Quasisymmetric Koebe Uniformization

# Quasisymmetric Koebe Uniformization

S. Merenkov Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green Street, Urbana, IL 61801, USA  and  K. Wildrick Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
###### Abstract.

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface homeomorphic to a finitely connected domain in the standard 2-sphere is quasisymmetrically equivalent to a circle domain in if and only if is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.

###### 2000 Mathematics Subject Classification:
30C65
S. M. was supported by NSF grants DMS-0653439 and DMS-1001144
K. W. was supported by Academy of Finland grants 120972 and 128144

## 1. Introduction

Uniformization problems are amongst the oldest and most important problems in mathematical analysis. A premier example is the measurable Riemann mapping theorem, gives a robust existence theory for quasiconformal mappings in the plane. A quasiconformal mapping between domains in a Euclidean space is a homeomorphism that sends infinitesimal balls to infinitesimal ellipsoids of uniformly bounded ellipticity. The theory of quasiconformal mappings has been one of the most fruitful in analysis, yielding applications to hyperbolic geometry, geometric group theory, complex dynamics, partial differential equations, and mathematical physics.

In the past few decades, many aspects of the theory of quasiconformal mappings have been extended to apply to abstract metric spaces. A key factor in these developments has been the realization that in metric spaces with controlled geometry, the infinitesimal condition imposed by quasiconformal mappings actually implies a stronger global condition called quasisymmetry [15]. The fact that quasisymmetric mappings are required to have good behavior at all scales makes them well suited to metric spaces that a priori have no useful infinitesimal structure.

A homeomorphism of metric spaces is quasisymmetric if there is a homeomorphism such that if are distinct points of , then

 dY(f(x),f(y))dY(f(x),f(z))≤η(dX(x,y)dX(x,z)).

The homeomorphism is called a distortion function of . If we wish to emphasize that a quasisymmetric mapping has a particular distortion function , we will call it -quasisymmetric.

Despite the highly developed machinery for quasiconformal analysis on metric spaces, an existence theory for quasisymmetric mappings on metric spaces analogous to that of conformal mappings on Riemann surfaces has only recently been explored. The motivation for such results arises from geometric group theory [5], the dynamics of rational maps on the sphere [7], and the analysis of bi-Lipschitz mappings and rectifiable sets in Euclidean space [4].

More than a decade after foundational results of Tukia and Väisälä in dimension one [24], Bonk and Kleiner [5] gave simple sufficient conditions for a metric space to be quasisymmetrically equivalent to the standard 2-sphere .

###### Theorem 1.1 (Bonk–Kleiner).

Let be an Ahlfors -regular metric space homeomorphic to the sphere . Then is quasisymmetrically equivalent to the sphere if and only if is linearly locally connected.

The condition that is linearly locally connected (LLC), which heuristically means that does not have cusps, is a quasisymmetric invariant. Ahlfors -regularity, which states that the two-dimensional Hausdorff measure of a ball is uniformly comparable to the square of its radius, is not. See Section 3 for precise definitions. A version of Theorem 1.1 for metric spaces homeomorphic to the plane was derived in [26], and a local version given in [27].

In this paper, we seek a version of Theorem 1.1 for domains in . The motivation for our inquiry comes from the Kapovich–Kleiner conjecture of geometric group theory, described in Section 2, and from analogous classical conformal uniformization theorems onto circle domains.

A circle domain is a domain such that each component of is either a round disk or a point. In 1909 [18], Koebe posed the following conjecture, known as the Kreisnormierungsproblem: every domain in the plane is conformally equivalent to a circle domain. In the 1920’s [19], Koebe was able to confirm his conjecture in the finitely connected case.

###### Theorem 1.2 (Koebe’s uniformization onto circle domains).

Let be a domain with finitely many complementary components. Then is conformally equivalent to a circle domain.

We first state a quasisymmetric version of Koebe’s theorem, which we attain as a consequence of our main result. Denoting the completion of a metric space by , we define the metric boundary of by . We say that a component of is non-trivial if it contains more than one point.

###### Theorem 1.3.

Let be an Ahlfors -regular metric space that is homeomorphic to a domain in , and such that has finitely many non-trivial components. Then is quasisymmetrically equivalent to a circle domain if and only if is linearly locally connected and the completion is compact.

