Quasisymmetric uniformization

# Quantitative quasisymmetric uniformization of compact surfaces

Lukas Geyer  and  Kevin Wildrick Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717
###### Abstract.

Bonk and Kleiner showed that any metric sphere which is Ahlfors 2-regular and linearly locally contractible is quasisymmetrically equivalent to the standard sphere, in a quantitative way. We extend this result to arbitrary metric compact orientable surfaces.

###### 2010 Mathematics Subject Classification:
Primary 30C65; Secondary 30C62, 51F99

## 1. Introduction and statement of results

Through isothermal coordinates, every Riemannian metric on a compact orientable surface determines a Riemann surface structure on . By the classical uniformization theorem, carries a conformally equivalent Riemannian metric of constant curvature (in the case of a sphere), (for a torus), or (for higher genus surfaces). Hence, the original Riemannian metric on can be conformally deformed to a metric of constant curvature.

The purpose of this note is to extend the above discussion to certain classes of possibly non-smooth distances on compact orientable surfaces. In this setting we have a metric, but no smooth structure, and the appropriate category replacing the class of conformal mappings is the class of quasisymmetric mappings.

In a metric space we will denote the distance between and by , or if the metric space is clear from the context. Let and be metric spaces. An embedding , i.e., a homeomorphism from onto its image , is a quasisymmetric embedding if there is a homeomorphism such that for all triples of distinct points , , and in ,

 |ϕ(a)−ϕ(b)|Y|ϕ(a)−ϕ(c)|Y≤η(|a−b|X|a−c|X),

The homeomorphism is called a distortion function for the mapping . If a quasisymmetric embedding has a distortion function , it will be called an -quasisymmetric embedding. The inverse of an -quasisymmetric map is -quasisymmetric with . A proof of this fact can be found in the book of Heinonen [Hei01, Proposition 10.6], which also serves as an excellent introduction to the theory of quasisymmetric mappings on metric spaces. A quasisymmetric mapping distorts relative distances by a controlled amount. While it may distort distances, it must do so more-or-less isotropically.

A motivating example arises from Cannon’s Conjecture in geometric group theory. The boundary at infinity of a Gromov hyperbolic group carries a natural family of visual metrics, any two of which are quasisymmetrically equivalent. For this reason, one might say that the metric on the boundary of a Gromov hyperbolic group is defined only up to quasisymmetry. Cannon’s Conjecture can be phrased as follows: If the boundary of a Gromov hyperbolic group is homeomorphic to the sphere , then each visual metric on is quasisymmetrically equivalent to the standard metric on . If Cannon’s conjecture is true, then a Gromov hyperbolic group whose boundary is homeomorphic to acts discretely and co-compactly by isometries on 3-dimensional hyperbolic space. See [Bon06] and the references therein for a more detailed discussion of this problem.

The following quasisymmetric uniformization theorem of Bonk and Kleiner [BK02] led to significant progress on Cannon’s Conjecture:

###### Theorem 1 (Bonk-Kleiner).

Let be an Ahlfors -regular metric space that is homeomorphic to . Then is quasisymmetrically equivalent to equipped with the smooth metric of constant curvature if and only if is linearly locally contractible.

A metric space is linearly locally contractible, if there is a constant such that for each point and radius , the ball is contractible inside the dilated ball . The metric space is Ahlfors -regular if there is a constant such that for each and radius , the two-dimensional Hausdorff measure satisfies

 r2K≤H2(B(x,r))≤Kr2.

Theorem 1 is quantitative in the sense that the quasisymmetric map can be chosen to have a distortion depending only on the data of , i.e., the constants and appearing in the linear local contractibility and Ahlfors -regularity conditions.

The linear local contractibility condition is a quantitative quasisymmetric invariant, i.e., if is -quasisymmetric and is linearly locally contractible with constant , then is linearly locally contractible with a constant that depends only on and . In contrast, Ahlfors -regularity is not a quasisymmetric invariant.

Given a Gromov-hyperbolic group with homeomorphic to , each visual metric on is linearly locally contractible. However, it is not known whether there always is an Ahlfors -regular visual metric on . Simple examples show that Theorem 1 fails without the assumption of Ahlfors -regularity.

