BELL REPRESENTATIONS OF FINITELY CONNECTED PLANAR DOMAINS

JEONG, MOONJA AND MASAHIKO TANIGUCHI

Abstract. In this paper, we solve a conjecture of S. Bell in [2] affirmatively. Actually, we prove that every non-degenerate n-connected planar domain Ω, where n > 1, is representable as Ω = {|f| < 1} with a suitable rational function f of degree n. This result is considered as a natural generalization of the classical Riemann mapping theorem for simply connected planar domains.

1. Introduction and the main theorem

Recently, S. Bell posed the following problem ([2] and [3]).

Problem 1.1. Can every non-degenerate n-connected planar domain with n > 1 be mapped biholomorphically onto a domain of the form

\{|z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} | < r\}

with complex numbers a_{k} and b_{k}, and a positive r ?

Here and in the sequel, a non-degenerate n-connected planar domain is a subdomain Ω of the Riemann sphere ℂ̂ such that ℂ̂ − Ω consists of exactly n connected components each of which contains more than one point. In this note, we solve this problem affirmatively. Actually, we give a proof of the following assertion.

Theorem 1.2. Every non-degenerate n-connected planar domain with n > 1 is mapped biholomorphically onto a domain defined by

\{|z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} | < 1\}

with suitable complex numbers a_{k} and b_{k}.

Recall that every domain defined as in Theorem 1.2 has algebraic kernel functions. See Theorem 4.4 in [2]. This is one of the reasons why we consider such domains.

Remark 1.3. It is well known (cf. for instance [5]) that the reduced Teichmüller space T (Ω) of a non-degenerate n-connected planar domain Ω can be identified with the Fricke space of a Fuchsian model G of Ω. Since G is a free real Möbius group with n − 1 hyperbolic generators, T (Ω) is real (3n − 6)-dimensional.

Date: March 1, 2002.

1991 Mathematics Subject Classification. Primary 32G10, 32G15; Secondary 30C20, 30F60.

Key words and phrases. conformal representation, Ahlfors maps.

The second author was supported in part by Grant-in-Aid for Scientific Research (B)(2) 2001-13440047.

Such a Bell representation as in Theorem 1.2 contains 2n−2 complex, i.e. 4n−4 real, parameters. The reason why we need many more number of parameters in a Bell representation than Teichmüller parameters for T (Ω) is that every Bell representation of a domain is actually associated with an n-sheeted branched covering of the unit disk by Ω.

Remark 1.4. Such a space as H_{0,n} consisting of all branched coverings of ℂ̂ induced by rational functions of degree n with n > 1 is called a Hurwitz space ([6]). This space H_{0,n} is parametrized by 2n − 2 critical values (the images of critical points), and hence complex (2n − 2)-dimensional.

Actually, we show that every Ahlfors map on Ω can be considered as the restriction of a rational function of degree n to a suitable domain which can be identified with Ω. On the other hand, the set of all branched coverings of ℂ̂ induced from Bell representations can be considered as a subdomain of H_{0,n}. Thus to solve the following problem would be interesting.

Problem 1.5. Find the sublocus A_{0,n} of H_{0,n} which corresponds to the set of all Bell representations of non-degenerate n-connected planar domains such that the restrictions of the associated rational functions give Ahlfors maps.

The authors also thank the referee for his/her valuable suggestions and comments.

2. Proof of Theorem 1.2

First let a non-degenerate n-connected planar domain Ω be given. Then by using the classical Riemann mapping theorem n times if necessary, we can assume that the boundary of Ω consists of exactly n smooth simple closed curves. Fix a point a in Ω, and let f_{a} be the Ahlfors map associated to the pair (Ω, a). Here for the definition and properties of the Ahlfors maps, see for instance, [1]. In particular, f_{a} maps Ω properly and holomorphically onto the unit disk U. Moreover, f_{a} can be extended to a continuous map of the closure \overline{Ω} of Ω onto the closed unit disk so that every component γ_{j} of the boundary of Ω, where j = 1, ⋯, n, is mapped homeomorphically onto the unit circle.

Lemma 2.1. There is a compact Riemann surface R (without boundary) of genus 0 and a holomorphic injection ι of Ω into R such that

f_{a} ∘ ι^{−1}

can be extended to a meromorphic function, say F, on R.

Proof. Since there are only a finite number of zeros of f′_{a}, there is a positive constant ρ such that ρ < 1 and that

D = {ρ < |ζ| < 1},

where ζ is the complex coordinate on the target plane of the map f_{a}, contains no critical values (i.e. no images of the zeros of f′_{a} by f_{a}). Hence every component W_{j}, where j = 1, ⋯, n, of f_{a}^{−1}(D) is mapped biholomorphically onto D by the restriction f_{a}|_{W_{j}}, of f_{a} to W_{j}.

