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Revista Colombiana de Matemáticas

versión impresa ISSN 0034-7426

Rev.colomb.mat. vol.51 no.1 Bogotá ene./jun. 2017

https://doi.org/10.15446/recolma.v51n1.66832 

Originals articles

On analytic families of conformai maps

Sobre familias analíticas de mapeos conformes

Jochen Becker1 

Christian Pommerenke2 

1 Institut für Mathematik, Technische Universitát Berlin, D-10623 Berlin, Germany. e-mail: becker@math.tu-berlin.de

2 Institut für Mathematik, Technische Universitát Berlin, D-10623 Berlin, Germany. e-mail: pommeren@math.tu-berlin.de


ABSTRACT:

Let Λ be a domain in ℂ and let ... be meromorphic in We assume that is holomorphic in for fixed z.

The main theorem states: Let Λ 0 be a subdomain of Λ such that f λ is univalent in for . If has a quasiconformal extension to the closure of for one then f λ has a quasiconformal extension for all .

This result is related to a theorem of Mañé, Sad and Sullivan (1983) where the assumptions are however different. The main tool of our proof is the Grunsky inequality for univalent functions.

Key words and phrases: Univalent function; quasiconformal extension; analytic parameter; Grunsky inequality

RESUMEN:

Sea Λ a dominio en ℂ y sea ... meromorfa en :=. Suponemos que es holomorfa en para z fijo.

El teorema principal dice: Sea Λ0 un subdominio de Λ tal que es univalente en para. Si tiene una extensión cuasiconforme a la clausura de para un entonces f λ tiene una extensión cuasiconforme para todo .

Este resultado está relacionado a un teorema de Mañe, Sad y Sullivan (1983) donde sin embargo las hipótesis son diferentes. Para nuestra demostración la herramienta principal es la desigualdad de Grunsky para funciones univalentes.

Palabras y frases clave: Funciones univalentes; extensión cuasiconforme; para-metro analítico; desigualdad de Grunsky

1. Introduction

In 1983, Mané, Sad and Sullivan proved the following surprising result. Suppose that

Then each fλ can be extended to a quasiconformal homeomorphism of into . See [6] and see [10] and [1] for further results.

We shall consider a different but related set of assumptions. Let Λ be a domain in C. We write :=. Let the function be defined for. We assume that

We will often write instead of . The assumption (3) corresponds to (b) in (1) whereas assumption (2) is quite different from (a). The initial condition (c) has no counterpart.

We need the Hartogs theorem of the theory of several complex variables, see e.g. [2, p.140]. We write it in a form adapted to our present context.

Proposition 1.1. Let (2) and (3) be satisfied. Then the function

is holomorphic except in z = ∞ and therefore continuous in every compact subset of.

2. Univalence

A complex-valued function is called univalent if it is injective and meromorphic in a domain in We define

Since by assumption (2) we can therefore write

Since the coefficients ak in (2) are holomorphic in A it follows that the coefficients are defined and holomorphic for see [7, p.58]. The Grunsky inequality states that

see [4] [7, p.60] [3, p.122]. It easily follows from (5) and (6) that

see e.g. [7, p.59].

Theorem 2.1. Let (2) and (3) be satisfied. Then U is relatively closed. For every component C of the open set Λ \ U we have

Proof. (i) Let Then there are such that as It follows from Proposition 1.1 that

locally uniformly in .

Now fλn is univalent by (4). Since fλo is non-constant by assumption (2) it follows that fλ0 is univalent in so that.

(ii) If C is unbounded then obviously. Now let C be a bounded component and suppose that so that. Since C is a component of Λ \ U we have.It follows, by (4) and (7), that 1 for. Since the are holomorphic in C we conclude by the maximum principle that the inequality 1 also holds for. Hence we obtain from (7) that f λ is univalent in D * so that by (4). This contradicts.

Remark 2.2. . If A is simply connected then Theorem 2.1 implies that every component of the open kernel U° is simply connected. If in addition then Λ \ U is a domain.

Example 2.3. Let p(λ) be a non-constant entire function and let. Then (2) is satisfied with . We have

If 1 then has the zero in so that fλ is not univalent. If 1 then 0 so that fλ is univalent. Hence we have. Since p is arbitrary this provides us with a huge variety of closed sets where fλ is univalent. If p is a non-constant polynomial then U is compact.

If we choose p(λ) = λ2- 1 then U is the classical lemniscate . If we choose p(λ) = exp(λ2) then U is the unbounded closed set which consists of two quarter-planes that meet at 0.

3. Quasiconformality

We assume that our conditions (2) and (3) are satisfied. Let U be defined by (4) and b k,l (λ) by (5).

Proposition 3.1. (Schiffer and Springer). Let. The function fλ has a quasiconformal extension to if and only if there exists K < 1 such that

See [8] [9] [5] [7,Th.9.12,Th.9.13].

Theorem 3.2. Let Λo be a subdomain of Λ and let fλ be univalent infor. If there exists such that has a quasiconformal extension to then fλ has a quasiconformal extension tofor every.

Let V be any component of . An obvious consequence of this theorem is that f λ has a quasiconformal extension either for all or for no. This raises an interesting question: If f λ has a quasiconformal extension for some component V, does it follow that f λ has quasiconformal extension for all

Proof. Let. Then there is a simply connected domain G with G and . Let g map conformally onto G and let.

Now let with and

Since fλ is univalent in we obtain from (6) that for. Furthermore, since and φ is holomorphic in Λo it follows that for. Finally we have

by our assumption and by Proposition 3.1.

The hyperbolic metric in is defined by

We have because and therefore

It follows that

References

[1] L. Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), 259-286. [ Links ]

[2] S. Bochner and W. T. Martin, Several complex variables, Princeton Univ. Press, 1948. [ Links ]

[3] P. L. Duren, Univalent functions, Springer, New York, 1983. [ Links ]

[4] H. Grunsky, Koeffizientenbedingungen fur schlicht abbildende meromorphe Funktionen, Math. Z. 45 (1939), 29-61. [ Links ]

[5] R. Kiihnau, Verzerrungssatze und Koeffizientenbedingungen vom Grunskyschen Typ fur quasikonforme Abbildungen, Math. Nachr. 48 (1971), 77-105. [ Links ]

[6] R. Mañe, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 (1983), 193-217. [ Links ]

[7] Ch. Pommerenke, Univalent functions, with a chapter on quadratic differentials by G.Jensen, Vandenhoeck and Ruprecht, Giittingen, 1975. [ Links ]

[8] M. Schiffer, Fredholm eigenvalues and conformai mapping, Rend. Mat. 22 (1963), 447-468. [ Links ]

[9] G. Springer, Fredholm eigenvalues and conformal mapping, Acta Math . 111 (1964), 121-142. [ Links ]

[10] D. Sullivan and W. P. Thurston, Extending holomorphic motions, Acta Math . 157 (1986), 243-257. [ Links ]

2010 Mathematics Subject Classification. 30C55, 30C62, 32A10.

Received: July 2016; Accepted: November 2016

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