Problems outside of geometric group theory, such as the search for an intrinsic characterization of up to bi-Lipschitz equivalence (note that Ahlfors -regularity is a bi-Lipschitz invariant), led to the development of versions of Theorem 1 for non-compact simply connected surfaces [Wil08] and for a large class of planar domains [MW13], as well as a local version [Wil10]. In this work, we provide a version of Theorem 1 that applies to all orientable compact surfaces, including those of higher genus. This is part of an ongoing program to complete the classification of Ahlfors -regular and linearly locally contractible metric surfaces up to quasisymmetry.

###### Theorem 2.

Let be a metric compact orientable surface. Assume that is Ahlfors -regular and linearly locally contractible. Then there exists a Riemannian metric of constant curvature , , or on such that the identity map is quasisymmetric with a distortion function depending only on the data of .

Here the data of consists of the topological genus of , and the constants in the Ahlfors -regularity and linear local contractibility conditions.

Except for the statement of dependence of the distortion function only on the data of , Theorem 2 follows immediately from the local uniformization theorem of [Wil10] and an elementary local-to-global result for quasisymmetric mappings due to Tukia and Väisälä [TV80, Theorem 2.23]. While a quantitative proof is significantly more involved, it provides much more information about the “space” of quasisymmetric structures on surfaces. For example, let us suppose that satisfies the assumptions of Theorem 2 and is homeomorphic to a torus. All smooth Riemannian metrics of constant curvature on the torus are quasisymmetrically equivalent, but not with a uniform distortion function. In essence, the quantitative Theorem 2 allows us to assign to a point in the Riemann moduli space of the torus that is “roughly optimal”. Finding a smooth metric on the torus that is quasisymmetrically equivalent to with “minimal” quasisymmetric distortion seems to be both a difficult and ill-defined question, but one of obvious interest.

Our proof of Theorem 2 proceeds as follows. Let be a metric surface as given in the theorem. By the local uniformization result of [Wil10], has an atlas of uniformly quasisymmetric mappings. Our first step is to create a compatible conformal atlas, giving the structure of a Riemann surface. This step can be thought of as creating “quasi-isothermal” coordinates; while the chart transitions are conformal, the chart mappings themselves are only quasisymmetries. The classical uniformization theorem then provides a globally defined conformal homeomorphism to a Riemann surface that is the quotient of the sphere, plane, or disk by an appropriate group of Möbius transformations. The Riemann surface inherits a Riemannian metric of constant curvature , , or , respectively. Although conformal maps are locally quasisymmetric, this does not immediately give us global information about the metric properties of the uniformizing map . We show that the uniformizing mapping is in fact globally quasisymmetric with a distortion function that depends only on the data of . The key idea in this step is a type of Harnack inequality. The claim in the theorem then follows by taking to be the pull-back of the metric under .

## 2. Background and preliminary results

### 2.1. Notation

In a metric space , we will denote the distance between points and of by or when the space is understood. We denote by

 BX(x,r)=B(x,r)={y:|x−y|

the open ball of radius centered at and by the corresponding closed ball. For an open or closed ball of radius and a number , the notation denotes the same type of ball with the same center and radius .

We denote the complex plane with the standard Euclidean metric by , and the disk model of hyperbolic space equipped with the standard hyperbolic metric of curvature by . The open ball of radius centered at is denoted by in and by in , i.e.,

 Cα=BC(0,α), and Dα=BD(0,α).

Note that is the Euclidean radius for , whereas it is the hyperbolic radius for .

### 2.2. Quasisymmetric mappings

Let be an embedding of metric spaces. For and , define

 Lϕ(x,r) :=sup{|ϕ(x)−ϕ(y)|Y:|x−y|X≤r}, and lϕ(x,r) :=inf{|ϕ(x)−ϕ(y)|Y:|x−y|X≥r}.

If is an -quasisymmetric embedding, then for and ,

 Lϕ(x,r1)lϕ(x,r2)≤η(r1r2).

We will need a statement about the equicontinuity of quasisymmetric maps, which is a slightly more general version of [Hei01, Corollary 10.27].

