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The ball in \({\mathbb{C}}^ n\) is a closed complex submanifold of a polydisc. (English) Zbl 0617.32039

The author constructs a proper holomorphic map from the unit ball \(B_ n\) in \({\mathbb{C}}^ n\) (n\(\geq 2)\) to some higher dimensional polydisc \(\Delta_{4M(n)}\) in \({\mathbb{C}}^{4M(n)}\), where M(n) depends only on n and is given in terms of a covering property of \(B_ n\). The construction is by an inductive procedure employing the peak function \(g(z)=\exp \{-m(1-<z,w>)\},\) with \(w\in \partial B_ n\) and m a large integer. The extra dimensions are needed to adjust for the oscillations of g(z), and an auxiliary weight function is used in the proof. The result is in contrast to the fact that there exists no proper holomorphic map from \(\Delta_ n\) to any \(B_ m\), nor from \(B_ n\) to \(\Delta_ n\). The author poses the interesting question whether 4M(n) can be improved to \(n+1\).
Reviewer: H.-S.Luk

MSC:

32H35 Proper holomorphic mappings, finiteness theorems
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References:

[1] Gunning, R., Rossi, H.: Analytic functions of several complex variables. Englewood Cliffs, NJ: Prentice-Hall 1965 · Zbl 0141.08601
[2] Hakim, M., Sibony, N.: Fonctions holomorphes bornee sur la boule unite de ? n . Invent. Math.67, 223-229 (1982) · Zbl 0491.32014 · doi:10.1007/BF01393814
[3] L?w, E.: A construction of inner functions on the unit ball in ? p . Invent. Math.67, 223-229 (1982) · Zbl 0528.32006 · doi:10.1007/BF01393815
[4] Rudin, W.: Function theory in the unit ball of ? n . New York: Springer 1980 · Zbl 0495.32001
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