# Koebe uniformization for 'almost circle domains.'

1. Introduction. In a recent paper [HS], we have proved the Koebe uniformization conjecture for countably connected domains: Any domain in the Riemann sphere [Mathematical Expression Omitted] with countably many boundary components is conformally homeomorphic to a circle domain; i.e. to a domain in [Mathematical Expression Omitted] whose boundary components are all circles and points.

Since our proof uses Baire category, it cannot be generalized directly to domains with uncountably many boundary components. However, if all boundary components of a domain are circular except those in a countable and closed subfamily, then the Koebe uniformization can still be proved. Precisely, we have the following theorem.

MAIN THEOREM. Let [Omega] be a domain in [Mathematical Expression Omitted] and let [B.sub.nc] ([Omega]) be the collection of all boundary components of [Omega] which are not circles or points. If the closure of [B.sub.nc]([Omega]) in the space B([Omega]) of all boundary components of [Omega] is (at most) countable, then [Omega] is conformally homeomorphic to a circle domain in [Mathematical Expression Omitted].

Let A be any domain [Mathematical Expression Omitted]. A domain [Omega] is contained in A is called a relative circle domain in A if each (connected) component of A - [Omega] is either a closed disk or a point. Clearly, a relative circle domain in A is a circle domain if and only if A is a circle domain. The main theorem is equivalent to the following statement: If A has at most countably many boundary components, then any relative circle domain in A is a conformally homeomorphic to a circle domain. In the case that A is a simply connected domain in the plane which is not the whole plane, our argument yields the following generalization to the Riemann mapping theorem:

RIEMANN MAPPING THEOREM FOR RELATIVE CIRCLE DOMAINS. Let A [subset or equal to] C be a simply connected domain with A [not equal to] C. Let [Omega] be a relative circle domain in A. Then there are a relative circle domain [[Omega].sup.*] in the unit disk U = {z : [absolute value of z] [less than] 1} and a conformal homeomorphism f : [Omega] [right arrow] [[Omega].sup.*] such that [f.sup.B] ([Delta]A) = [Delta]U (i.e. the boundary components [Delta]A and [Delta]U correspond under f).

The proof of this theorem is contained in the proof of Lemma 6.3. In a subsequent paper [HS2], we will prove the following inverse theorem.

INVERSE RIEMANN MAPPING THEOREM FOR RELATIVE CIRCLE DOMAINS. Let [Omega] be a relative circle domain in the unit disk U; and let A [subset or equal to] C be a simply connected domain with A [not equal to] C. Then there are a relative circle domain [[Omega].sup.*] in A and a conformal homeomorphism g: [Omega] [right arrow] [[Omega].sup.*] such that [g.sup.B] ([Delta]U) = [Delta]A.

The proof of the Main Theorem uses the same method as the proof in [HS]. The major point here is that all the rigidity results which were essential in the proof of the existence require some extra hypotheses when there are uncountably many components. Here the Ahlfors-Bers quasiconformal deformation theory for Kleinian groups and rigidity theorems of Sullivan [Su] will be used to show that the Fixed Point Index Lemma of [HS] holds for conformal mappings of circle domains with a certain extension property. As in [HS], this lemma yields all the necessary rigidity results.

We begin in section 2 by recalling the Ahlfors-Bers theory of quasiconformal deformations of Schottky groups associated to circle domains. A basic uniqueness result (Lemma 2.2) is derived as a consequence of Sullivan's rigidity theorems [Su]. In section 3, we will study various convergence properties for circle domains and quasiconformal mappings between them. In particular, we will show that a quasiconformal homeomorphism of arbitrary circle domains with a certain extension property can be approximated by quasiconformal mappings between finitely connected circle domains. This prepares us to the generalization of the Fixed Point Index Lemma done in section 4. In section 5, we will present the generalized Schwarz Pick Lemma, the Angle Lemma, the Uniqueness Theorem, and derive several immediate consequences of these. Finally, the proof of the Main Theorem will be completed in section 6.

We will assume that the readers are familiar with the notation and methods of the paper [HS].

2. Quasiconformal deformations of circle domains. Let [Omega] be a circle domain in [Mathematical Expression Omitted]. For any circle C in [Delta][Omega], let [Mathematical Expression Omitted] be the inversion on C. Then {[[Gamma].sub.c]; C [subset or equal to][Delta][Omega]} generate a free, discrete group of Mobius transformations(1), called the Schottky group associated to the circle domain [Omega]. It is denoted by [Gamma] = [Gamma]([Omega]).

Let [Mathematical Expression Omitted] be a Beltrami coefficient on [Mathematical Expression Omitted]; i.e., a measurable function with [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted] defines a "complex structure" on [Mathematical Expression Omitted]. Using the action of the group [Gamma]([Omega]), [Lambda] can be extended to [Mathematical Expression Omitted] with [Mathematical Expression Omitted] such that the actions of [Gamma] are conformal or anti-conformal with respect to the complex structure determined by [Mathematical Expression Omitted]. We will say such a Beltrami coefficient [Mathematical Expression Omitted] is invariant by [Gamma]. Extend [Mathematical Expression Omitted] to [Mathematical Expression Omitted] by letting [Mathematical Expression Omitted] outside [Mathematical Expression Omitted], then [Mathematical Expression Omitted] is still invariant by [Gamma].

Let [Mathematical Expression Omitted] be a quasiconformal homeomorphism which satisfies:

[Mathematical Expression Omitted].

It is well-known (see e.g. [AhB], [LV]) that such a map exists and is unique up to post-compositions with Mobius transformations. Let [Mathematical Expression Omitted]. Then by invariance of [Mathematical Expression Omitted] and uniqueness of [Mathematical Expression Omitted] is also a discrete group of Mobius transformations. We have:

LEMMA 2.1. [f.sub.[Lambda]]([Omega]) is a circle domain and [Mathematical Expression Omitted] is the associated Schottky group.

Proof. If C is a boundary component of [Omega], then [f.sub.[Lambda]](C) is a boundary component of [f.sub.[Lambda]]([Omega]), and vice versa. So it is enough to show that [f.sub.[Lambda]](C) is a point or a circle when C is a point or a circle in [Delta][Omega]. This is trivial if C is a point. So assume C is a circle. Then C is fixed by the inversion [[Gamma].sub.c] on C and [[Gamma].sub.c] [element of] [Gamma]. Since C is the fixed point set of [[Gamma].sub.c], [f.sub.[Lambda]](C) is the fixed point set of [Mathematical Expression Omitted]. As [Mathematical Expression Omitted] is Mobius and orientation reversing, it follows that [f.sub.[Lambda]](C) must be a circle. The second part of the lemma is clear, because [Mathematical Expression Omitted].