###### Lemma 3.

Let be an -quasisymmetric embedding of metric spaces and let and be positive real numbers. If and , then

 ω(t)=max{3DVtDU,DVη(3tDU)}

is a modulus of continuity for .

###### Proof.

Given with , there exists such that . Then

 |ϕ(x1)−ϕ(x2)|≤|ϕ(x1)−ϕ(x3)|η(|x1−x2||x1−x3|)≤DVη(3|x1−x2|DU).

If , the trivial estimate yields the desired estimate. ∎

Note that the modulus of continuity provided by Lemma 3 is not scale-invariant. If the metrics on and are scaled by the same quantity, the resulting modulus of continuity may still change; see subsection 2.4.

The following result, which is a slight variation of [TV80, Theorem 2.23], gives a local-to-global result for quasisymmetric homeomorphisms between compact spaces in terms of Lebesgue numbers. We include a proof for the reader’s convenience. Recall that is a Lebesgue number for a covering if for every set with there exists such that .

###### Theorem 4 (Tukia-Väisälä).

Let be a homeomorphism between compact connected metric spaces and . Suppose that

• is a finite open covering of with Lebesgue number .

• satisfies the implication

 |x−x′|=LX/2⟹|F(x)−F(x′)|≥δ,
• there is a quasisymmetric distortion function such that for each , the restricted mapping is -quasisymmetric.

Then is quasisymmetric with distortion function depending only on and the ratios and .

###### Proof.

Let , , and be distinct points of . Set

 ρ=|a−b||a−c|, and ρ′=|F(a)−F(b)||F(a)−F(c)|.

We consider four cases.

1. Suppose that Then is contained in some , and so

2. Suppose that but . Since is connected with , there exists with . Then there exists such that and

 |F(a)−F(x)|≥δ and |a−b||a−x|≤2ρdiamXLX.

This implies that

 ρ′=|F(a)−F(b)||F(a)−F(x)||F(a)−F(x)||F(a)−F(c)|≤η(2ρdiamXLX)diamYδ.
3. Suppose that but By the same argument as in the previous case, there exists with , as well as an index such that . This gives

 |F(a)−F(x)|≥δ and |a−x||a−c|≤2ρdiamXLX.

which implies that

 ρ′=|F(a)−F(b)||F(a)−F(x)||F(a)−F(x)||F(a)−F(c)|≤diamYδη(2ρdiamXLX).
4. Suppose that Then and , so

 ρ′≤2(diamXLX)(diamYδ)ρ

Combining these estimates on in terms of yields the desired result. ∎

### 2.3. Conformality and quasisymmetry

A conformal mapping between domains in is quasisymmetric when restricted to a relatively compact subdomain , with quasisymmetric distortion depending only on and . This fact, which should be compared to Koebe’s distortion theorem, will play a key role in the proof of Theorem 12. A more general statement is even true: one may consider quasiconformal mappings in higher dimensional Euclidean spaces, although the quasisymmetric distortion then also depends on the maximal quasiconformal dilatation. See [Väi81, Theorem 2.4] for a proof of this fact, which we will use in the following form:

###### Proposition 5.

Let be a conformal embedding. For each , the restriction is a quasisymmetric embedding with distortion function that depends only on .

An easy consequence of Proposition 5 is the following.

###### Corollary 6.

There is a universal constant such that for any conformal embedding ,

 G(Cβ1)⊆BC(G(0),lG(0,1/2)6).
###### Proof.

By Proposition 5, there is a universal quasisymmetric distortion function for . Hence, if , then

 LG(0,β)≤η(2β)lG(0,1/2)

Thus, choosing so small that fulfills the requirements of the statement. ∎

We will also need a hyperbolic version of this result. Recall that the hyperbolic disk is equipped not just with a conformal structure, but also with the standard hyperbolic metric. Our proof employs Koebe’s distortion theorem for specificity, but could also be carried out using Proposition 5 alone.

###### Proposition 7.

Let be a conformal embedding of the Euclidean unit disk into the hyperbolic plane. Then the restriction is a quasisymmetric embedding with a universal distortion function.