Theorem 1.3 is only quantitative in the sense that the distortion function of the quasisymmetric mapping may be chosen to depend only on the constants associated to the various conditions on and the ratio of the diameter of to the minimum distance between components of .

In 1993, He and Schramm confirmed Koebe’s conjecture in the case of countably many complementary components [13]. In full generality the conjecture remains open. A key tool in He and Schramm’s proof was transifinte induction on the rank of the boundary of a domain in , which measures the complexity with which components of the boundary converge to one another. The rank of a collection of boundary components is defined via a canonical topology on the set of components of the boundary; see Section 4. It is shown there that if a metric space is homeomorphic to a domain in and is linearly locally connected, then the natural topology on the set of components of the metric boundary is homeomorphic to the natural topology on the set of boundary components of . This allows us to define rank as in the classical setting. We denote the topologized collection of components of by .

In the following statement, which is our main result, we consider quasisymmetric uniformization onto circle domains with the property that is uniformly relatively separated, meaning there is a uniform lower bound on the relative distance

 △(E,F)=dist(E,F)min{diam(E),diam(F)},

between any pair of non-trivial boundary components. This condition is appears naturally in both classical quasiconformal analysis and geometric group theory. Moreover, we employ annular linear local connectedness (ALLC), which is more natural than the condition in this setting.

###### Theorem 1.4.

Let be a metric space, homeomorphic to a domain in , such that the closure of the collection of non-trivial components of is countable and has finite rank. Moreover, suppose that

1. is Ahlfors -regular,

2. setting, for each integer ,

 nk=supcard{E∈C(X):E∩B(x,r)≠∅ and 2−kr

where the supremum is taken over all and , it holds that

 ∞∑k=0nk2−2k<∞.

Then is quasisymmetrically equivalent to a circle domain such that is uniformly relatively separated if and only if has the following properties:

1. the completion is compact,

2. is annularly linearly locally connected,

3. is uniformly relatively separated.

Theorem 1.4 is quantitative in the sense that the distortion function of the quasisymmetric mapping may be chosen to depend only on the constants associated to the various conditions on , and vice-versa.

A key tool in our proof is the following similar uniformization result of Bonk, which is valid for subsets of [1].

###### Theorem 1.5 (Bonk).

Let be a collection of uniformly relatively separated uniform quasicircles in that bound disjoint Jordan domains. Then there is a quasisymmetric homeomorphism such that for each , the set is a round circle in .

This result, which is also quantitative, allows us to conclude that if a metric space as in the statement of Theorem 1.4 satisfies conditions (1)-(5), then it is quasisymmetrically equivalent to a circle domain with uniformly relatively separated boundary circles as soon as there is any quasisymmetric embedding of into . Thus, producing such an embedding is the main focus of this paper.

It is of great interest to know if conditions (1) and (2) can be replaced with conditions that are quasisymmetrically invariant. By snowflaking, i.e., raising the metric to power , the sphere in one direction only, say, in the direction of -axis, one produces a metric space homeomorphic to that satisfies all the assumptions of Theorem 1.4 (and Theorem 1.1) except for Ahlfors -regularity, but fails to be quasisymmetrically equivalent to . On the other hand, not every quasisymmetric image of is Ahlfors -regular, as is seen by the usual snowflaking of the standard metric on .

Our second main result is the existence of a metric space satisfying all assumptions of Theorem 1.4, except for condition (2), that fails to quasisymmetrically embed in .

###### Theorem 1.6.

There is a metric space , homeomorphic to a domain in , with the following properties

• has rank ,

• is Ahlfors -regular,

• the completion is compact,

• is annularly linearly locally connected,

• the components of are uniformly relatively separated,

• there is no quasisymmetric embedding of into .

