Proper holomorphic maps between balls in one co-dimension
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 \(B^ N\) [\textit{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.
scientific article; zbMATH DE number 4146397
scientific article