Extension of an analytic disc and domains in ℂ²with noncompact automorphism group

Abstract

Let be a smoothly bounded domain in ‫ރ‬ 2 such that the Bergman representative map near the boundary continues to be diffeomorphic up to the boundary. If such a domain admits a holomorphic automorphism group orbit accumulating at a boundary point of finite D'Angelo type 2m, we show that the domain is biholomorphic to the Thullen domain {(z, w) ∈ ‫ރ‬ 2 : |z| 2m + |w| 2 < 1}. This result refines the well-known theorem of E. Bedford and S. Pinchuk.

Automorphisms of normal quasi-circular domains

“…After its discovery, the Bergman mapping appeared in many studies (cf. [11,12,14,21,25,27]) and it played a substantial role in their studies. We will see that the Bergman mapping also plays an essential role in our study.…”