Stably-interior points and the Semicontinuity of the Automorphism group

Abstract

We present the new semicontinuity theorem for automorphism groups: If a sequence {Ωj } of bounded pseudoconvex domains in ℂ2 converges to Ω0 in C∞-topology, whereΩ0 is a bounded pseudoconvex domain in ℂ2 with its boundary ℂ∞ and of theD’Angelo finite type and with Aut (Ω0) compact, then there is an integer N > 0 such that, for every j > N, there exists an injective Lie group homomorphism ψj : Aut (Ωj) → Aut (Ω0). 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.