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Extending proper holomorphic mappings of positive codimension. (English) Zbl 0633.32017

Let \(M\subset {\mathbb{C}}^ n\) and \(M'\subset {\mathbb{C}}^ N\) \((N>n>1)\) be smooth real-analytic pseudoconvex hypersurfaces, M of finite type, and M’ strictly pseudoconvex. Let \(D\subset {\mathbb{C}}^ n\) be a domain which contains M in its boundary and is pseudoconvex along M. Assume that \(f: D\cup M\to {\mathbb{C}}^ N\) is a mapping of class \({\mathcal C}^{\infty}\) that is holomorphic on D and maps M to M’. We associate to f an integer valued, upper semicontinuous function \(\nu\) on M, called the deficiency of f. If \(\nu\) is constant in a neighborhood of a point \(z\in M\) in M, then f extends holomorphically to a neighborhood of z in \({\mathbb{C}}^ n\). This holds in an open, everywhere dense subset \(M_ 0\) of M. If M’ is the unit sphere in \({\mathbb{C}}^ N\), then the same is true if f is merely of class \({\mathcal C}^{N-n+1}(D\cup M)\). If both M and M’ are unit spheres, then f extends to a rational mapping of \({\mathbb{C}}^ n\) to \({\mathbb{C}}^ N\). In particular, every proper holomorphic map \(f: {\mathfrak B}^ n\to {\mathfrak B}^ N\) \((N>n>1)\) that is of class \({\mathcal C}^{N- n+1}(\bar {\mathfrak B}^ n)\) near a boundary point \(p\in b{\mathfrak B}^ n\) is rational of degree at most \(N^ 2(N-n+1).\) This generalizes the theorem of H. Alexander [Math. Ann. 209, 249-256 (1974; Zbl 0272.32006)] which asserts that every proper holomorphic map \(f: {\mathfrak B}^ n\to {\mathfrak B}^ n\) for \(n>1\) is an automorphism of \({\mathfrak B}^ n\) and thus a Möbius map. Special cases have been obtained previously by Webster, Faran, Cima and Suffridge, and the author.

MSC:

32D15 Continuation of analytic objects in several complex variables
32H35 Proper holomorphic mappings, finiteness theorems
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References:

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