We now outline the proof of Theorem 1.4 and the structure of the paper. In Section 4 we establish a topological characterization of the boundary components of a metric space as the space of ends of the underlying topological space, at least in the presense of some control on the geometry of the space. This allows us to develop a notion of rank, and in Section 6, a theory of cross-cuts analogous to the classical theory. A key tool in this development is the following purely topological statement: every domain in is homeomorphic to a domain in with totally disconnected complement. This folklore theorem is proven in Section 5. In Section 7, we use cross-cuts and a classical topological recognition theorem for to uniformize the boundary components of . The resulting theorem, which generalizes [26, Theorem 1.3], may be of independent interest:

###### Theorem 1.7.

Suppose that is a metric space that is homeomorphic to a domain in , has compact completion, and satisfies the - condition for some . Then each non-trivial component of is a topological circle satisfying the - condition for some depending only on . In particular, if the space is additionally assumed to be doubling, then each non-trivial component of is quasisymmetrically equivalent to with distortion function depending only on and the doubling constant.

Suppose that is an Ahlfors -regular and metric space that is homeomorphic to a domain in . Section 8 shows that the completion is homeomorphic to the closure of an appropriately chosen circle domain. In Section 9 we prove general theorems implying that each non-trivial component of has Assouad dimension strictly less than , and hence, up to a bi-Lipschitz mapping, is the boundary of a planar quasidisk. We describe a general gluing procedure in Section 10, and use it to “fill in” the non-trivial components of . The resulting space is , but not always Ahlfors -regular. In order to guarantee this, we impose condition (2) of Theorem 1.4. The assumption of finite rank allows us to reduce to the case that there are only finitely many components of . We show that is homeomorphic to , and apply Theorem 1.1. The problem is now planar, and the above mentioned theorem of Bonk now yields the desired result. Section 11 summarizes our work and provides a formal proof of Theorems 1.3 and 1.4.

Section 12 is dedicated to proving Theorem 1.6. We conclude in Section 13 with discussion of related open problems.

### 1.1. Acknowledgements

We wish to thank Mario Bonk, Peter Feller, Pekka Koskela, John Mackay, Daniel Meyer, Raanan Schul, and Jeremy Tyson for useful conversations and critical comments. Some of the research leading to this work took place while the second author was visiting the University of Illinois at Urbana-Champaign and while both authors were visiting the State University of New York at Stony Brook. We are very thankful for the hospitality of those institutions. Also, the first author wishes to thank the Hausdorff Research Institute for Mathematics in Bonn, Germany, for its hospitality during the Rigidity program in the Fall 2009.

## 2. Relationship to Gromov hyperbolic groups

In geometric group theory, there is a natural concept of a hyperbolic group due to Gromov. These abstract objects share many features of Kleinian groups, including a boundary at infinity. The boundary of a Gromov hyperbolic group is equipped with a natural family of quasisymmetrically equivalent metrics. The structure of this boundary is categorically linked to the structure of the group: a large-scale bi-Lipschitz mapping between Gromov hyperbolic groups induces a quasisymmetric mapping between the corresponding boundaries at infinity, and vice versa.

One of the premier problems in geometric group theory is Cannon’s conjecture, which states that for every Gromov hyperbolic group with boundary at infinity homeomorphic to , there exists a discrete, co-compact, and isometric action of on hyperbolic 3-space. In other words, if a Gromov hyperbolic group has the correct boundary at infinity, then it arises naturally from the corresponding situation in hyperbolic geometry. By a theorem of Sullivan [23], Cannon’s conjecture is equivalent to the following statement: if is a Gromov hyperbolic group, then is homeomorphic to if and only if is quasisymmetrically equivalent to . If is homeomorphic to , then each natural metric on is and Ahlfors -regular for some . If , then Theorem 1.1 confirms Cannon’s conjecture. Indeed, even stronger statements are known [6].

Closely related to Cannon’s conjecture is the Kapovich–Kleiner conjecture, which states that for every Gromov hyperbolic group with boundary at infinity homeomorphic to the Sierpiński carpet, there exists a discrete, co-compact, and isometric action of on a convex subset of hyperbolic 3-space with totally geodesic boundary. The Kapovich–Kleiner conjecture is implied by Cannon’s conjecture, and is equivalent to the following statement: if is a Gromov hyperbolic group, then is homeomorphic to the Sierpiński carpet if and only if is quasisymmetrically equivalent to a round Sierpiński carpet, i.e., to a subset of that is homeomorphic to the Sierpiński carpent and has peripheral curves that are round circles.

