Boundary regularity of correspondences in \(\mathbb{C}^2\)
It has been previously proved by \textit{K. Diederich} and \textit{S. Pinchuk} [Indiana Univ. Math. J. 44, No. 4, 1089-1126 (1995; Zbl 0857.32015)], that a proper holomorphic mapping between two bounded domains with smooth real analytic boundaries in \(\mathbb C^2\) extends holomorphically to an open neighbourhood crossing the boundary. The present article attempts to answer a similar question for proper holomorphic correspondence between domains in \(\mathbb C^2\). The main theorem asserts the following: Let \(\widehat f: D\to D'\) be a proper holomorphic correspondence between domains \( D\) and \(D'\) in \(\mathbb C^2\) with inverse correspondence \(\widehat g\), and let \(\Gamma_1\subset\subset \Gamma \subset \partial D\) and \(\Gamma' \subset \partial D'\) be open pieces of the boundaries such that \(\partial D\) (resp. \(\partial D'\)) is real analytic and of finite type in an open neighbourhood of \(\bar \Gamma\) (resp. \(\bar \Gamma'\)). Assume that the cluster point of \(\Gamma_1\) under \(\widehat f\) is a subset of \(\Gamma'\) and the cluster point of \(\Gamma'\) under \(\widehat g\) is a subset of \(\Gamma\). Then \(\widehat f\) extends as a holomorphic correspondence near each \(z_0\in \Gamma_1\).
scientific article; zbMATH DE number 1336887
scientific article