Some new results on domains in complex space with non-compact automorphism group. (Q1399329)

Let \(\Omega\subset\mathbb C^n\) be a bounded domain which is strictly pseudoconvex in a neighborhood of a given point \(p\in\partial \Omega \). Let \(G\) be the group of biholomorphic automorphisms of \(\Omega\) and assume that \(p\) is an orbit accumulation boundary point, i.e. \(p\) is contained in the closure of some \(G\)-orbit in \(\Omega \). By a theorem of \textit{B. Wong} [Invent. Math. 41, 253--257 (1977; Zbl 0385.32016)] and \textit{J.-P. Rosay} [Ann. Inst. Fourier 29, 91--97 (1979; Zbl 0402.32001)], \(\Omega \) and \(\mathbb B\) are biholomorphically equivalent. The authors give a new proof of this fact, based on the theorem of L. Bers that two pseudoconvex domains in \(\mathbb C^n\) are biholomorphically equivalent if their \(\mathbb C\)-algebras of holomorphic functions are isomorphic. In the above situation \(\Omega\) is globally pseudoconvex [see \textit{R. E. Greene} and \textit{S. G. Krantz}, Editoria Elettronica, Semin. Conf. 8, 108--135 (1991; Zbl 0997.32012)]. The method of establishing an isomorphism between function algebras can also be applied in the following situation: Without the above boundary condition on \(\Omega\) let \(p'\) be a smooth boundary point of a bounded symmetric domain \(E\) and \(X\) and \(Y\) the maximal analytic varieties in \(\partial \Omega\) and \(\partial E\) passing through \(p\) and \(p'\) respectively. If \(U\cap\overline\Omega\) and \(U'\cap\overline E\) are biholomorphically equivalent for suitable neighborhoods \(U\) of \(X\) and \(U'\) of Y, then \(\Omega\) and \(E\) are biholomorphically equivalent. It is also shown that every hyperbolic orbit boundary point of a bounded domain in \(\mathbb C^n\) with finite boundary type is a peak point, and that under some additional assumptions every parabolic orbit accumulation boundary point of a pseudoconvex domain in \(\mathbb C^n\) with \(C^\infty\)-boundary is of finite type.