The domain [f.sub.[Lambda]]([Omega]) will be called the circle domain obtained from [Omega] by quasi-conformal deformation determined by [Mathematical Expression Omitted]. We will denote [[Omega].sub.[Lambda]] = [f.sub.[Lambda]]([Omega]). Note that for any given [Lambda], [[Omega].sub.[Lambda]] is well-defined up to Mobius equivalences.

The following fundamental uniqueness result follows easily from the rigidity results of Sullivan [Su].

LEMMA 2.2. Let [Omega] be a circle domain. Suppose [Mathematical Expression Omitted] is a quasiconformal mapping which maps [Omega] onto a circle domain [[Omega].sup.*]. Let [Lambda] be the complex dilation of g restricted to [Mathematical Expression Omitted]. Then, up to post-compositions with Mobius transformations, [Mathematical Expression Omitted].

Proof. From the hypothesis, we have

[Mathematical Expression Omitted]

We will change the map g such that it will conjugate [Gamma]([Omega]) to [Gamma]([[Omega].sup.*]). Let [Mathematical Expression Omitted] and let [Mathematical Expression Omitted] be the collection of circles on [Mathematical Expression Omitted]. Define [Mathematical Expression Omitted] by

[Mathematical Expression Omitted].

Next, let [Mathematical Expression Omitted] and let [Mathematical Expression Omitted] be the collection of circles on [Mathematical Expression Omitted]. Define [Mathematical Expression Omitted] by

[Mathematical Expression Omitted].

We repeat this process. For successive j, we construct a quasiconformal mapping [Mathematical Expression Omitted] and a subset [Mathematical Expression Omitted] by

[Mathematical Expression Omitted];

and [Mathematical Expression Omitted]. We have [g.sub.l] = [g.sub.j] on [Mathematical Expression Omitted] if l [greater than or equal to] j, and [g.sub.j] converge

to some quasiconfomal mapping [Mathematical Expression Omitted]. Clearly f([Omega]) = g([Omega]) = [[Omega].sup.*], and f conjugates the Schottky group [Gamma]([Omega]) to [Gamma]([[Omega].sup.*]): f[Gamma]([Omega])[f.sup.-1] = [Gamma]([[Omega].sup.*]).

Let [Mathematical Expression Omitted] be the complex dilatation of f. Since f = g on [Mathematical Expression Omitted], we have [Mathematical Expression Omitted] on [Mathematical Expression Omitted]. And because f conjugates [Gamma]([Omega]) to [Gamma]([[Omega].sup.*]), [Mathematical Expression Omitted] is invriant by the action of [Gamma]([Omega]). We will show that [Mathematical Expression Omitted] vanishes outside [Mathematical Expression Omitted]. Consider [Mathematical Expression Omitted] as the boundary at infinity of hyperbolic 3-space. The group [Gamma] = [Gamma]([Omega]) then acts by isometries on hyperbolic space. Then [Mathematical Expression Omitted] is equal to the "intersection" of a fundamental domain in the hyperbolic 3-space of [Gamma] with the sphere at infinity. Then by the first corollary of Theorem III of [Su, pp. 486-487], the action of [Gamma] on [Mathematical Expression Omitted] is conservative. By Theorem I (and II) of [Su], there is no invariant measurable tangent line field on the conservative part of the action of any Kleinian group on [Mathematical Expression Omitted]. Thus [Mathematical Expression Omitted] in [Mathematical Expression Omitted].

It follows that up to post-compositions with Mobius transformations, f = [f.sub.[Lambda]] and [[Omega].sup.*] = [[Omega].sub.[Lambda]] By construction, [Mathematical Expression Omitted].

COROLLARY 2.3. Let [Mathematical Expression Omitted] and [Mathematical Expression Omitted] be conformal homeomorphisms of circle domains which extend to some quasiconformal homeomorphisms [Mathematical Expression Omitted] respectively. If the complex dilatations of [g.sub.1] and [g.sub.2] are identical a.e. in [Mathematical Expression Omitted], then [f.sub.2] = T [convolution] for some Mobius transformation T of [Mathematical Expression Omitted].

The following lemma was proved by Sibner [Si].

LEMMA 2.4. Let [Omega][prime] be a domain which is quasiconformally homeomorphic to a circle domain. Then [Omega][prime] is conformally homeomorphic to a circle domain.

Proof. Let [Omega] be a circle domain and let g: [Omega] [right arrow] [Omega][prime] be a quasiconformal homeomorphism. Let [Lambda] be the complex dilatation of g. Extend [Lambda] to a function defined on [Mathematical Expression Omitted] by letting [Lambda] = 0 on [Delta][Omega]. Let [Mathematical Expression Omitted] and [[Omega].sub.[Lambda]] = [f.sub.[Lambda]]([Omega]) be constructed as in Lemma 2.1. Then [f.sub.([Lambda]]) is a circle domain and [f.sub.[Lambda]][g.sup.-1]:[Omega][prime] [right arrow] [f.sub.[Lambda]]([Omega]) is a conformal homeomorphism.

3. Convergence of circle domains. Let [[Omega].sub.k] be a sequence of circle domains which are conformally homeomorphic to a fixed circle domain [Omega]. Let [f.sub.k]: [Omega] [approaches] [[Omega].sub.k] be a sequence of conformal homeomorphisms. We say that the sequence [f.sub.k] is normalized, if there are three distinct points [z.sub.1], [z.sub.2], [z.sub.3] in [Omega] such that [f.sub.k]([z.sub.j]) = [z.sub.j] for each k and j = 1, 2, 3. If that is the case, then a subsequence of [f.sub.k] converges to some conformal injective mapping [Mathematical Expression Omitted]. (When we speak of convergence of a sequence of quasiconformal or conformal maps, we mean uniform converence on compacts in the spherical metric.) We are interested to find conditions which would imply that the image of [f.sub.[infinity]] is also a circle domain.

