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

Liczberski, Piotr; Starkov, Victor V.

doi:10.4064/ap85-2-3

Cited by 6 publications

(22 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It turns out that there exist domains, for instance C \ {0}, that admit no finitely valent locally biholomorphic mapping onto C. However, Ligocka proved that every finitely connected domain D = C \ {a}, with a ∈ C, admits an m-valent and locally biholomorphic mapping onto C; she has been unable to estimate the valency m. The result of Ligocka is considerably strengthened in [2].…”

Section: Introductionmentioning

confidence: 87%

“…Definition [2]. Consider some domain D in C. Say that D has an isolated boundary fragment if at least one of the following three conditions is fulfilled:…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 97%

See 2 more Smart Citations

Locally biholomorphic mappings of multiconnected domains

Starkov

2007

Sib Math J

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 87%

“…Definition [2]. Consider some domain D in C. Say that D has an isolated boundary fragment if at least one of the following three conditions is fulfilled:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Locally biholomorphic mappings of multiconnected domains

Starkov

2007

Sib Math J

View full text Add to dashboard Cite

show abstract

“…It was shown in [4] that the domain D from Theorem 1 can be mapped onto C locally biholomorphically and at most 3-valently. The mapping function F : D → C is the composition of the univalent function ϕ from Theorem 1 and an 3-valent function g. Here 3-valence of a function F means that for every w ∈ C the equation F (z) = w has at most three solutions and it has exactly three solutions for some w ∈ C. For the polynomial Q(z) = z 3 − 3z the Riemann surface Q(C) contains all rays Γ a = Γ a ∩ {w ∈ C : |w| > ρ} for sufficiently large ρ > 2, here Γ a are rays from Gross' example.…”

mentioning

confidence: 99%

“…the point 0 ∈ ∂D 1 corresponds to the z 0 and some interval (α, β) 0 corresponds to Γ (see [4] for details). It follows from the Caratheodory theorem that ψ can be extended to homeomorphism from (D ∪ Γ) onto (ψ (D) ∪ (α, β)).…”

mentioning

confidence: 99%

Asymptotic Values of Functions, Analytic in Planar Domains

Ganenkova¹,

Starkov²

2013

Issues of Analysis

View full text Add to dashboard Cite

Abstract. In [1] W. Gross constructed the example of an entire function of infinite order whose set of asymptotic values is equal to the extended complex plain. We obtain an analog of Gross' result for functions, analytic in planar domains of arbitrary connectivity with isolated boundary fragment. Key words: analytic function, asymptotic value, isolated boundary fragment.2010 Mathematical Subject Classification: 30D40.In 1918 W. Gross constructed the example of an entire function Φ of infinite order whose set of asymptotic values is equal to C (see [1]). This means that for every a ∈ C there exists an open arc Γ a with endpoint at infinity such that limIn this note, using Gross' example, we construct a function, analytic in preassigned planar domain, having C as the set of asymptotic values at a given boundary point.

show abstract

Finitely valent locally biholomorphic mappings of multiconnected domains onto the plane

Graf

Starkov

2009

Sib Math J

View full text Add to dashboard Cite

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...

show abstract

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

Cited by 6 publications

References 3 publications

Locally biholomorphic mappings of multiconnected domains

Locally biholomorphic mappings of multiconnected domains

Asymptotic Values of Functions, Analytic in Planar Domains

Finitely valent locally biholomorphic mappings of multiconnected domains onto the plane

Contact Info

Product

Resources

About