Uniformization of strictly pseudoconvex domains. II. (English. Russian original) Zbl 1106.32011

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

Izv. Math. 69, No. 6, 1203-1210 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 6, 131-138 (2005).

Let \(D\), \(D\) be strictly pseudoconvex Stein domains with real analytic boundaries.
The aim of the paper and the former article [part I, Izv. Math. 69, No. 6, 1189–1202 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 6, 115–130 (2005; Zbl 1106.32010)], 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 “only if” part of the above equivalence: If the universal coverings of \(D\) and \(D'\) are not biholomorphic to the unit ball, then any biholomorphism between them extends to a biholomorphism of the universal coverings of \(\overline D\) and \(\overline D'\). If \(D\) is covered by the unit ball, then \(\partial D\) is spherical.

Reviewer: Marek Jarnicki (Kraków)

MSC:

32D15	Continuation of analytic objects in several complex variables
32E35	Global boundary behavior of holomorphic functions of several complex variables

Citations:

Zbl 1106.32010