Now we construct a compact Riemann surface R by using the Ahlfors map f_{a} to attach disks to the exterior of Ω along each boundary curve. More precisely, we consider the disjoint union R of Ω and n copies V_{j} (j = 1, ⋯, n) of

V = {ρ < |ζ|} ∪ {∞}.

Identify every subdomain W_{j} of Ω with the subdomain D_{j} of V_{j} corresponding to D by the biholmorphic map corresponding to f_{a}|_{W_{j}}. Then the resulting set, which we denote by R = R/f_{a}, has a natural complex structure induced from those on Ω and on every V_{j}, and hence is a Riemann surface. Here the natural inclusion map ι of Ω into R is a holomorphic injection, and using the complex coordinate ζ_{j} on the copy V_{j} corresponding to ζ on V, we have

f_{a} ∘ ι^{−1}(ζ_{j}) = ζ

on D_{j} by the definition.

Now, since topologically R is obtained from Ω by attaching a disk along each boundary curves of Ω, R is a simply connected compact Riemann surface without boundary, and hence in particular, is of genus 0. Also we can extend F = f_{a} ∘ ι^{−1} to a meromorphic function on the whole R by setting F (ζ_{j}) = ζ and F (∞) = ∞ on the whole V_{j} for every j.

Here the following uniformization theorem (which is also called the generalized Riemann mapping theorem) is classical and well-known. As references, we cite for instance [4] and [5].

Proposition 2.2 (Klein, Koebe and Poincaré). Every simply connected Riemann surface is mapped biholomorphically onto one of

• the unit disk U,

• the complex plane ℂ, and

• the Riemann sphere ℂ̂.

Corollary 2.3. There is a biholomorphic map h of the above Riemann surface R onto the Riemann sphere ℂ̂, and hence F ∘ h^{−1} is a rational function.

Proof. Since R is compact, R is mapped by a biholomorphic map h onto the Riemann sphere. Set f = F ∘ h^{−1}. Then f is meromorphic on the whole ℂ̂, which implies that f is a rational function.

Here and in the sequel, we may assume that

f(∞) = ∞

by applying to f the pre-composition of a Möbius transformation S which sends ∞ to a pole of f, i.e. by replacing h by S^{−1} ∘ h, if necessary.

Lemma 2.4. Let w be the complex variable of the above rational function f. Then f has the following partial fraction decomposition:

f(w) = Cw + D + ∑_{k=1}^{n−1} \frac{A_{k}}{w − B_{k}} .

Here A_{k}, B_{k}, C and D are complex constants, every A_{k} and C are non-zero, and {B_{k}} are mutually distinct.

Proof. Since f has exactly n simple poles, as is seen from the construction, and one of them is ∞ by the above assumption, f is of degree exactly n and has n − 1 finite, mutually distinct, simple poles, say B_{1}, ⋯, B_{n−1}. Hence we can write f(w) as

f(w) = \frac{P (w)}{Q(w)}

with polynomials P (w) of degree exactly n and

Q(w) = (w − B_{1}) ⋯ (w − B_{n−1}).

Thus it is easy to see that the partial fraction decomposition of f is as claimed.

Proof of Theorem 1.2. We replace the complex variable w of f by

z = T (w) = Cw + D

by applying to f the precomposition by an affine transformation T. Further set

a_{k} = CA_{k}, b_{k} = CB_{k} + D

for every k. Then we conclude that

f ∘ T^{−1}(z) = z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} .

Thus the Bell representation

\{|z + ∑_{k=1}^{n−1} \frac{a_{k}}{z − b_{k}} | < 1\}

is the subdomain

{|f ∘ T^{−1}(z)| < 1} = {|F ∘ h^{−1} ∘ T^{−1}(z)| < 1} = (T ∘ h ∘ ι)(Ω)

of ℂ̂, which is mapped biholomorphically onto Ω by the holomorphic injection (T ∘ h ∘ ι)^{−1}.

References

1. S. R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping, CRC Press, Boca Raton, 1992.

2. S. R. Bell, Finitely generated function fields and compelxity in potential theory in the plane, Duke Math. J., 98 (1999), 187–207.

3. S. R. Bell, A Riemann surface attached to domains in the plane and complexity in potential theory, Houston J. Math., 26, (2000), 277–297.

4. H. M. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag, 1980.

5. Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces, Springer-Verlag Tokyo, 1992.

6. S. M. Natanzon, Hurwitz spaces, Topics on Riemann surfaces and Fuchsian Groups, LMS Lecture notes 287, (2001), 165–177.

Department of Mathematics, The University of Suwon, Suwon P.O.Box 77, Kyungkido, 440-600, Korea

  E-mail address: mjeong@mail.suwon.ac.kr

Department of Mathematics, Graduate school of Science, Kyoto University, Kyoto 606, Japan

  E-mail address: tanig@kusm.kyoto-u.ac.jp