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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
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