LEMMA 3.1. Suppose that the sequence [f.sub.k] above is normalized, and that there is a constant K [greater than or equal to] 1, such that each [f.sub.k] : [Omega] [approaches] [[Omega].sub.k] extends to a K-quasiconformal homeomorphism of [Mathematical Expression Omitted]. Then a subsequence of [f.sub.k] converges to a conformal homeomorphism [f.sub.[infinity]] : [Omega] [approaches] [[Omega].sub.[infinity]] where [[Omega].sub.[infinity]] is also a circle domain. Moreover, [f.sub.[infinity]] extends to a K-quasiconformal homeomorphism of [Mathematical Expression Omitted].

Proof. Let [Mathematical Expression Omitted] be the quasiconformal homeomorphism which extends [f.sub.k]. Let [Mathematical Expression Omitted] be the complex dilatation of [g.sub.k] on [Mathematical Expression Omitted]. Then by Lemma 2.2, we have [[Omega].sub.k] = [[Omega].sub.[[Lambda].sub.k]]. Replacing [g.sub.k] by the mapping [f.sub.[[Lambda].sub.k]], we may assume that [g.sub.k] conjugates the Schottky group [Gamma]([Omega]) to [Gamma]([[Omega].sub.k]).

A subsequence of [g.sub.k], still denoted by [g.sub.k], converges to a K-quasiconformal homeomorphism [Mathematical Expression Omitted], and then for each [Gamma] [element of] [Gamma]([Omega]), [g.sub.k][Gamma][[g.sub.k].sup.-1] converge to [Mathematical Expression Omitted]. As each [Mathematical Expression Omitted] is Mobius, [Mathematical Expression Omitted] must also be Mobius. It follows that [Mathematical Expression Omitted] is a group of Mobius transformations and an argument similar to Lemma 2.1 shows that [g.sub.[infinity]] ([Omega]) is a circle domain. The lemma follows because the corresponding subsequence [f.sub.k] converges to [g.sub.[infinity]][where][Omega] which can be taken as [f.sub.[infinity]].

We will show below that a certain class of quasiconformal mappings between circle domains with infinitely many boundary components can be approximated by quasiconformal mappings between circle domains with finitely many boundary components. This will be essential to the generalization of the Fixed Point Index Lemma of [HS] to mappings of circle domains with uncountably many boundary components. We begin with the following elementary lemma.

LEMMA 3.2. Let A be a domain in [Mathematical Expression Omitted]. Let [Mathematical Expression Omitted] be a sequence of uniformly quasiconformal mappings which converge to a quasiconformal mapping [Mathematical Expression Omitted]. Let [[Lambda].sub.k] : A [approaches] C denote the complex dilatation of [g.sub.k]. Suppose that lim [[Lambda].sub.k](z) where k [approaches] [infinity] exists for a.e. z [element of] B; where B [subset or equal to] A. Then the complex dilatation of [g.sub.[infinity]] equals lim [[Lambda].sub.k](z) where k [approaches] [infinity] for a.e. z [element of] B.

Proof. Since this is a local problem, we may assume that A and [g.sub.k](A) are all contained in the unit disk. Then the [L.sup.2]-norms of the first-order partial derivatives of [g.sub.k] are uniformly bounded (see e.g. [LV, Chap. IV]). So, these partial derivatives of [g.sub.k] converge weakly in [L.sup.2](A) to the corresponding derivatives of [g.sub.[infinity]]. On the other hand, by hypothesis, [Mathematical Expression Omitted] converge pointwise to some function, say, [Lambda][prime] : B [approaches] C. Since [absolute value of [[Lambda].sub.k](z)] [less than] 1, the Dominated Convergence Theorem implies that [[Lambda].sub.k] converge strongly in [L.sup.2](B) to [Lambda][prime]. As product of a strongly convergent sequence and a weakly convergent sequence in [L.sup.2](B), the sequence of functions [[Lambda].sub.k](z) [Delta][g.sub.k](z)/[Delta]z converges weakly-star to [Lambda][prime](z) [Delta][g.sub.[infinity]](z)/[Delta]z in [L.sup.1] (B). Letting k [approaches] [infinity] in the Beltrami equations

[Mathematical Expression Omitted], a.e. z [element of] B,

we obtain as functions in [L.sup.1](B),

[Mathematical Expression Omitted].

It follows that [Lambda][prime] is equal to the conformal dilatation of [g.sub.[infinity]] in B.

Let [[Omega].sub.[infinity]] be a circle domain in [Mathematical Expression Omitted], and let [C.sub.1], [C.sub.2],... be all the circles in [Delta][Omega]. For each k, let [[Omega].sub.k] be the domain bounded by [C.sub.1], [C.sub.2],..., [C.sub.k]. We have [[Omega].sub.k+1] [subset or equal to] [[Omega].sub.k].

Let [Mathematical Expression Omitted] be a quasiconformal homeomorphism between [[Omega].sub.[infinity]] and some other circle domain [Mathematical Expression Omitted]. Suppose that f extends to a quasiconformal homeomorphism g of [Mathematical Expression Omitted] such that the complex dilatation [Mathematical Expression Omitted] of g restricted to [Mathematical Expression Omitted] vanishes a.e. on [Delta][[Omega].sub.[infinity]]. Extend [Lambda] to a Beltrami coefficient [[Lambda].sub.k] : [Mathematical Expression Omitted] by letting

(3.1) [Mathematical Expression Omitted].

Note that [[Lambda].sub.k] = 0 a.e. in [Mathematical Expression Omitted]. As in [section]2, extend [[Lambda].sub.k] to [Mathematical Expression Omitted] by the Schottky group [Gamma]([[Omega].sub.k]) and let [Mathematical Expression Omitted] be a quasiconformal homeomorphism whose complex dilitation is [Mathematical Expression Omitted]. Then by Lemma 2.1, [g.sub.k]([[Omega].sub.k]) = [([[Omega].sub.k]).sub.[[Lambda].sub.k]] is a circle domain.

LEMMA 3.3. Let f and [g.sub.k] be maps constructed as above, and assume that they are normalized (in the sense of Lemma 3.1). Then [g.sub.k] [where] [[Omega].sub.[infinity]] converges uniformly to f (in the spherical metric).