Theorem 1.4 can be seen as a uniformization result for domains that might approximate a Sierpiński carpet arising as the boundary of a hyperbolic group. If is homeomorphic to the Sierpiński carpet, then it is and the peripheral circles are uniformly separated uniform quasicirlces. Recent work of Bonk established the Kapovich–Kleiner conjecture in the case that can be quasisymmetrically embedded in ; see [1]. As noted by Bonk–Kleiner in [2], this is true when the Assouad dimension of is strictly less than two, a hypothesis analgous to condition (2) in Theorem 1.4. This observation and its proof provided ideas that will be used in Section 10.

## 3. Notation and basic results

### 3.1. Metric Spaces

We are often concerned with conditions on a mapping or space that involve constants or distortion functions. These constants or distortion functions are refered to as the data of the conditions. A theorem is said to be quantitative if the data of the conclusions of theorem depend only on the data of the hypotheses. In the proof of quantitative theorems, given non-negative quantities and , we will employ the notation if there is a quantity , depending only on the data of the conditions in the hypotheses, such that . We write if and .

We will often denote a metric space by . Given a point and a number , we define the open and closed balls centered at of radius by

 B(X,d)(x,r)={y∈X:d(x,y)

For , we denote the open annulus centered at of inner radius and outer radius by

 A(X,d)(x,r,R)={y∈X:r

Note that when , this corresponds to

Where it will not cause confusion, we denote by , , or . A similar convention is used for all other notions which depend implicitly on the underlying metric space.

We denote the completion of a metric space by , and define the metric boundary of by . These notions are not to be confused with their topological counterparts.

For , the -neighborhood of a subset is given by

 Nϵ(E)=⋃x∈EB(x,ϵ).

The diameter of is denoted by and the distance between two subsets is denoted by If at least one of and has finite diameter, then the relative distance of and is defined by

 △(E,F)=dist(E,F)min{diamE,diamF},

with the convention that if at least one of and has diameter .

###### Remark 3.1.

If is a quasisymmetric homeomorphism of metric spaces, and and are subsets of , then

 △(f(E),f(F))≃△(E,F).

This is easily seen using [14, Proposition 10.10].

Let be the standard 2-sphere equipped with the restriction of the Euclidean metric on . We say that is a domain in if it is an open and connected subset of . We always consider a domain in as already metrized, i.e., equipped with the restriction of the standard spherical metric.

### 3.2. Dimension and measures

A metric space is doubling if there is a constant such that for any and , the ball can be covered by at most balls of radius . This condition is quantitatively equivalent to the existence of constants and such that is -homogeneous, meaning that for every , and , the ball can be covered by at most balls of radius . The infinimum over all such that is -homogeneous for some is called the Assouad dimension of . Hence, doubling metric spaces are precisely those metric spaces with finite Assouad dimension.

In a doubling metric space, some balls may have lower Assouad dimension than the entire space. To rule out this kind of non-homogeneity, one often employs a much stricter notion of finite-dimensionality. The metric space is Ahlfors -regular, , if there is a constant such that for all and ,

 (3.1) rQK≤HQX(¯¯¯¯B(x,r))≤KrQ,

where denotes the -dimensional Hausdorff measure on . It is quantitatively equivalent to instead require that (3.1) hold for all open balls of radius less that . The existence of any Borel regular outer measure on that satisfies (3.1) quanitatively implies that is Ahlfors -regular; see [14, Exercise 8.11].

###### Remark 3.2.

Suppose that is Ahlfors -regular, . Given , the space is again Ahlfors -regular, quantitatively. This is proven as in [26, Lemma 2.11].

### 3.3. Connectivity conditions

Here we describe various conditions that control the existence of “cusps” in a metric space by means of connectivity. The basic concept of such conditions arose from the theory of quasiconformal mappings in the plane, where they play an important role as invariants.

Let . A metric space is -linearly locally connected (-) if for all and , the following two conditions are satisfied:

• for each pair of distinct points , there is a continuum such that ,

• for each pair of distinct points , there is a continuum such that .

Individually, conditions and are referred to as the - and - conditions, respectively.

The condition extends in a particularly nice way to the completion of a metric space. We say that a metric space is - if for all and the following conditions are satisfied:

• For each pair of distinct points , there is an embedding such that , , , and

• For each pair of distinct points , there is an embedding such that , , , and

Individually, conditions and are referred to as the - and - conditions, respectively.

