Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls
This paper contains some interesting results on the topics stated in the title. In an earlier paper the author had shown that the unit ball \(B_ n\) in \({\mathbb{C}}^ n\) can be embedded as a closed complex submanifold of a polydisc in some higher dimension. In the first part of the present paper he generalizes this result from the ball to an arbitrary strictly pseudoconvex domain with \(C^ 2\) boundary. This result is based on the Hakim-Sibony-Løw construction of inner functions on the ball, and its extension to strictly pseudoconvex domains [the author, Math. Zeitschr. 185, 191-210 (1984; Zbl 0508.32005)]. In the second part the author proves that every strictly pseudoconvex domain with \(C^ 2\) boundary also admits a proper holomorphic embedding into some higher dimensional ball, which extends continuously to the boundary. Again, the construction is quite explicit, based on modifications of the techniques used for the embedding into polydiscs. This latter result was also obtained independently by \textit{F. Forstneric} [Preprint Ser. Dept. Math., Univ. Ljubljana, 23, no. 153 (1985)] and Hakim and Sibony (unpublished); these authors also give examples of proper holomorphic maps \(B_ n\to B_ N\), \(N>n\), which do not extend continuously to the boundary.
proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls
Hakim-Sibony-Løw construction of inner functions
scientific article; zbMATH DE number 3935476