Proof. Since [Mathematical Expression Omitted], [g.sub.k] are uniformly quasiconformal. By passing to a subsequence, we may assume that [g.sub.k] converges uniformly to some quasiconformal map [Mathematical Expression Omitted] (see e.g. [LV]). We have to show that [g.sub.[infinity]] [where] [[Omega].sub.[infinity]] = f.

We first claim that [g.sub.[infinity]] conjugates [Gamma]([Omega]) to a discrete group of Mobius transformations. In fact, for any circle [C.sub.k] on [Delta][Omega], the mapping [Mathematical Expression Omitted] is a Mobius transformation if j [greater than or equal to] k. Letting j [approaches] [infinity], we see that the mapping [Mathematical Expression Omitted] is Mobius; hence the claim.

By the argument of Lemma 2.1, it follows that [g.sub.[infinity]] ([Omega]) is a circle domain. By Lemma 2.2, in order to prove [g.sub.[infinity]] [where] [[Omega].sub.[infinity]] = f it is enough to show that the complex dilatation [Mathematical Expression Omitted] of [g.sub.[infinity]] is identical to [Lambda] when restricted to [Mathematical Expression Omitted]. But for any [Mathematical Expression Omitted], we have [Mathematical Expression Omitted], and hence by Lemma 3.2,

[Mathematical Expression Omitted].

4. The fixed point index lemma. Let A be a domain in [Mathematical Expression Omitted]. Recall that a domain [Omega] contained in A is a relative circle domain in A if each component of A - [Omega] is either a closed disk or a point. The following result is a restatement of Corollary 3.4 of [HS].

LEMMA 4.1 (Fixed Point Index Lemma). Let A and [A.sup.*] be two bounded finitely connected domains in C bounded by Jordan curves. Let [Omega] and [[Omega].sup.*] be relative circle domains in A and [A.sup.*] respectively. Suppose that [Omega] and [[Omega].sup.*] have at most countably many boundary components. Let [Mathematical Expression Omitted] be a conformal homeomorphism with no fixed point on [Delta]A. Let n be the index of f on [Delta]A. Then f has at most n fixed points (counting multiplicities) in [Omega].

This lemma can be generalized using Lemma 3.3.

LEMMA 4.2 (Generalized Fixed Point Index Lemma). Let A and [A.sup.*] be bounded domains in C bounded by finitely many Jordan curves. Let [Omega] and [[Omega].sup.*] be relative circle domains in A and [A.sup.*] respectively. Let [Mathematical Expression Omitted] be a conformal homeomorphism with no fixed point on [Delta]A. Let n be the index of f on [Delta]A. Suppose that f extends to a locally quasiconformal homeomorphism G : A [approaches] [A.sup.*], such that the complex dilatation of G vanishes a.e. in [Delta][Omega] [intersection] A, then the number of fixed points of f in [Omega], counting multiplicities, is at most n.

Remark. A homeomorphism g : A [approaches] [A.sup.*] of subsets in [Mathematical Expression Omitted] is called locally quasiconformal if it is quasiconformal in any domain B with [Mathematical Expression Omitted] interior(A). It is called quasiconformal if there is a uniform bound on its conformal dilatation in interior (A). It is called conformal if it is conformal in interior (A).

Proof. Replacing A by an appropriate subdomain of A, we may assume that: i) all boundary components of A and [A.sup.*] are smooth Jordan curves; ii) the components of [Delta]A are isolated boundary components of [Omega]; iii) the locally quasiconformal map G : A [approaches] [A.sup.*] is (globally) quasiconformal. Note that the index of f on [Delta]A is unchanged if the subdomain is close to A.

Since [Delta]A and [Delta][A.sup.*] are smooth, G : A [approaches] [A.sup.*] can be extended to a quasiconformal homeomorphism [Mathematical Expression Omitted]. Let [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Then clearly [[Omega].sub.[infinity]] and [Mathematical Expression Omitted] are circle domains, and [Mathematical Expression Omitted]. Let [Mathematical Expression Omitted] be the complex dilatation of g in [Mathematical Expression Omitted]. Then by the hypothesis of the lemma,

(4.1) [Mathematical Expression Omitted].

Let [C.sub.1], [C.sub.2],... be all the circles in [Delta][[Omega].sub.[infinity]]. Let [[Omega].sub.k] be the circle domain bounded by [C.sub.1], [C.sub.2],..., [C.sub.k]. Let [[Lambda].sub.k] : [[Omega].sub.k] [approaches] C be the complex dilatation defined by (3.1), and let [Mathematical Expression Omitted] and [Mathematical Expression Omitted] be constructed as in Lemma 3.3. Then by the same lemma, [g.sub.k] converge uniformly to f in [Mathematical Expression Omitted].

Since f has no fixed point on [Delta]A and its index on [Delta]A is n, for k big enough [g.sub.k] has no fixed point on [Delta]A and its index on [Delta]A is also equal to n. By (4.1) and (3.1), [g.sub.k] is conformal in A [intersection] [[Omega].sub.k]. On the other hand, by Lemma 2.1, [g.sub.k] maps the circle domain [[Omega].sub.k] onto a circle domain; hence it maps the relative circle domain A [intersection] [[Omega].sub.k] onto a relative circle domain in [g.sub.k](A). As [[Omega].sub.k] [intersection] A is finitely connected, we may use Lemma 4.1 to conclude that the number of fixed points of [g.sub.k] in A [union] [[Omega].sub.k], counting multiplicities, is at most n. As [Omega] [subset or equal to] [[Omega].sub.k] [intersection] A, the number of fixed points of [g.sub.k] in [Omega], counting multiplicities, is at most n. Since [Mathematical Expression Omitted], we deduce that the number of fixed points of f in [Omega], counting multiplicities, is at most n.

5. Schwarz Pick Lemma, the Angle Lemma and other corollaries. In [HS], the Schwarz Pick Lemma, the Angle Lemma, the Uniqueness Theorem, etc., were derived as corollaries of the Fixed Point Index Lemma. By the same argument, Lemma 4.2 implies the following generalizations.

