Analytic sets extending the graphs of holomorphic mappings between domains of different dimensions (Q456625)

The aim of the paper is to generalize the following theorem due to \textit{K. Diederich} and \textit{S. I. Pinchuk} [J. Geom. Anal. 14, No. 2, 231--239 (2004; Zbl 1078.32012)]:NEWLINENEWLINELet \(f:D\to D'\) be a proper holomorphic mapping between bounded domains in \(\mathbb C^n\) with smooth boundaries. If, for \(p\in \partial D\) and \(q\in\partial D'\) with \(q\) an accumulation point for the values of \(f\) near \(p\), the graph of \(f\) extends as an analytic set near \((p,q)\), then \(f\) extends holomorphically in a neighbourhood of \(p\).NEWLINENEWLINEIn the paper, the case is considered when \(D'\) is a domain of higher dimension, with boundary smooth real-algebraic. Moreover \(f\) is no longer asked to be proper, nor the domains to be bounded. The following is proved NEWLINENEWLINENEWLINE Let \(1<n\leq N\), let \(D\subset\mathbb C^n\), \(D'\subset\mathbb C^N\) be any two domains, and let \(M\subset\partial D\), \(M'\subset\partial D'\) be open subsets of the boundaries such that \(\partial D\) is smooth real-analytic and minimal in a neighbourhood of \(\overline M\), and \(\partial D'\) is smooth real-algebraic and minimal in a neighbourhood of \(\overline M'\). Let \(f:D\to D'\) be a holomorphic mapping such that the cluster set \(\text{cl}_f(M)\) does not intersect \(D'\). If the cluster set \(\text{cl}_f(p)\) of a point \(p\in M\) contains some point \(q\in M'\) such that the graph of \(f\) extends as an analytic set to a neighbourhood of \((p,q)\), then \(f\) extends as a holomorphic map near \(p\).NEWLINENEWLINEThe proof of the theorem is based on analytic continuation along Segre varieties.