Extension of an analytic disc and domains in \(\mathbb C^2\) with noncompact automorphism group (Q411897)

A smooth domain \(\Omega\subset\mathbb C^n\) satisfies condition BR if for every boundary point \(q\in \partial\Omega\) there exists an interior point \(p\in\Omega\) where the Bergman representative map gives a smooth coordinate system on a relatively compact neighborhood of \(p\) that includes the point \(q\).NEWLINENEWLINEIn the present paper smooth domains \(\Omega\subset \mathbb C^2\) that satisfy condition BR are considered. If such a domain has a boundary point \(q\) of finite D'Angelo type \(2m\) and there exists a point \(p\in\Omega\) and a sequence \(\{ \varphi_j \} \subset \text{Aut}(\Omega)\) such that \(\varphi_j(p)\) converges to \(q\), then \(\Omega\) is biholomorphic to the Thullen domain NEWLINE\[NEWLINEE_{2m} := \big\{ (z,w)\in\mathbb C^2 \;\big|\; |z|^{2m}+|w|^2 < 1 \big\}.NEWLINE\]