LEMMA 5.1 (Schwarz Pick Lemma for relative circle domains). Let [Mathematical Expression Omitted] denote the open unit disk, and let A and [A.sup.*] be simply connected domains in [Mathematical Expression Omitted] such that A [contains or equal to] U [contains or equal to] [A.sup.*]. Let [Omega] and [[Omega].sup.*] be relative circle domains in A and [A.sup.*] respectively. Suppose that f : [Omega] [approaches] [[Omega].sup.*] is a conformal homeomorphism, and that [Delta]A corresponds to [Delta][A.sup.*] under f. If f extends to a locally quasiconformal homeomorphism g : A [approaches] [A.sup.*] whose complex dilatation vanishes a.e. on [Delta][Omega] [intersection] A, then for any p, q [element of] [Omega] [intersection] U,

(5.1) [d.sub.hyp](f(p), f(q)) [less than or equal to] [d.sub.hyp](p, q);

where [d.sub.hyp] denotes the Poincare metric in the unit disk. Furthermore, if equality holds for one pair p [not equal to] q, then A = [A.sup.*] and f is the restriction of a hyperbolic isometry of U.

LEMMA 5.2 (Angle Lemma). Let [Omega] be a relative circle domain in some Jordan domain [Mathematical Expression Omitted], and let [[Omega].sup.*] be a relative circle domain in a disk D (hence [[Omega].sup.*] is a circle domain). Let f : [Omega] [approaches] [[Omega].sup.*] be a conformal homeomorphism which extends to a locally quasiconformal homeomorphism [Mathematical Expression Omitted] whose complex dilatation vanishes a.e. in [Delta][Omega][intersection]A. Suppose that [z.sub.0] [element of] [Omega][intersection]D, and the boundary of the connected component of D [intersection] A which contains [z.sub.0] is the union of an arc [Alpha] [subset or equal to] [Delta]A [intersection] D and an arc [Beta] [subset or equal to] [Delta]D [intersection] A, as in Figure 5.1. then

angle (f([z.sub.0]), f([Alpha])) [greater than or equal to] angle ([z.sub.0], [Eta]);

where [Eta] is the arc of [Delta]D complementary to [Beta].

LEMMA 5.3 (Uniqueness Theorem). Let A and [A.sup.*] be circle domains with countably many boundary components. Let [Omega] and [[Omega].sup.*] be relative circle domains in A and [A.sup.*] respectively. Suppose that f : [Omega] [approaches] [[Omega].sup.*] is a conformal homeomorphism which extends to a locally quasiconformal homeomorphism F : A [approaches] [A.sup.*] whose complex dilatation vanishes in [Delta][Omega] [intersection] A. then f is the restriction of a Mobius transformation.

As a direct consequence of the Uniqueness Theorem, we have the following result.

COROLLARY 5.4. Let A and [A.sup.*] be circle domains with countably many boundary components. Let [Omega] and [[Omega].sup.*] be relative circle domains in A and [A.sup.*], respectively. Let g : A [approaches] [A.sup.*] be a locally quasiconformal mapping which maps [Omega] onto [[Omega].sup.*]. Let [Mathematical Expression Omitted] be the complex dilatation of g restricted to [Mathematical Expression Omitted]. Suppose that [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted] and [Mathematical Expression Omitted]; where [Mathematical Expression Omitted] is the quasiconformal homeomorphism which maps [Omega] onto [[Omega].sub.[Lambda]] as constructed in Lemma 2.1, and T is some Mobius transformation.

Proof. Since [Delta]A [subset or equal to] [Delta][Omega], [f.sub.[Lambda]](A) is also a circle domain. [[Omega].sub.[Lambda]] is clearly a relative circle domain in [f.sub.[Lambda]](A). Consider the mappings [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. It is easy to see that f and F satisfies the hypotheses of Lemma 5.3. It follows that [Mathematical Expression Omitted] is the restriction of a Mobius transformation, and the Corollary follows.

LEMMA 5.5 (Extension Property). Let J and [J.sup.*] be two smooth Jordan domains. Let A and [A.sup.*] be relative circle domains in J and [J.sup.*] respectively such that [Delta]A and [Delta][A.sup.*] have at most countably many boundary components. Let [Omega] and [[Omega].sup.*] be relative circle domains in A and [A.sup.*] respectively, such that [Delta]J is an isolated component of [Delta][Omega].

Let h : [Omega] [approaches] [[Omega].sup.*] be a conformal homeomorphism which extends to a locally quasiconformal homeomorphism H : A [approaches] [A.sup.*] such that [H.sup.B] ([Delta]J) = [Delta][J.sup.*] and the complex dilatation of H vanishes in [Delta][Omega] [intersection] J. Then h extends to a quasiconformal homeomorphism [Mathematical Expression Omitted] whose conformal dilatation depends only on the domain [Omega] and on the smoothness of [J.sup.*].

Proof. H extends to a quasiconformal homeomorphism [Mathematical Expression Omitted], because [Delta]J and [Delta][J.sup.*] are smooth and isolated in [Delta][Omega] and [Delta][[Omega].sup.*], respectively, and H is conformal in J near [Delta]J.

Let [Mathematical Expression Omitted]. Since [Delta]J is isolated in [Delta][Omega], and [Omega] is a relative circle domain in J, we deduce that [[Omega].sup.1] is a circle domain. Similarly, [Mathematical Expression Omitted] is also a circle domain. Let [Mathematical Expression Omitted] be the complex dilatation of g restricted to [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted].

Let [Mathematical Expression Omitted]. By Corollary 5.4, we have [Mathematical Expression Omitted], and hence [Mathematical Expression Omitted]. It is clear from the proof that the conformal dilatation of [Mathematical Expression Omitted] depends only on the conformal dilatation of the extension of H in [Mathematical Expression Omitted] which, in turn, depends only on the domain [Omega] and on the smoothness of [J.sup.*].

The following convergence result follows essentially by combining ideas of Lemmas 5.5 and 3.1.

LEMMA 5.6. Let J be a simply connected domain in [Mathematical Expression Omitted], let A be a subdomain with countably many boundary components such that [Delta]J is also a boundary component of A, and let [Omega] be a relative circle domain in A. Let [Mathematical Expression Omitted] be a sequence of simply connected domains and let [Mathematical Expression Omitted] be relative circle domains in [Mathematical Expression Omitted]. Suppose that [Mathematical Expression Omitted] is a normalized sequence of conformal homeomorphisms with [Mathematical Expression Omitted] and each [f.sub.k] extends to a locally quasiconformal mapping [Mathematical Expression Omitted] whose complex dilatation vanishes in [Delta][Omega] [intersection] A. Then a subsequence of [f.sub.k] converges to a conformal mapping f: [Omega] [approaches] [[Omega].sup.*] ; where [[Omega].sup.*] is a relative circle domain in some simply connected domain, say [J.sup.*], with [Delta][J.sup.*] = [f.sup.B]([Delta]J).