###### Proof.

The map is conformal in , so by Koebe’s distortion theorem [Dur83], the image contains the Euclidean disk , and the image of the Euclidean disk is contained in the Euclidean disk . This implies

 BC(G(0),|G′(0)|4)⊆G(C1)⊆D,

and

 G(C1/10)⊆BC(G(0),|G′(0)|8)⊆BC(G(0),1−|G(0)|2).

It follows from the definition of the hyperbolic metric that the identity mapping

 id:BC(G(0),1−|G(0)|2)→D

is, up to scaling, a bi-Lipschitz embedding with universal constants, so it is quasisymmetric with universal distortion. Since the composition of quasisymmetric maps is quasisymmetric, quantitatively, the result follows from Proposition 5. ∎

The following statement is an easy corollary of Proposition 7.

###### Corollary 8.

There is a universal constant such that for any conformal embedding ,

 G(Cβ2)⊆BD(G(0),lG(0,1/2)6).
###### Proof.

By Proposition 7, there is a universal quasisymmetric distortion function for . Hence, if , then

 LG(0,β)≤η(10β)lG(0,1/10)≤η(10β)lG(0,1/2).

Choosing so small that , we get the claim of the corollary. ∎

For the remainder of the paper, for convenience we define .

### 2.4. The data of X, scalings, and normalization

Let be a metric space that is Ahlfors -regular with constant , linearly locally contractible with constant , and homeomorphic to a compact orientable surface. We will refer to and as the data of .

For , we may form a new metric space by multiplying the original metric on by . Then is again Ahlfors -regular and linearly locally contractible, and has the same data as . Moreover, the identity mapping from to is -quasisymmetric with . Hence, in the proof of Theorem 2, we may scale the domain as we see fit.

For the remainder of this article, let be a metric space that is Ahlfors -regular with constant , linearly locally contractible with constant , homeomorphic to a compact orientable surface, and has unit diameter. When we state that a quantity depends only the data of , we are implicitly assuming this normalization.

The main reason for this convention (aside from notational convenience) is that Lemma 3 is not scale-invariant. Without this normalization, we would not be able to say that certain moduli of continuity depend only on the data of .

## 3. Finding a conformal structure

By [Wil10, Theorem 4.1], possesses a quasisymmetric atlas:

###### Theorem 9.

There is a quantity and a quasisymmetric distortion function , each depending only on the data of , such that for each there is a neighborhood of such that

1. ,

2. there exists an -quasisymmetric map with ,

The original theorem in [Wil10] does not include the normalization . However, since is -quasisymmetric, the basic distortion estimates [Hei01, Proposition 10.8] imply that where depends only on . As the Möbius transformation has quasisymmetric distortion depending only , we may assume that . Accordingly, given a pair where is a homeomorphism, we will call the center of .

We use the atlas provided by Theorem 9 to produce a conformal atlas on that is adapted to its metric. We have separated the construction into two lemmas. The first is purely metric; the second modifies the output of the first.

###### Lemma 10.

Let be given. Then there exists a quasisymmetric distortion function , radii , , and a positive integer , such that the following statements hold:

1. There exists an atlas of , where each mapping is an -quasisymmetric homeomorphism with center denoted by .

2. The collection is pairwise disjoint.

3. The collection covers .

4. For each , it holds that , and

 Cα⊆ϕj(B(xj,r0))⊆ϕj(B(xj,10r0))⊆Cρ.

Moreover, depends only on the data of , while , , and depend only on the data of and .

Note that the collection forms an open cover of for which is a Lebesgue number; cf. Theorem 4. Moreover, Lemma 3 implies that for each , the restriction and its inverse have moduli of continuity that depend only on and the data of .

###### Proof.

For each , let be the -quasisymmetric mapping provided by Theorem 9 with , so that

 B(x,1/A20)⊆Ux⊆B(x,1/A0).

Applying Lemma 3 to and its inverse, we see that there is a common modulus of continuity for all of the mappings , depending only on the data of .

Hence, there is a radius and a number , each depending only on and the data of , such that for each ,

 (1) ϕx(B(x,10r0))⊆Cρ, and ϕx(B(x,r0))⊇Cα.

