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.


32D15 Continuation of analytic objects in several complex variables
32H35 Proper holomorphic mappings, finiteness theorems
Full Text: DOI EuDML


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