If a metric space is -, then it is also -. The next proposition states that the two conditions are quantitatively equivalent for the spaces in consideration in this paper.

###### Proposition 3.3.

Let , and let be a locally compact and locally path-connected metric space that satisfies the - condition. Then is -, where depends only on .

###### Proof.

The key ingredient is the following statement: If is an open subset of , and is a continuum, then any pair of points are contained in an arc in . The details are straightforward and left to the reader. ∎

Let . A metric space is -annularly linearly locally connected (-) if for all points and all , each pair of distinct points in the annulus is contained in a continuum in the annulus

The condition forbids local cut-points in addition to ruling out cusps. For example, the standard circle is but not . In our setting, the condition is a more natural assumption, and is in some cases equivalent to the condition.

We omit the proofs of the following three statements. The first is based on decomposing an arbitrary annulus into dyadic annuli. The second uses the fact that in a connected space, any distinct pair of points is contained in annulus around some third point. The third states that the condition extends to the boundary as in Proposition 3.6.

###### Lemma 3.4.

Let . Suppose that a connected metric space satisfies the condition that for all points and all , each pair of disctinct points in the annulus is contained in a continuum in the annulus Then satisfies the - condition.

###### Lemma 3.5.

Suppose that is a connected metric space that satisfies the - condition. Then satisfies the - condition.

###### Proposition 3.6.

Suppose that is a locally compact and locally path-connected metric space that satisfies the - condition. Then there is a quantity , depending only on , such that for all and , for each pair of distinct points , there is an embedding such that , , , and

 imγ⊆A¯¯¯¯X(a,r/λ′,λ′R).

There is a close connection between the condition and the uniform relative separation of the components of the boundary of a given metric space. The following statement addresses only circle domains, but a more general result is probably valid. We postpone the proof until Section 5.

###### Proposition 3.7.

Let be a circle domain. Then satisfies the condition if and only if the components of are uniformly relatively separated, quantitatively.

In the case of metric spaces that are homeomorphic to domains in with finitely many boundary components, the -condition may be upgraded to the -condition. Again, we postpone the proof until Section 5.

###### Proposition 3.8.

Let be a metric space homeomorphic to a domain in , and assume that the boundary has finitely many components. If is -, , then it is -, where depends on and the ratio of the diameter of to the minimum distance between components of .

## 4. The space of boundary components of a metric space

In this section we assume that is a connected, locally compact metric space with the additional property that the completion is compact. Note that as is locally compact, it is an open subset of . Hence is closed in and hence compact.

### 4.1. Boundary components and ends

Of course, the topological type of depends on the specific metric . However, the goal of this section is to show that under a simple geometric condition, the collection of components of depends only on the topological type of .

We define an equivalence relation on by declaring that if and only if and are contained in the same component of . Then there is a bijection between and the quotient , and hence we may endow with the quotient topology. Since is compact, the space is compact as well. Given a compact set and a component of , denote

 C(K,U)={E⊆C(X):E⊆(∂U∩∂X)}.

Let denote the collection of sequences with the property that for every compact set , there is a number and a connected subset of such that

 {xi}i≥N⊆U.

Define an equivalence relation on by if and only if the sequence is in . An equivalence class defined by is called an end of , and we denote the collection of ends of by .

Given a compact subset and a component of , define

 E(K,U)={[{xi}]:{xi}∈U(X) and ∃ N∈N such that {xi}i≥N⊆U},

That this set is well-defined follows from the definition of the equivalence relation on . Let be the collection of all such sets.

###### Proposition 4.1.

The collection generates a unique topology on such that every open set is a union of sets in .

###### Proof.

We employ the standard criteria for proving generation [22, Section 13]. As is connected, taking shows that contains . Thus it suffices to show that given compact subsets and of and components and of and respectively, and given an end

 E∈E(K1,U1)∩E(K2,U2),

there is a compact set and a component of such that

 (4.1) E∈E(K,U)⊆(E(K1,U1)∩E(K2,U2)).

Let represent the end . By definition of , there is a component of , where , such that for some . This implies that . By assumption, there is a number such that is contained in . Since is a connected subset of and , it follows that . This implies (4.1). ∎

We set the topology on to be that given by Proposition 4.1. It follows quickly from the definitions that is a Hausdorff space.

