## Proper holomorphic maps between balls in one co-dimension.(English)Zbl 0699.32014

A proper holomorphic map cannot lower the dimension. When $$f: B^ N\to B^ N$$ is proper (here, $$B^ N$$ denotes the unit ball in $${\mathbb{C}}^ N)$$ then f is an inner automorphism of $$B^ N$$ [H. Alexander, Indiana Univ. Math. J. 26, 137-146 (1977; Zbl 0391.32015)], and in particular it extends holomorphically to a neighbourhood of $$B^ N$$. Thus for proper $$f: B^ N\to B^ M$$ with bad boundary behavior, $$M>N$$. Such maps have been constructed by various authors, but always with M very big relative to N. The author constructs a proper holomorphic map $$f: B^ N\to B^{N+1}$$ $$(N\geq 2)$$ that extends continuously to $$\overline{B^ N}$$, but that cannot be extended in a $$C^{\infty}$$ way to any open, nonempty subset of the boundary.
Reviewer: E.Straube

### MSC:

 32H35 Proper holomorphic mappings, finiteness theorems 32E35 Global boundary behavior of holomorphic functions of several complex variables

### Keywords:

proper mapping between balls; boundary behavior

Zbl 0391.32015
Full Text:

### References:

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