Convergence of automorphisms and semicontinuity of automorphism groups (Q416403)

Some results on convergence of automorphisms of smoothly bounded domains in \(\mathbb C^n\) are given.NEWLINENEWLINEThe author then studies how automorphisms groups of domains of finite type in \(\mathbb C^2\) behave under small \(C^k\) perturbations and proves the following result. Suppose \(\Omega\) is a smoothly bounded, finite type domain in \(\mathbb C^2\) with \(\text{Aut}(\Omega)\) compact in the compact-open topology. If \(\widehat{\Omega}\) is a smoothly bounded, finite type domain with \(C^k\) distance (for suitably large enough \(k\)) less than \(\epsilon\) from \(\Omega\), then there exists a smooth diffeomorphism \(\Phi: \widehat{\Omega} \to \Omega\) such that \(\varphi \mapsto \Phi\circ\varphi\circ\Phi^{-1}\) is a monomorphism of \(\text{Aut}(\widehat{\Omega})\) into \(\text{Aut}(\Omega)\).NEWLINENEWLINESome examples are also given.