Holomorphic mappings between domains in \(\mathbb C^2\) (Q2882891)

The main result of this article is the following theorem.NEWLINENEWLINETheorem. Let \(D\), \(D^{\prime}\) be domains in \(\mathbb{C}^{2}\), both possibly unbounded, and \(f:D \rightarrow D^{\prime}\) a holomorphic mapping. Let \(M \subset \partial D\) and \(M^{\prime} \subset D^{\prime}\) be open pieces, which are smooth real analytic and of finite type, and fix \(p \in M\). Suppose there is a neighborhood \(U\) of \(p\) in \(\mathbb{C}^{2}\) such that the cluster set of \(U\cap M\) does not intersect \(D^{\prime}\). Then \(f\) extends holomorphically across \(p\) if one of the following conditions holds:NEWLINENEWLINE(i) \(p\) is a strongly pseudocnvex point, and the cluster set of \(p\) contains a point in \(M^{\prime}\);NEWLINENEWLINE(ii) \(M\) is pseudoconvex near \(p\), and the cluster set of \(p\) is bounded and contained in \(M^{\prime}\);NEWLINENEWLINE(iii) \(D\) is bounded, \(f:D \rightarrow D^{\prime}\) is proper, and the cluster set of \(M\) is contained in \(M^{\prime}\).NEWLINENEWLINEThe proof entails considering carefully several cases. The reflection principle and Segre varieties play an important role.