The automorphism group of domains with boundary points of infinite type. (English) Zbl 1065.32016

Landucci, Mario

Ill. J. Math. 48, No. 3, 875-885 (2004).

Let \(D\) be a bounded domain in \({\mathbb C}^n\). It is classical that \(\operatorname{Aut}(D)\) is noncompact if and only if there exists a boundary orbit accumulation point in \(\partial D\), i.e., a point \(p\in\partial D\) for which there exists a point \(x\in D\) and a sequence \(\{ f_{n} \} \subset \operatorname{Aut}(D)\) such that \(f_{n}(x)\to p\). 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)] have conjectured that if \(D\) has smooth boundary, then a boundary orbit accumulation point of \(D\) must be of finite type and there has been some progress on this, e.g., see loc cit.
In this paper the author shows that if \(\partial D\) contains a special smooth curve of points of infinite type, then \(\operatorname{Aut}(D)\) is compact. This then verifies the Greene-Krantz conjecture for this type of domain.

Reviewer: Bruce Gilligan (Regina)

MSC:

32M05	Complex Lie groups, group actions on complex spaces
32T25	Finite-type domains

Keywords:

automorphism group; finite type domain

Citations:

Zbl 0779.32017