###### Remark 4.2.

As ends are defined purely in topological terms, a homeomorphism to some other topological space induces a homeomorphism . This homeomorphism is natural in the sense that a sequence represents the end if and only represents the end .

To relate the ends of a metric space to the components of its metric boundary, we need an elementary result regarding connectivity. Let and let and be points in any metric space . An -chain connecting to in is a sequence of points in such that for each . In compact spaces, the existence of arbitrarily fine chains can detect connectedness. We leave the proof of the following statement to that effect to the reader.

###### Lemma 4.3.

Let and be points in a metric space . If and lie in the same component of , then for every there is an -chain in connecting to . The converse statement holds if is compact.

An end always defines a unique boundary component.

###### Proposition 4.4.

If and are Cauchy sequences representing the same end of , then they represent points in the same component of .

###### Proof.

Denote by and the points in defined by and , respectively. By Lemma 4.3, in order to show that and are contained in a single component of , it suffices to find an -chain in connecting to for every . To this end, fix . Let be the compact subset of defined by

 (4.2) K=X∖N¯¯¯¯X(∂X,ϵ/3).

By assumption, we may find so large that and lie in a connected subset of , and that and are both less than . By Lemma 4.3, we may find an -chain in . By (4.2), for each we may find a point such that . The triangle inequality now implies that is an -chain in , as required. ∎

###### Remark 4.5.

Proposition 4.4 allows us to define a map as follows. Let and let be a sequence representing . As is compact, we may find a Cauchy subsequence of that represents a point in some boundary component . Set . This is well defined by Proposition 4.4.

We now consider when a boundary component corresponds to an end. For subspaces of , this is always the case. The key tool in the proof of this is the following purely topological fact, which is mentioned in the proof of [5, Lemma 2.5].

###### Proposition 4.6.

Each domain in may be written as a union of open and connected subsets of such that for each , the closure of is a compact subset of and is a finite collection of pairwise disjoint Jordan curves.

###### Proposition 4.7.

Let be a domain in . If and are points in the same component of , then any Cauchy sequences representing and are in and represent the same end of .

###### Proof.

The metric boundary coincides with the usual topological boundary of in . Let denote the exhaustion of provided by Proposition 4.6. Since and are in the same component of , for each they belong to a single simply connected component of . Let and be Cauchy sequences representing and respectively.

Suppose that is a compact subset of . We may find so large that . Then does not intersect . Moreover, there is a number such that

 ({xi}i≥N∪{yi}i≥N)⊆Ui0∩Ω.

It now suffices to show that is connected. Let and be points in . We may find so large that and are contained in . The set is a simply connected domain with finitely many disjoint closed topological disks removed from its interior, and is therefore path-connected. Thus and may be connected by a path inside , and hence inside . ∎

The proof given above does not even pass to metric spaces that are merely homeomorphic to a domain in , as it need not be the case that the completion of such a space embeds topologically in . However, under an additional assumption controlling the geometry of , we can give a different proof.

###### Proposition 4.8.

Suppose that satisfies the condition. If and are points in the same component of , then any Cauchy sequences representing and are in and represent the same end of .

###### Lemma 4.9.

Suppose that satisfies the - condition for some . Let be a connected subset of and let . Then is contained in a connected subset of .

###### Proof.

It suffices to show that if and are points in , then there is a continuum containing to inside of . Let and be points in such that and . By Lemma 4.3, there is an -chain in . For each , find a point such that . The triangle inequality implies that for ,

 zj+1∈BX(zj,3ϵ).

Repeatedly applying the - condition and concatenating now yields the desired result. ∎

###### Proof of Proposition 4.8.

Let and be points in a connected subset of , and let and be Cauchy sequences in corresponding to and , respectively. Let be a compact subset of . As is compact, we may find such that . Let be so large that

 {xi}i≥N∪{yi}i≥N⊆N¯¯¯¯X(E,ϵ)∩X.

Lemma 4.9 now implies the desired results. ∎

###### Remark 4.10.

Proposition 4.8 is not true without some control on the geometry of . The following example was pointed out to us by Daniel Meyer. Let denote cylindrical coordinates on , and set

 X={(r,θ,z):(r−1)2+z2=1}∖{(0,0,0)}.

