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Biholomorphic maps of the direct product of domains. (English. Russian original) Zbl 0629.32024
Math. Notes 41, 469-472 (1987); translation from Mat. Zametki 41, 824-828 (1987).
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 [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 D. K. Tishabaev [Tes. Dokl. Vs. Sem. Molod. Uchen., Tashkent (1985)].
Reviewer: J.Davidov

MSC:
32H99 Holomorphic mappings and correspondences
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References:
[1] W. Rudin, Theory of Functions in a Polydisc [Russian translation], Mir, Moscow (1974). · Zbl 0294.32001
[2] E. Ligocka, ?How to prove Fefferman’s theorem without use of differential geometry,? Ann. Polon. Math.,39, 117-130 (1981). · Zbl 0489.32016
[3] Dzh. K. Tishabaev, ?Invariant metrics and biholomorphically equivalent domains in Cn,? in: Abstracts of Reports to the All-Union Seminar of Young Scholars ?Current Questions of Complex Analysis? [in Russian], Takent (1985).
[4] B. V. Shabat, Introduction to Complex Analysis [in Russian], Part 2, Nauka, Moscow (1985). · Zbl 0574.30001
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