Finitely valent locally biholomorphic mappings of multiconnected domains onto the plane

Abstract

We give a final solution to the problem of the possibility of a finitely valent locally biholomorphic mapping from an arbitrary multiconnected domain on a complex plane onto the entire complex plane with the indication of the least valency constant.As is well known from Riemann's Theorem, every two simply connected domains D 1 and D 2 with boundary larger than a singleton are conformally equivalent; i.e., there exists a univalent conformal mapping of one domain onto the other. For multiconnected domains, it is impossible to guarantee the existence of a univalent conformal mapping from D 1 onto D 2 even if their connection orders coincide. However, the famous Poincaré Uniformization Theorem states that the universal covering of an arbitrary Riemann surface (including any multiconnected domain) is conformally equivalent to the unit disk, complex plane, or Riemann sphere depending on the conformal type of the surface. This, in particular, implies the existence of a locally biholomorphic mapping of the circle, plane, or Riemann sphere onto the surfaces corresponding to them by the conformal type of the surface. Here the inverse mappings are in general not single-valued.In this article, we consider the problem of the possibility of a finitely valent locally biholomorphic mapping from an arbitrary multiconnected domain onto the complex plane C.Below by a locally biholomorphic mapping of a domain D we will understand a single-valued mapping that is conformal and univalent at some neighborhood of every point z ∈ D. A mapping f from D onto Ω is called m-valent if f −1 (w) consists of at most m points for all w ∈ Ω.The Uniformization Theorem does not give an estimate of the valency of locally biholomorphic mappings from the universal coverings onto Riemann surfaces. Moreover, under these mappings, each point on the Riemann surface must be covered the same number of times, which implies that, for example, the universal coverings of any not simply connected hyperbolic Riemann surfaces (in particular, of the multiconnected domains different from the punctured plane) are countably valent.It is also known that some multiconnected domains (for example, C\{0}) admit no finitely valent and locally biholomorphic mapping onto C [1].The problem of estimating the minimal valency of a locally biholomorphic mapping of an arbitrary domain D ⊂ C onto the complex plane C was raised in [1] by Ligocka in connection with the question whether every open Riemann surface is a Riemann domain over the entire plane C. Ligocka proved that every finitely connected domain D different from a punctured plane admits a finitely valent locally biholomorphic mapping onto C. However, the valency constant was not estimated.Afterwards, this result by Ligocka was considerably strengthened in [2,3]. In particular, it is proved in [2] (see Theorem A below) that, for some class of domains D with an isolated boundary fragment, defined in the sequel, there exist 3-valent mappings of D onto C.Definition [2]. Let D be a domain in C. Say that D has isolated boundary fr...