by their noncompact automorphism groups
Ninh Van Thu - University of Natural Sciences, National University
Nguyen Thi Tuyet Mai - College of Education - Thai Nguyen University
Let Ω be a domain (connected, open set) in a complex manifold M. Let the automor-, phism group of Ω (denoted Aut(Ω)) be the collection of biholomorphic self-maps of Ω with, composition of mappings as its binary operation. The topology on Aut(Ω) is that of uniform, convergence on compact sets (i.e., the compact-open topology)., One of the important problems in several complex variables is to study the interplay between, the geometry of a domain and the structure of its automorphism group. More precisely, we, wish to see to what extent a domain is determined by its automorphism group., It is a standard and classical result of H. Cartan that if Ω is a bounded domain in Cn and, the automorphism group of Ω is noncompact then there exist a point x ∈ Ω, a point p ∈ ∂Ω, and automorphisms ϕj ∈ Aut(Ω) such that ϕj(x) → p. In this circumstance we call p a, boundary orbit accumulation point., Work in the past twenty years has suggested that the local geometry of the so-called ”bound-, ary orbit accumulation point” p in turn gives global information about the characterization of, model of the domain. There have been several contributions by several authors concerning, this line research. We refer readers to the recent survey  and references therein for the, development in related subjects. For instance, in the case of domains in C2, (not necessary, bounded), F. Berteloot  proved the following theorem.