Proof. By taking a subsequence, we may assume that [f.sub.k] converges to a conformal homeomorphism f : [Omega] [approaches] [[Omega].sup.*]; where [[Omega].sup.*] is some domain in [Mathematical Expression Omitted]. Let [J.sup.*] be the simply connected domain containing [[Omega].sup.*] such that [f.sup.B]([Delta]J) = [Delta][J.sup.*]. We need show that any boundary component of [[Omega].sup.*] other than [Delta][J.sup.*] is either a circle or a point.

Let us first assume that A is a relative circle domain in J. By replacing J with an appropriate subdomain, we may assume that [Delta]J is a smooth Jordan domain and it is an isolated component of [Delta][Omega]. We may also assume that the conformal mappings [f.sub.k] extend conformally to a fixed neighborhood of [Delta]J, and hence [Mathematical Expression Omitted] are uniformly smooth curves. Since [F.sub.k] is identical to [f.sub.k] in a neighborhood of [Delta]J in [Mathematical Expression Omitted], it extends to a locally quasiconformal homeomorphism [Mathematical Expression Omitted], which is K quasiconformal in [Mathematical Expression Omitted], and the constant K can be chosen independent of k.

Let [Mathematical Expression Omitted]. Since [Delta]J is isolated in [Delta][Omega], and [Omega] is a relative circle domain in J, we deduce that [[Omega].sup.1] is a circle domain. Similarly, because [g.sub.k]([Omega]) = [f.sub.k]([Omega]) is a relative circle domain in [Mathematical Expression Omitted], we conclude that [g.sub.k]([[Omega].sup.1]) is a circle domain. Let [Mathematical Expression Omitted] be the complex dilatation of [g.sub.k] restricted to [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted]. We also have that [Mathematical Expression Omitted] are uniformly bounded away from 1.

By Corollary 5.4, we have [Mathematical Expression Omitted]. By passing to a subsequence, we may assume that [f.sub.[[Lambda].sub.k]] converges to some quasiconformal mapping [Mathematical Expression Omitted]. By Lemma 3.1, [f.sub.[[Lambda].sub.[infinity]]] maps [[Omega].sup.1] onto a circle domain, and hence maps [Omega] onto a relative circle domain in [f.sub.[[Lambda].sub.[infinity]]] (J). The lemma follows since [Mathematical Expression Omitted].

When A is not a relative circle domain in J, we may replace [Omega] by [Mathematical Expression Omitted], J by [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and A by [F.sub.1](A). Since [F.sub.1] ([Omega]) is a relative circle domain in [Mathematical Expression Omitted], [F.sub.1] (A) must be a relative circle domain in [Mathematical Expression Omitted].

6. Proof of the main theorem. Let [Omega] be a domain in [Mathematical Expression Omitted]. Let [B.sub.nc]([Omega]) be the collection of boundary components of [Omega] which are not circles or points. Let [Mathematical Expression Omitted] be the closure of [B.sub.nc]([Omega]) in B([Omega]). It is easy to see that there is a domain A such that [Omega] [subset or equal to] A and [Mathematical Expression Omitted]. Then [Omega] is a relative circle domain in A. If [Omega] satisfies the hypothesis of the Main Theorem, then A has countably many boundary components.

We will prove the following stronger result.

THEOREM 6.1. Let A be a domain in [Mathematical Expression Omitted], with countably many boundary components and let [Omega] be a relative circle domain in A. Then there is a circle domain [[Omega].sup.*] and a conformal homeomorphism f : [Omega] [approaches] [[Omega].sup.*]. Moreover, f extends to some locally quasiconformal mapping [Mathematical Expression Omitted] whose complex dilatation vanishes in [Delta][Omega] [intersection] A.

The proof of Theorem 6.1 will be divided into several steps.

LEMMA 6.2. Let A be a domain in [Mathematical Expression Omitted] bounded by a finite number of smooth Jordan curves. Let [Omega] be a relative circle domain in A such that each component of [Delta]A is isolated in [Delta][Omega]. Then there is a circle domain [[Omega].sup.*] and a conformal homeomorphism f : [Omega] [approaches] [[Omega].sup.*] which extends to a quasiconformal homeomorphism [Mathematical Expression Omitted] whose complex dilatation vanishes a.e. in [Delta][Omega].

Proof. Let [Mathematical Expression Omitted] be a diffeomorphism onto [Mathematical Expression Omitted] such that h = id outside a small tubular neighborhood of [Delta]A in A disjoint from [Delta][Omega] [intersection] A, and such that h maps the components of [Delta]A onto circles or points. Then h([Omega]) and h(A) are both circle domains. Clearly, h extends to a quasiconformal homeomorphism [Mathematical Expression Omitted]. Let [Mathematical Expression Omitted] be the complex dilatation of [Mathematical Expression Omitted]. We have [Lambda] = 0 in [Delta]h([Omega]) [intersection] h(A) = [Delta][Omega] [intersection] A because h = id near [Delta][Omega] [intersection] A. So [Lambda] = 0 a.e. in [Delta](h([Omega])) = ([Delta][Omega] [intersection] A) [union] h([Delta]A). Let [[Omega].sup.*] = [(h([Omega])).sub.[Lambda]] and [Mathematical Expression Omitted] be the corresponding quasiconformal homeomorphism which conjugate [Gamma](h([Omega])) to [Gamma]([[Omega].sup.*]). Then [f.sub.[Lambda]]h([Omega]) = [[Omega].sup.*] and [Mathematical Expression Omitted] is a conformal homeomorphism of [Omega] onto a circle domain. Moreover, [Mathematical Expression Omitted] extends f and its complex dilatation vanishes in [Delta][Omega].

The following is the key step in the proof of our main theorem.

LEMMA 6.3. Theorem 6.1 holds if A is simply connected.

Proof of Lemma 6.3. If [Mathematical Expression Omitted] consists of a point, then any relative circle domain in A is itself a circle domain and the theorem is trivial. So let us assume that [Mathematical Expression Omitted] has more than one point.

