Journal of the
Korean Mathematical Society JKMS

The paper is devoted to the description of changes of the structure of the holomorphic automorphism group of a bounded domain in $\mathbb C^n$ under small perturbation of this domain in the Hausdorff metric. We consider a number of examples when an arbitrary small perturbation can lead to a domain with a larger group, present theorems concerning upper semicontinuity property of some invariants of automorphism groups. We also prove that the dimension of an abelian subgroup of the automorphism group of a bounded domain in ${\mathbb C}^n$ does not exceed $n$.

Keywords: automorphism groups, perturbation of domains, Hausdorff distance, abelian subgroups

MSC numbers: Primary: 32M05, 54H15