] showed that under certain conditions on \(\partial\Omega\) the automorphism group \(\text{Aut}(\Omega)\) is compact. One of these conditions is that \(P_{\infty}(\partial\Omega)\) is a certain type of smooth curve in \(\partial\Omega\).
In the present paper the authors give a sufficient condition on the boundary of a smooth, bounded, convex domain \(\Omega\subset{\mathbb C}^{n}\) in order that \(\text{ Aut}(\Omega)\) be compact, namely, a connected component of \(P_{m}(\partial\Omega)\) is an interval \(I\) that is transversal to \(T^{\mathbb C}_{z} (\partial\Omega)\) at some point \(z\in I\). This is a corollary of the following Theorem. Suppose \(\Omega\) is a bounded pseudoconvex domain in \({\mathbb C}^{n}\) that satisfies condition (R) of Bell and Ligocka. (Condition (R) asserts that a certain Bergman projection operator is bounded and was introduced and used by S. Bell and E. Ligocka [Invent. Math. 57, 283–289 (1980; Zbl 0411.32010)] to give an elementary proof of C. Fefferman’s extension result [Invent. Math. 26, 1–65 (1974; Zbl 0289.32012)].) Assume there are open sets \(U\) and \(V\) with \(\overline{U} \subset V\) such that a connected component of \(P_{m}(\partial\Omega\cap V)\) is an interval \(I\subset U\) transversal to the complex tangent space \(T_{z}^{\mathbb C} (\partial\Omega)\) at some point \(z\in I\). Then there are no automorphism orbits in \(\Omega\) accumulating at any convex part of the boundary of \(\Omega\).

Reviewer: Bruce Gilligan (Regina)

MSC:

32M05	Complex Lie groups, group actions on complex spaces
32V15	CR manifolds as boundaries of domains

Keywords:

automorphism group; compactness

Citations:

Zbl 0779.32017; Zbl 1065.32016; Zbl 0411.32010; Zbl 0289.32012

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