Biholomorphic maps of the direct product of domains
The following theorem is proved: Let \(D_ i\subset {\mathbb{C}}^{k_ i}\), \(1\leq i\leq m\), and \(G_ j\subset {\mathbb{C}}^{l_ j}\), \(1\leq j\leq n\), be bounded domains with \(C^ 2\)-boundary. Let \(f=(f^ 1,...,f^ n)\) be a biholomorphic mapping from \(D_ 1\times...\times D_ m\) onto \(G_ 1\times...\times G_ n\). Then \(m=n\) and the domains \(G_ j\) can be renumbered in such a way that \(l_ j=k_ j\), \(f^ j\) depends on \(z^ j=(z^ j_ 1,...,z^ j_{k_ j})\) only, and \(f^ j\) biholomorphically maps \(D_ j\) onto \(G_ j\). The proof is based on a study of the boundary behaviour of \(f^ j\) on intersections of \(D_ j\) with complex lines. This theorem was obtained by E. Ligocka in the case when each \(D_ j\) and each \(G_ j\) belongs to one of the following classes: plane domains with \(C^ 2\)-boundary, circular complete strictly starlike domains with \(C^ 2\)-boundary, strictly pseudoconvex domains with \(C^{\infty}\)-boundary, pseudoconvex domains with real analytic boundary [\textit{E. Ligocka}, Bull. Acad. Pol. Sci., Sér. Sci. Math. 28, 319-323 (1980; Zbl 0488.32009)]. For convex domains with \(C^{\infty}\)- boundary the same result was proved by \textit{D. K. Tishabaev} [Tes. Dokl. Vs. Sem. Molod. Uchen., Tashkent (1985)].
scientific article; zbMATH DE number 4023703
scientific article