On locally biholomorphic mappings from multi-connected onto simply connected domains

Abstract

Ligocka [Lig] studied the problem: which domains D C can be mapped locally biholomorphically onto C or : She nedeed such mapping from D onto C to decide if each open Riemann surfaceX is a Riemann domain over the whole plane C. Ligocka generalized a GunningNarasimhan result from [GN] and proved that for every domain D C there exists a locally biholomorphic mapping from D onto C. Moreover, if D is nitely connected, not biholomorphic to C n f0g; then there exists an m-valent;m 2 N; locally biholomorphic mapping from D onto C. During the talk the author showed that for a class of domains (with an isolated boundary fragment of type I; II orIII), wider than the class of nitely connected domains, there exist a universal bound m M for the m valence of locally biholomorphic mapping from D onto C, and M = 3 is the best possible such constant [LS1], [Sta]. The case f(D) = refers to the following Fornaess-Stout result [FS]: For every paracompact connected n dimensional complex manifold X there exists a locally biholomorphic mapping from the open unit polydisc n onto X with the property that every bre f (x); x 2 X; consists of not more than (2n + 1)4 + 2 points. Ligocka [Lig] replaced the polydisc n in this result by a Cartesian product D1 ::: Dn; of multi-connected domains Dj; j = 1; :::; n; but at a cost of worse estimation of the valence: m (24)[(2n+1)4+2]: This result follows from her theorem that each domain D C, whose complement C nD has an isolated component not a singleton, can be mapped onto locally biholomorphically and m valently, where m 24: During the talk the author showed that for a class of domains with an isolated boundary fragment of the type I or II there exist a universal bound m M for the m valence of locally biholomorphic mapping from D onto , and M = 3 (see [LS2], [Sta]). Hence, also the result: If X = D1 ::: Dn; where domains Dj; j = 1; :::; n; ful l the assumptions of the previous result, and Y is a connected paracompact n dimensional complex manifold, then there exists a locally biholomorphic and m valent mapping f from domain X onto manifold Y and m 3[(2n+ 1)4 + 2]):