Equipped with the standard metric inherited from , the space is homeomorphic to a punctured disk, and hence has two ends. However, the metric boundary of consists only of the point .

The following statement is the main result of this section. We note that in the case that is a domain in , the statement is mentioned in [13].

###### Theorem 4.11.

Suppose that is either a domain in or satisfies the condition. Then the map defined in Remark 4.5 is a homeomorphism that is natural in the sense that a Cauchy sequence represents the end if and only if it represents a point on the boundary component .

In the proof of the following lemma, we consider only the case that satisfies the condition. If is a domain in , a proof is easily constructed using Proposition 4.6.

###### Lemma 4.12.

Suppose that is a domain in or satisfies the condition. Let be a compact subset of and let be a component of . Then the following statements hold:

• for each , there is a number such that ,

• if intersects , then

• the set is open in .

###### Proof.

We assume that satisfies the - condition for some .

Suppose that statement (i) is not true. Then for all sufficiently small we may find points such that and is in some other component of . Using the - condition to connect to inside of produces a point of in the ball . Letting tend to produces a contradiction with the assumption that is a compact subset of .

Statement (i) implies that collection

 {E∩∂V∩∂X:V is a component of X∖K}

consists of pairwise disjoint open subsets of . Hence the connectedness of proves statement (ii).

Now, recall that is endowed with the quotient topology. Hence, by statement (ii), in order to show that is open in , it suffices to show that is open in . This follows from statements (i) and (ii). ∎

###### Proof of Theorem 4.11.

Proposition 4.7 or 4.8 shows that is injective. Given and a Cauchy sequence representing a point in , Proposition 4.7 or 4.8 also state that is in and hence represents an end . By definition, this implies that , and so is surjective.

We now check that the bijection is a homeomorphism. Since is compact and is Hausdorff, this is true if is continuous. Hence, by Lemma 4.12 (iii) and the defintion of the toplogy on , it suffices to show that for any compact set and any component of ,

 Φ(E(K,U))=C(K,U).

Let be an end in . By definition, the limit of any Cauchy sequence representing lies in . Hence, intersects , and so Lemma 4.12 (ii) shows that . Now, let and choose a Cauchy sequence representing a point in . Lemma 4.12 (i) implies that there is such that is contained in . Again, Proposition 4.7 or 4.8 states that is in and hence represents an end . Thus, by definition, and . ∎

### 4.2. Rank

We breifly recall the notion of rank as discussed in [13]. Let be a countable, compact, and Hausdorff topological space. Set , and for each , set , to be the set of non-isolated points of , and endow with the subspace topology. This process can be continued using transfinite induction to define for each ordinal , though we will not have need for this. For each ordinal , the space is again countable, compact, and Hausdorff. By the Baire category theorem, there is a unique ordinal such that is finite and non-empty; this ordinal is defined to be the rank of .

Let be a metric space that is either a domain in or satisfies the - condition. By Theorem 4.11, the space of boundary components is homeomorphic to the space of ends . As mentioned above, the former is clearly compact and the latter clearly Hausdorff, hence both are compact Hausdorff spaces. Hence, if is closed and countable subset of , the rank of is defined.

## 5. Domains with totally disconnected complement

The simplest possible structure of a boundary component is that it consists of a single point. The aim of this section is to show that if we are only concerned with the topological properties, we may always assume this is the case.

###### Proposition 5.1.

Every domain in is homeomorphic to a domain in that has totally disconnected complement.

To prove Proposition 5.1, we employ the theory of decomposition spaces [9]. A decomposition of a topological space is simply a partition of . The non-degenerate elements of a decomposition are those elements of the partition that contain at least two points. The decomposition space associated to a decomposition of a topological space is the topological quotient of obtained by, for each , identifying the points of . A decomposition is an upper semi-continuous decomposition if each element is compact, and given any and any open set containing , there is another open set containing with the property that if intersects , then . If is an upper semi-continuous decomposition of a separable metric space , then is a separable and metrizable space. [9, Proposition I.2.2].

We will use one powerful theorem from classical decomposition space theory. It identifies decompositions of that are homeomorphic to itself [21].

###### Theorem 5.2 (Moore).

Suppose that is an upper semi-continuous decomposition of with the property that for each , both and are connected. Then is homeomorphic to .