Let be a maximal -separated set in . Then the open balls are pairwise disjoint, while the open balls cover . Since is Ahlfors -regular and we have assumed that , this implies that is comparable to and therefore depends only on and the data. Moreover, as , for each , it holds that . In particular is also a cover of . This shows that is the desired atlas. ∎

We now adapt the atlas given in Lemma 10 so that the transition mappings are conformal. This step is similar to the proof that a quasiconformal structure on a surface has a compatible conformal structure; see [Kuu67] and [Can69].

We recall that is a universal constant given by Corollaries 6 and 8.

###### Lemma 11.

There exists a quasisymmetric distortion function , radii , , and a positive integer , all depending only on the data of , such that the following statements hold:

1. There exists an atlas of , where each mapping is an -quasisymmetric homeomorphism with center denoted by .

2. The collection is pairwise disjoint.

3. The collection covers .

4. For each , it holds that , and

 Cα⊆ψj(B(xj,r0))⊆ψj(B(xj,10r0))⊆Cβ.
5. The transition maps are conformal wherever defined.

###### Proof.

We begin by letting be a number which will be determined below and will depend only on the data of . Consider the atlas provided by Lemma 10. Let us say that this atlas is given by -quasisymmetric charts where depends only on the data of . The associated radii , and the number of charts depend only on the data of and .

Since is connected, the charts can be relabeled to satisfy

 Uj+1∩(U1∪…∪Uj)≠∅

for .

Define by setting . We will iteratively construct for as follows. For , we assume that the already constructed map is quasisymmetric on , has center , and that if , then the transition functions are conformal where defined.

In the following, we will write for indices in .

For , define . Then is quasisymmetric and hence quasiconformal on . Therefore, the complex dilatation of is well-defined (up to a.e. equivalence) on . Given another index , it holds that on . It therefore follows from the conformality of that a.e. on the intersection . This shows that there exists a measurable Beltrami coefficient with a. e. on for each index , and on . We extend to the whole plane by for . By the Measurable Riemann Mapping Theorem there exists a unique quasiconformal map , normalized by , , with complex dilatation a.e. By the symmetry of and the chosen normalization, , and so . We define . The transformation formula for Beltrami coefficients (see, e.g., [LV73, IV.5.1]) shows that if , then is conformal. Moreover, .

We claim that for each , the mapping has a quasisymmetric distortion function that depends only on the data of ; as this is true of , it suffices to prove the same of , and we may also assume that . As a normalized quasiconformal self-map of the unit disk, is quasisymmetric with distortion controlled by the maximal dilatation , see e. g. [Väi81, Theorem 2.4]. By an inductive argument, it is easy to show that this dilatation is bounded by a constant depending only on (which depends only on the data of ) and the number of charts (which depends on the data and ). However, the bound is actually independent of , by the following argument.

Fix . By the uniform quasisymmetry of , there is a quantity , depending only on the data of , such that for all indices with , the complex dilatation of satisfies For , define

 Fj,k=Dj,k∖k−1⋃l=1Dj,l, Fk=C1∖k−1⋃l=1Dk,l, and Fj=C1∖j−1⋃k=1Dj,k.

If , then . If , then , and hence

 μk|ϕk∘ϕ−1j(Fj,k)=0.

The transformation formula for Beltrami coefficients now shows that for almost-every point ,

 μj(z)=μj,k(z)=μψk∘ϕ−1j(z)=μhk∘ϕk∘ϕ−1j(z)=μϕk∘ϕ−1j(z).

Since we see that As discussed above, we may now conclude that each of the mappings has a quasisymmetric distortion function that depends only on the data of .

We have now seen the atlas of satisfies conditions (1) and (5) of the statement. Moreover, setting , the conditions (2) and (3) follow directly from the corresponding statements for the atlas . Note that only condition (4) involves the constant . We now show how to choose so that condition (4) is satisfied.

