## 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

### Keywords:

proper holomorphic map; holomorphic extension

### Citations:

Zbl 0272.32006; Zbl 0281.32019
Full Text:

### References:

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