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**A local extension of proper holomorphic maps between some unbounded domains in \({\mathbb C}^n\).**
*(English)*
Zbl 1205.32015

The boundary regularity of proper holomorphic maps between smooth domains in \(\mathbb C^n\) is still an open problem in full generality. For domains with real-analytic smooth boundaries some related partial results are given in [R. Shafikov and K. Verma, J. Geom. Anal. 13, No. 4, 697–714 (2003; Zbl 1049.32023)] and [K. Diederich and S. Pinchuk, J. Geom. Anal. 14, No. 2, 231–239 (2004; Zbl 1078.32012)]. Here the authors give a positive answer in some special cases. The main result of the paper is the following theorem:

Let \(D,D'\) be arbitrary domains in \(\mathbb C^n\), \(n>1\), and \(f:D\rightarrow D'\) be a proper holomorphic mapping. Let \(M\subset\partial D\), \(M'\subset\partial D'\) be open pieces of the boundaries. Suppose that \(\partial D\) (resp. \(\partial D'\)) is smooth real-analytic and nondegenerate in an open neighborhood of \(\bar M\) (resp. \(\bar M'\)). If the cluster set \(cl_f(p)\) of a point \(p\in M\) contains a point \(q\in M'\) and the graph of \(f\) extends as an analytic set to a neighborhood of \((p,q)\in\mathbb C^n\times\mathbb C^n\), then \(f\) extends holomorphically to a neighborhood of \(p\).

As an application of this theorem the authors prove the following result:

Let \(D,D'\) be smooth algebraic domains in \(\mathbb C^n\), \(n>1\), with nondegenerate boundaries and \(f:D\rightarrow D'\) be a proper holomorphic mapping.

(a) If the cluster set \(cl_f(p)\) of a point \(p\in\partial D\) contains a point \(q\in\partial D'\), then \(f\) extends holomorphically to a neighborhood of \(p\) and the set of holomorphic extendability of \(f\) is an open dense subset of \(\partial D\).

(b) If either \(D\) or \(D'\) has a global holomorphic peak function at infinity, then the set of holomorphic extendability of \(f\) is an open dense subset of \(\partial D\).

Moreover, the authors give also a local version of the theorem above, which is a generalization of the result contained in [R. Shafikov and K. Verma, J. Geom. Anal. 13, No. 4, 697–714 (2003; Zbl 1049.32023)].

Let \(D,D'\) be arbitrary domains in \(\mathbb C^n\), \(n>1\), and \(f:D\rightarrow D'\) be a proper holomorphic mapping. Let \(M\subset\partial D\), \(M'\subset\partial D'\) be open pieces of the boundaries. Suppose that \(\partial D\) (resp. \(\partial D'\)) is smooth real-analytic and nondegenerate in an open neighborhood of \(\bar M\) (resp. \(\bar M'\)). If the cluster set \(cl_f(p)\) of a point \(p\in M\) contains a point \(q\in M'\) and the graph of \(f\) extends as an analytic set to a neighborhood of \((p,q)\in\mathbb C^n\times\mathbb C^n\), then \(f\) extends holomorphically to a neighborhood of \(p\).

As an application of this theorem the authors prove the following result:

Let \(D,D'\) be smooth algebraic domains in \(\mathbb C^n\), \(n>1\), with nondegenerate boundaries and \(f:D\rightarrow D'\) be a proper holomorphic mapping.

(a) If the cluster set \(cl_f(p)\) of a point \(p\in\partial D\) contains a point \(q\in\partial D'\), then \(f\) extends holomorphically to a neighborhood of \(p\) and the set of holomorphic extendability of \(f\) is an open dense subset of \(\partial D\).

(b) If either \(D\) or \(D'\) has a global holomorphic peak function at infinity, then the set of holomorphic extendability of \(f\) is an open dense subset of \(\partial D\).

Moreover, the authors give also a local version of the theorem above, which is a generalization of the result contained in [R. Shafikov and K. Verma, J. Geom. Anal. 13, No. 4, 697–714 (2003; Zbl 1049.32023)].

Reviewer: Pawel Zapalowski (Kraków)