As in [HS], let [J.sub.k] be a sequence of smooth Jordan domains satisfying [Delta][J.sub.k] [subset or equal to] [Omega], [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Let [[Omega].sub.k] = [J.sub.k] [intersection] [Omega]. Then [[Omega].sub.k] satisfies the hypotheses of Lemma 6.2 and therefore there is a conformal homeomorphism [Mathematical Expression Omitted] onto some circle domain [Mathematical Expression Omitted] which extends to a quasiconformal homeomorphism [Mathematical Expression Omitted] whose complex dilatation vanishes in [Delta][[Omega].sub.k] = ([Delta][Omega] [intersection] [J.sub.k]) [union] [Delta][J.sub.k]. Normalizing [Mathematical Expression Omitted] and [f.sub.k] by Mobius transformations, we may assume that each [Mathematical Expression Omitted] is a relative circle domain in U, [Mathematical Expression Omitted] and [f.sub.k]([z.sub.0]) = 0 for some fixed [z.sub.0] [element of] [[Omega].sub.1] = [Omega][intersection][J.sub.1]. By taking a subsequence, we may also assume that [f.sub.k] converges to some mapping f : [Omega] [approaches] U. Clearly, f is either a conformal homeomorphism onto its image or a constant.

By the argument of [HS, Theorem 8.1] (but using the generalized Angle Lemma 5.2 and Schwarz Pick Lemma 5.1), we may conclude that f is not a constant and f([Omega]) is a relative circle domain in U, hence a circle domain. (In particular, the Riemann mapping theorem for relative circle domains stated in the introduction is proved.) It remains to show that f extends to a locally quasiconformal homeomorphism g of A onto U, whose complex dilatation vanishes a.e. in [Delta][Omega] [intersection] A.

Let [k.sub.0] be a fixed positive integer. For k [greater than or equal to] [k.sub.0] + 1, the functions [Mathematical Expression Omitted] are conformal and they converge uniformly to [Mathematical Expression Omitted]. In particular, these functions are uniformly smooth in an open neighborhood of [Delta][J.sub.[k.sub.0]]. Therefore [Mathematical Expression Omitted] can be extended to some quasiconformal homeomorphisms [Mathematical Expression Omitted] whose conformal dilatations are uniformly bounded in [Mathematical Expression Omitted]. Since the complex dilatation of [Mathematical Expression Omitted] vanishes in [Mathematical Expression Omitted], we conclude that there is some constant K [greater than or equal to] 1 such that the conformal dilatations of [g.sub.k] in [Mathematical Expression Omitted] are bounded by K.

Let [Mathematical Expression Omitted]. This is a circle domain with [Mathematical Expression Omitted]. Let [Mathematical Expression Omitted] be the complex dilatation of [g.sub.k] on [Mathematical Expression Omitted]. Then by Lemma 2.2, [Mathematical Expression Omitted] is Mobius equivalent to [Mathematical Expression Omitted] and there is a K-quasiconformal homeomorphism [Mathematical Expression Omitted] which extends [Mathematical Expression Omitted]. Thus the limit of a subsequence of [Mathematical Expression Omitted] is a K-quasiconformal map [Mathematical Expression Omitted], on [J.sub.[k.sub.0]] such that

[Mathematical Expression Omitted].

Since [[Lambda].sub.k] = 0 in [Delta][Omega] [intersection] [J.sub.[k.sub.0]], Lemma 3.2 implies that the complex dilatation of [Mathematical Expression Omitted] also vanishes in [Delta][Omega] [intersection] [J.sub.[k.sub.0]].

We have shown that for any [k.sub.0], [Mathematical Expression Omitted] can be extended to a quasiconformal homeomorphism [Mathematical Expression Omitted] whose complex dilatation vanishes in [Delta][Omega] [intersection] [J.sub.[k.sub.0]]. As [Omega] is a relative circle domain in A, it follows that f extends to a locally quasiconformal map g : A [approaches] U whose complex dilatation vanishes in [Delta][Omega] [intersection] A.

LEMMA 6.4. Theorem 6.1 holds if A is finitely connected.

Proof. The proof proceeds by induction on the number N of boundary components of A. For N = 1, this follows by Lemma 6.3. So let N [greater than or equal to] 2. Let [[Gamma].sub.1], [[Gamma].sub.2],..., [[Gamma].sub.N] be the boundary components of A. Let [A.sub.1] be the simply connected domain with [Delta][A.sub.1] = [[Gamma].sub.N] and [A.sub.1] [contains or equal to] A and let [A.sub.2] be the (N - 1)-connected domain with [Delta][A.sub.2] = [[Gamma].sub.1] [union] ... [union] [[Gamma].sub.N-1] and [A.sub.2] [contains or equal to] A. Then [A.sub.1] [intersection] [A.sub.2] = A.

Let [J.sub.1], [J.sub.2] be smooth Jordan domains in [Mathematical Expression Omitted] with disjoint closures such that the closed annulus [Mathematical Expression Omitted] is contained in [Omega], and such that [[Gamma].sub.N] [subset or equal to] [J.sub.1] and [[Gamma].sub.i] [subset or equal to] [J.sub.2] for i [less than] N. Let [[Omega].sub.1] = [Omega] [union] [J.sub.1] and [[Omega].sub.2] = [Omega] [union] [J.sub.2]. Then [[Omega].sub.1] [intersection] [[Omega].sub.2] = [Omega]. Clearly [[Omega].sub.1], is a relative circle domain in [A.sub.1] and [[Omega].sub.2] is a relative circle domain in [A.sub.2]. Then by the inductive hypothesis, there are circle domains [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and conformal homeomorphisms [Mathematical Expression Omitted] and [Mathematical Expression Omitted] which extend to locally quasiconformal homeomorphisms [Mathematical Expression Omitted] and [Mathematical Expression Omitted] where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are circle domains (with finitely many boundary components). Moreover, the complex dilatation of [g.sub.1] vanishes in [Delta][[Omega].sub.1] [intersection] [A.sub.1] and that of [g.sub.2] vanishes in [Delta][Omega] [intersection] [A.sub.1].

