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On locally biholomorphic finitely valent mappings from multi-connected domains onto the open disc
Abstract
In this paper we offer new results of research presented in the referred papers of J.E. Fornaess and E.L. Stout, E. Ligocka and the authors, and concerning the existence of m-valent locally biholomorphic mappings from product domains of Cn onto n-dimensional complex manifolds. In particular, we confirm an own conjecture about the estimation of the valentness m of locally biholomorphic mappings from multi-connected domains onto the open unit disc.
In 1954 M. Heins proved that, for every analytic set A containing the infinity, there exists an entire function whose set of asymptotic values at the infinity equals A. We obtain analogs of this result for functions analytic in planar domains of arbitrary connectivity.
We continue E. Ligocka’s [Ann. Pol. Math. 82, No. 2, 127–135 (2003; Zbl 1057.30019)] investigations concerning the existence of m-valent locally biholomorphic mappings from multi-connected onto simply connected domains. We decrease the constant m, and also give the minimum of m in the case of mappings from a wide class of domains onto the complex plane ℂ.
Sufficient conditions were found in [1, 2] for a domain in the complex plane to admit a finitely valent locally biholomorphic
mapping of the whole plane. In this article we find the relevant necessary and sufficient conditions. Also, we answer the
question that was posed by Aksent’ev and Ul’yanov in connection with the problem under consideration. The answer yields a
lower bound for the valency while generalizing a result of [3] to polydisks.
Mapping a polydisc onto a complex manifold
- D H Bushnell
D.H. Bushnell, Mapping a polydisc onto a complex manifold, Senior Thesis, Princeton University Library, 1976.