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Approximation by proper holomorphic maps into convex domains. (English) Zbl 0777.32014
The paper deals with the problem of existence of a continuous proper holomorphic map from the unit ball of $$\mathbb{C}^ N$$ to an arbitrary $$C^ 2$$-smooth, bounded domain of higher dimension. The author proves two results. The first theorem (see below) shows that the proper holomorphic maps from the ball into a convex domain $$\Omega$$ of higher dimension are a dense subset of the holomorphic maps from the ball to the domain, in the topology of uniform convergence on compacta. The second theorem implies the same for $$C^ 2$$-smooth, bounded strongly convex domains with continuous proper holomorphic maps.
Theorem 1. Let $$2 \leq N \leq M-1$$, and take $$\Omega \subseteq \mathbb{C}^ M$$, an arbitrary convex domain, $$f: \overline B^ N \to \Omega$$ continuous and holomorphic in $$B^ N$$, and $$\varepsilon>0$$. Then there exists a proper holomorphic map $$F:B^ N \to \Omega$$ such that $$| F-f |< \varepsilon$$ on $$(1-\varepsilon)\overline B^ N$$.
Theorem 2. Let $$2 \leq N \leq M-1$$, and let $$\Omega \Subset \mathbb{C}^ M$$ be a strongly convex domain with $$C^ 2$$ boundary, $$f:\overline B^ N \to \Omega$$ continuous and holomorphic in $$B^ N$$, and $$\varepsilon>0$$. Then there exists a proper holomorphic map $$F:B^ N \to \Omega$$ which is continuous on $$\overline B^ N$$ and $$| F-f |< \varepsilon$$ on $$(1-\varepsilon) \overline B^ N$$.

##### MSC:
 32H35 Proper holomorphic mappings, finiteness theorems
##### Keywords:
proper holomorphic maps; convex domain
Full Text:
##### References:
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