## 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)].

### MSC:

 32H40 Boundary regularity of mappings in several complex variables 32H35 Proper holomorphic mappings, finiteness theorems

### Citations:

Zbl 1049.32023; Zbl 1078.32012
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