Complex scaling and domains with non-compact automorphism group. (English) Zbl 1065.32014

Kim, Kang-Tae; Krantz, Steven G.

Ill. J. Math. 45, No. 4, 1273-1299 (2001).

Suppose \(D\subset{\mathbb C}^n\) is a bounded domain. A classical result that can be proven using a normal family argument, e.g., see R. Narasimhan’s book [Several Complex Variables, (University of Chicago Press, Chicago) (1971; Zbl 0223.32001)], is 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 \(f_{n}(x)\to p\). A result due to B. Wong [Invent. Math. 41, 253–257 (1977; Zbl 0385.32016)] and J.-P. Rosay [Ann. Inst. Fourier 29, 91–97 (1979; Zbl 0402.32001)] characterizes the unit ball \({\mathbb B}_{n}\) as the domain that has a strictly pseudoconvex boundary orbit accumulation point. Interest has centered on boundary orbit accumulation points that are not strongly pseudoconvex and 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 every boundary orbit accumulation point of \(D\) must be of finite type.
This paper contains a step towards proving this conjecture in that the authors show that a boundary orbit accumulation point cannot be exponentially flat.
One technique that is used in the proofs is a sharpened form of scaling including a convergence result for Pinchuk scaling and an equivalence statement for the scaling methods of Pinchuk and Frankel.

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 0223.32001; Zbl 0385.32016; Zbl 0402.32001; Zbl 0779.32017