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Uniformization of strictly pseudoconvex domains. I. (English. Russian original) Zbl 1106.32010
Izv. Math. 69, No. 6, 1189-1202 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 6, 115-130 (2005).
Let \(D\), \(D'\) be strictly pseudoconvex Stein domains with real analytic boundaries.
The aim of the paper and its continuation [part II, Izv. Math. 69, No. 6, 1203–1210 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 6, 131–138 (2005; Zbl 1106.32011)] is to prove the following theorem. The universal coverings of \(D\) and \(D'\) are biholomorphic iff \(\partial D\) and \(\partial D'\) are locally biholomorphically equivalent. The main result of the present paper gives the “if” part of the above equivalence: Any local equivalence between \(\partial D\) and \(\partial D'\) extends to a biholomorphism from the universal covering of \(D\) to the universal covering of \(D'\). In the generic case when \(\partial D\) and \(\partial D'\) are non-spherical, any local equivalence between \(\partial D\) and \(\partial D'\) extends to a biholomorphism from the universal covering of \(\overline D\) to the universal covering of \(\overline D'\). If \(\partial D\) is spherical, then \(D\) is universally covered by the unit ball.

MSC:
32D15 Continuation of analytic objects in several complex variables
32D10 Envelopes of holomorphy
32E10 Stein spaces
Citations:
Zbl 1106.32011
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