Stably-interior points and the Semicontinuity of the Automorphism group

Abstract

We present the new semicontinuity theorem for automorphism groups: If a sequence $\{\Omega _j\}$ of bounded pseudoconvex domains in $\mathbb C^2$ converges to $\Omega _0$ in ${\mathcal C}^\infty$-topology, where $\Omega _0$ is a bounded pseudoconvex domain in $\mathbb C^2$ with its boundary ${\mathcal C}^\infty$ and of the D’Angelo finite type and with $\text {Aut}\,(\Omega _0)$ compact, then there is an integer $N>0$ such that, for every $j > N$, there exists an injective Lie group homomorphism $\psi _j:\text {Aut}\,(\Omega _j) \rightarrow \text {Aut}\,(\Omega _0)$. The method of our proof of this theorem is new that it simplifies the proof of the earlier semicontinuity theorems for bounded strongly pseudoconvex domains.