Let us make the convention that is the identity. As discussed above, the mappings are uniformly quasisymmetric with a distortion function depending only on the data of . Hence, by Lemma 3, there is a common modulus of continuity for all of the mappings that depends only on the data of . Since is a universal constant, we may choose depending only on the data of such that for each , it holds that Having so chosen , the radius depends only on the data of , and so we may also choose depending only on the data of such that for for each , This establishes condition (4). ∎

### 3.1. Uniformizing to a standard metric

The atlas given by Lemma 11 determines a conformal structure on the compact orientable surface , i.e., the pair determines a Riemann surface. By the classical uniformization theorem, is conformally equivalent to a standard Riemann surface , where denotes the standard Riemann surface structure on the sphere, the plane, or the unit disk, and is a discrete group of Möbius transformations acting freely and properly discontinuously on . The standard spherical, plane, or hyperbolic Riemannian metric on then descends to a Riemannian metric of constant curvature , , or on , compatible with the conformal structure. We fix a uniformizing conformal homeomorphism , and equip with the distance function arising from the Riemannian metric of constant curvature. Denote by the quotient mapping from to . Recall that may be expressed as where the infimum is taken over all smooth paths in such that the projected path connects and . A priori, it is not clear how the properties of the distance or the map depend on the original metric space . The following statement is the main result of this paper, and completes the proof of Theorem 2.

###### Theorem 12.

The uniformizing map is -quasisymmetric, with distortion depending only on the data of .

###### Proof of Theorem 12..

In the case that , then itself must be homeomorphic to , and so Theorem 1 implies Theorem 12. We consider the remaining planar and hyperbolic cases together.

Define by . Then is a conformal homeomorphism that lifts to a conformal embedding .

We use a sequence of claims to complete the proof.

###### Claim 1.

Let . The projection maps isometrically onto In other words, for each , .

###### Proof of Claim 1.

For , define

 r(u)=min{|u−γ(u)|U:γ∈Γ∖{id}}.

If , then is independent of and depends only on the group . If , then is bounded below by the minimal translation distance of a non-identity element of , and is a -Lipschitz function of , see e. .g. [Bea83, Section 7.35]. These facts imply that for any and ,

• if , then is an isometry,

• if is injective, then .

Since is injective, it follows that . Corollaries 6 and 8 now complete the proof of this claim. ∎

###### Claim 2.

For each , the mapping is quasisymmetric with distortion that depends only on the data of .

###### Proof of Claim 2.

By Propositions 5 and 7, the mapping restricted to is quasisymmetric with a universal distortion function. By Claim 1, this is also true of . Since we may write , and is -quasisymmetric where depends only on the data, it follows that for each , the mapping is quasisymmetric with distortion that depends only on the data of . According to Lemma 11, , implying the claim. ∎

###### Claim 3.

There is a constant depending only on the data of such that for each ,

 (2) lgj(0,α)≥diamYC.
###### Proof of Claim 3.

As covers and , it holds that also covers . Since is a homeomorphism, this implies that As is connected, it follows that

 (3) max{diamgj(Cβ):j=1,…,n}≥diamYn.

Recall that the number of charts depends only on the data of .

Consider indices and in , and suppose that . Then , and so On the other hand, , and so , implying . Moreover, since is a quasisymmetric embedding with universal distortion function, there is a quantity depending only on the ratio of to , and hence only on the data of , such that

 lgj(0,α)≥Lgj(0,β)C0≥diamgj(Cβ)2C0

 diamgj(Cβ)≥Lgj(0,β)≥lgj(0,α),

we have now shown that the quantities

 lgj(0,α), Lgj(0,β), diamgj(Cβ), and dY(F(xj),F(xk))

are all comparable with constants that depend only on the data of . The same is true with the roles of and reversed.

Since the open sets cover the connected space , for any pair of indices , there is a sequence of distinct indices of length at most so that for each Since depends only on the data of , (3) proves the claim. ∎

We complete the proof of Theorem 12 by employing Theorem 4. We consider the covering of given by . By Claim 2, the mapping is quasisymmetric on each element of this cover with a distortion function that depends only on the data of . Moreover, Lemma 11 implies that this cover has Lebesgue number . Suppose that satisfy . We may find indices and in such that and . Then , so and

 |F(x′)−F(x)|≥|F(x′)−F(xj)|η(|x′−x