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

##### MSC:
 32D15 Continuation of analytic objects in several complex variables 32E35 Global boundary behavior of holomorphic functions of several complex variables
Zbl 1106.32010
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