Uniformization of strictly pseudoconvex domains. I. (English. Russian original) Zbl 1106.32010

Nemirovskii, S. Yu.; Shafikov, R. G.

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.

Reviewer: Marek Jarnicki (Kraków)

MSC:

32D15	Continuation of analytic objects in several complex variables
32D10	Envelopes of holomorphy
32E10	Stein spaces

Citations:

Zbl 1106.32011