# zbMATH — the first resource for mathematics

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
##### Keywords:
product domains; biholomorphic mapping
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.