By post composing [f.sub.1] with a Mobius transformation, if necessary, we assume that in [Mathematical Expression Omitted] there is an annulus E bounded by the curves [f.sub.1]([Delta][J.sub.2]) and [f.sub.2]([Delta][J.sub.1]). Let F : [Omega] [approaches] [f.sub.1]([J.sub.2] [intersection] [Omega]) [union] [f.sub.2]([J.sub.1] [intersection] [Omega]) [union] E be some quasiconformal homeomorphism which is equal to [f.sub.1] on [J.sub.2] [intersection] [Omega] and is equal to [f.sub.2] on [J.sub.1] [intersection] [Omega]. Similarly define a locally quasiconformal extension of F to A by using [g.sub.1] and [g.sub.2]. Note that the image of F is a circle domain. Now a quasiconformal deformation argument as used in the last part of the proof of Lemma 6.2 shows that there are a circle domain [[Omega].sup.*] and a conformal homeomorphism f: [Omega] [approaches] [[Omega].sup.*] with the required extension property.

Proof of Theorem 6.1. The proof is by transfinite induction on the type ([Alpha], n) of A as is done in [HS]. If [Alpha] = 0, then A is finitely connected and the theorem follows by Lemma 6.4. Let [Alpha] [greater than] 0. We will show that [Omega] can be mapped by a conformal homeomorphism to a relative circle domain in a simply connected domain, and moreover, the conformal homeomorphism can be extended to some locally quasiconformal homeomorphism [Mathematical Expression Omitted] whose complex dilatation vanishes in [Delta][Omega] [intersection] A. The proof can then be completed by Lemma 6.3.

Let [[Sigma].sub.0] be a component rank [Alpha] in [Delta]A. Then [[Sigma].sub.0] is the boundary of a simply connected domain [J.sub.[infinity]] which contains A. Let [J.sub.k] be smooth Jordan domains such that [Delta][J.sub.k] [subset or equal to] [Omega], [J.sub.k] [subset or equal to] [J.sub.k+1] and [Mathematical Expression Omitted]. Let [A.sub.k] = A [intersection] [J.sub.k] and [[Omega].sub.k] = [Omega] [intersection] [J.sub.k]. Then each [[Omega].sub.k] is a relative circle domain in [A.sub.k]. Since [Alpha] [greater than] 0, the type of [A.sub.k] is strictly smaller than ([Alpha], n). So by the inductive hypothesis, there are circle domains [Mathematical Expression Omitted] and conformal homeomorphisms [Mathematical Expression Omitted] which extend to a locally quasiconformal homeomorphism [Mathematical Expression Omitted] whose complex dilatation vanishes in [Delta][[Omega].sub.k] [intersection] [A.sub.k] = [Delta][Omega] [intersection] [A.sub.k]. We also assume without loss of generality that the sequence [f.sub.k] is normalized at three points of [[Omega].sub.1]. Then the subsequence of [f.sub.k], still denoted by [f.sub.k], converges to a conformal map [Mathematical Expression Omitted]. Moreover, using Lemma 5.5, for any smooth Jordan domain B [subset or equal to] A with [Delta]B [subset or equal to] [Omega], and for any k big enough so that [Mathematical Expression Omitted], [Mathematical Expression Omitted] extends to a uniformly quasiconformal mapping [Mathematical Expression Omitted]. Let [B.sub.j] be a countable family of disjoint smooth Jordan domains in A with [Delta][B.sub.j] [subset] [Omega] such that A - [Omega] [subset] [[union].sub.j][B.sub.j]. Modify the definition of each [g.sub.k] by letting it equal to [F.sub.k,[B.sub.j]] on [B.sub.j], whenever [B.sub.j] [subset] [A.sub.k]. Then the [g.sub.k] are uniformly quasiconformal in any subdomain whose closure is contained in A. Thus by taking a subsequence, we can assume that [g.sub.k] converges to a locally quasiconformal map [Mathematical Expression Omitted] which extends f. The complex dilatation of g vanishes on [Delta][Omega] [intersection] A because that of [g.sub.k] vanishes in [Delta][Omega] [intersection] [A.sub.k] for each k (see Lemma 3.2).

Let [Mathematical Expression Omitted] be the simply connected domain containing f([Omega]) with [Mathematical Expression Omitted]. It remains to show that f([Omega]) is a relative circle domain in [Mathematical Expression Omitted]; that is, all boundary components of f([Omega]), except possibly [Mathematical Expression Omitted], are circles and points. Let [Sigma] be a boundary component of [Omega] different than [Delta][J.sub.[infinity]]. In order to show that [f.sup.B]([Sigma]) is either a circle or a point, we choose [k.sub.0] big enough, such that [Sigma] [subset or equal to] interior([J.sub.[k.sub.0]]). Let k [greater than] [k.sub.0], [Mathematical Expression Omitted], and let [Mathematical Expression Omitted]. Then the image of [h.sub.k] is a relative circle domain in a Jordan domain bounded by the curve [f.sub.k]([Delta][J.sub.[k.sub.0]]). Using Lemma 5.6, we conclude that the limit of [h.sub.k] maps [[Omega].sub.[k.sub.0]] onto a relative circle domain in a Jordan domain bounded by f([Omega][J.sub.[k.sub.0]]). It follows that f([[Omega].sub.[k.sub.0]]) is a relative circle domain in a Jordan domain bounded by f([Delta][J.sub.[k.sub.0]]). This implies in particular that [f.sup.B]([Sigma]) is a circle or a point.

UNIVERSITY OF CALIFORNIA AT SAN DIEGO, LA JOLLA, CA 92093-0112 Electronic mail: ZHE@UCSD.EDU

WEIZMANN INSTITUTE, REHOVOT 76100, ISRAEL Electronic mail: SCHRAMM@WISDOM.WEIZMANN.AC.IL

1 We allow orientation reversing Mobuius transformations.

REFERENCES

[AhB] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. 72 (1960), 385-404.

[Be] L. Bers, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215-228.

[HS] Z-X. He and O. Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 (1993), 369-406.

[HS2] -----, Inverse Riemann mapping theorem for relative circle domains, Pacific J. Math. (to appear).

[LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973.

[Si] R.J. Sibner, Uniformizations of symmetric Riemann surfaces by Schottky groups, Trans. Amer. Math. Soc. 116 (1965), 79-85.

[Su] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann Surfaces and Related Topics, Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Stud., vol. 97, Princeton University Press, Princeton, N.J., 1981, pp. 465-496.
COPYRIGHT 1995 Johns Hopkins University Press
No portion of this article can be reproduced without the express written permission from the copyright holder.