We also employ a powerful theorem of classical complex analysis. It states that any domain in can be mapped conformally (and hence homeomorphically) to a slit domain, i.e., to a domain in that is either complete, or whose complementary components are points or compact horizontal line segments in [11, V.2].

###### Theorem 5.3.

Any domain in is conformally equivalent to a slit domain.

###### Lemma 5.4.

If is a slit domain, then the components of containing at least two points form the non-degenerate elements of an upper semi-continuous decomposition of .

###### Proof.

Let denote the decomposition of whose non-degenerate elements are those components of that contain at least two points. Let , and let be an open set containing . Without loss of generality we may assume that and that is an open and bounded subset of containing .

Set

 l=max{x:(x,0)∈R2∖U and x<0} and r=min{x:(x,0)∈R2∖U and x>1}.

Since is open and is a compact and connected subset of , there are closed, non-degenerate intervals such that

 (L×{0})⊆((l,0)×{0})∩Ω∩U and (R×{0})⊆((1,r)×{0})∩Ω∩U.

If is not upper semi-continuous, then for every , we may find some horizontal line segment and points such that

 (xn,yn)∈gn∩N(g,1/n) and(x′n,yn)∈gn∩(S2∖U).

After passing to a subsequence, we may assume that tends to a point of . Moreover, passing to another subsequence if needed, we may assume that either

 limsupn→∞x′n≤l orliminfn→∞x′n≥r.

We consider the latter case; a similar argument applies in the former. For sufficiently large , the point is greater than any point in , while is less than any point in . By the connectedness of , we conclude that there is a point with . After passing to yet another subsequence, we may assume that converges to a point in . This is a contradiction as is open. See Figure 1.∎

###### Proof of Proposition 5.1.

Let be a domain in . By Theorem 5.3, there is a homeomorpism , where is a slit domain. By Lemma 5.4, the components of with at least two points form the non-degenerate elements of an upper semi-continuous decomposition of . As each element of is either a point or a compact line segment in , the hypotheses of Theorem 5.2 are satisfied. Thus there is a homeomorphism Let denote the standard projection map. By definition is a homeomorphism and is totally disconnected. Thus is a homeomorphism, and the image of under this map has totally disconnected complement. ∎

The ends of a domain in with totally disconnected complement are particularly easy to understand: they are in bijection with the points of the complement, which are precisely the boundary components.

###### Proposition 5.5.

Suppose is a domain in with totally disconnected complement. Then there is a homeomorphism with the property that a sequence represents the end if and only if it converges to .

###### Proof.

Since a totally disconnected subset of cannot have interior, we see that Moreover, it is clear that and are naturally homeomorphic. Hence Theorem 4.11 provides the desired homeomorphism. ∎

The following statement transfers the work of this section to the general setting. For the remainder of this section, we assume that is a metric space that has compact completion and is homeomorphic to a domain in .

###### Corollary 5.6.

Suppose that satisfies the condition or is a domain in . Then there is a continuous surjection such that is a homeomorphism onto a domain with totally disconnected complement. Moreover, the map is constant on each boundary component , and for any , there is such that

 (5.1) h−1(BS2(h(E),ϵ′))⊆N¯¯¯¯X(E,ϵ).

Finally, induces a homeomorphism from to .

###### Proof.

We address only the case that satisfies the condition. We have assumed that is homeomorphic to a domain in . Hence by Proposition 5.1, there is a homeomorphism , where is a domain with totally disconnected complement.

By Theorem 4.11 and Remark 4.2, there is a homeomorphism with the property that a Cauchy sequence converges to a point of the boundary component if and only if represents the end . Moreover, Proposition 5.5 provides a homeomorphism with the property that a sequence represents the end if and only if it converges to the boundary point . We define the extension of to by setting , where is the boundary component containing . The naturality properties of and ensure that so defined is continuous. As is a homeomorphism onto and , the definitions show that the extended map is a surjection.

Now, let and . By construction (or from the fact that is a homeomorphism and is totally disconnected), the set consists of a single point in . Suppose that there is no such that (5.1) holds. Then there is a sequence of points

 xn∈¯¯¯¯¯X∖N¯¯¯¯X(E,ϵ)

such that