Invariants of Bergman geometry and the automorphism groups of domains in \({\mathbb{C}^n}\). (English) Zbl 0997.32012

Greene, R. E.; Krantz, S. G.

Berenstein, Carlos A. (ed.) et al., Geometrical and algebraical aspects in several complex variables. Papers from the conference, Cetraro, Italy, June 1989. Rende: Editoria Elettronica, Semin. Conf. 8, 108-135 (1991).

In the frame of the study of the group of automorphisms \(\operatorname{Aut} (\Omega)\) of a bounded domain \( \Omega\) in \(\mathbb{C}^n\), the authors investigate the type of convexity of a point of accumulation of an orbit of \(\operatorname{Aut} (\Omega)\) . They also show that under certain conditions, the existence of such accumulation points implies that \(\Omega\) is a domain of holomorphy and, under some more general circumstances, diffeomorphic (but certainly not necessarily biholomorphic) to the ball. The behaviour of orbits near an orbit accumulation point is also under investigation here. Finally, they discuss the question of the location of orbits by an invariant involving only two derivatives of the Bergman kernel and deepen the study of this invariant by furnishing some very interesting estimates.
For the entire collection see [Zbl 0969.00052].

Reviewer: Eugen Pascu (Montreal)

MSC:

32F10	\(q\)-convexity, \(q\)-concavity
32M05	Complex Lie groups, group actions on complex spaces
32T15	Strongly pseudoconvex domains
32M17	Automorphism groups of \(\mathbb{C}^n\) and affine manifolds
32M10	Homogeneous complex manifolds
32E40	The Levi problem
32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)

Keywords:

Lie group of automorphisms; domains in \(\mathbb{C}^n\); orbit; pseudoconvexity; Bergman kernel