QR Mappings on sub-Riemannian Manifolds

# Quasiregular mappings on sub-Riemannian manifolds

Katrin Fässler Anton Lukyanenko  and  Kirsi Peltonen Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, FI-00014 Helsinki, Finland Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076, Aalto, Finland
July 6, 2019
###### Abstract.

We study mappings on sub-Riemannian manifolds which are quasi-regular with respect to the Carnot-Carathéodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint.

As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres.

One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis–Mostow derivative (which generalizes the classical and Pansu derivatives).

###### 2010 Mathematics Subject Classification:
30L10
K.F. was supported by the Academy of Finland, project number 252293. A.L. was supported by National Science Foundation grants DMS 0838434 and DMS 0901620 and Department of Mathematics and Systems Analysis of Aalto University.

## 1. Introduction

A homeomorphism between two metric spaces is quasiconformal if it has bounded infinitesimal distortion. That is, there exists a finite constant such that

 limsupr→0sup|x−y|≤r|fx−fy|inf|x−y|≥r|fx−fy|≤Hfor all x. (1.1)

In many contexts, one broadens the definition to include locally quasiconformal mappings, that is, covering mappings satisfying

 H(x,f):=limsupr→0sup|x−y|≤r|fx−fy|inf|x−y|=r|fx−fy|≤Hfor all x. (1.2)

While is a reasonable quantity to measure the dilatation or distortion also for a non-injective mapping , the expression appearing in (1.1) would be inappropriate in this regard since may vanish for arbitrarily small values of even in points where is locally injective.

The theory of quasiconformal mappings is well-studied in a variety of contexts, see [Mos73, KR85, Pan89, KR95, HK95, HK98, HKST01]. The focus of this paper is on branched covering mappings satisfying (1.2) in the context of sub-Riemannian manifolds.

Recall that a sub-Riemannian manifold is a smooth manifold with a choice of smooth subbundle satisfying the Hörmander condition. That is, the iterated brackets of span . A smooth choice of inner product on then gives a sub-Riemannian path metric (Carnot-Carathéodory distance). We will assume that every sub-Riemannian manifold is equipped with an inner product , corresponding norm , and distance function .

A sub-Riemannian manifold is equiregular if the dimension of and the bundles is constant over all of . One writes , so that the topological dimension of is given by . The Hausdorff dimension, on the other hand, is given by , and the Hausdorff measure with respect to the cc-distance in dimension is equivalent to Lebesgue measure on [Mit85a]. Without further specification, the term ‘measure’ on a sub-Riemannian manifold will refer to one of these equivalent measures. We often write for the measure of a set in a sub-Riemannian manifold.

We say that a mapping between manifolds is a branched cover if it is continuous, discrete, open, and sense-preserving (notice that we do not assume a branched cover to be onto). It is known that a branched cover between manifolds of dimension is a local homeomorphism away from a branch set of topological codimension at least two (Chernavskii’s theorem, see [Ric93]).

###### Definition 1.1.

Let and be two equiregular sub-Riemannian manifolds. We call a mapping -quasiregular if it is constant or if

1. is a branched cover onto its image
(i.e., continuous, discrete, open and sense-preserving),

2. is locally bounded on ,

3. for almost every ,

4. the branch set and its image have measure zero.

A mapping is said to be quasiregular if it is -quasiregular for some .

###### Remark 1.2.

We expect that the last condition in Definition 1.1 is unnecessary, but proving this is beyond the scope of the current paper, as it requires the development of a modulus definition of quasiregularity, see [Ric93, OR09]. Note also that while in this paper the distortion will always be computed with respect to Carnot-Carathéodory distances on the corresponding sub-Riemannian manifolds, the definition could be applied to more general metric spaces as well.

A non-injective quasiregular mapping acting on space is uniformly quasiregular (UQR for short) if, for all , we have that is -quasiregular for some not depending on (note that a priori it is not clear that the composition of two quasiregular maps is quasiregular). Injective UQR mappings are called uniformly quasiconformal.

Denote by the semigroup generated by a UQR map , that is

 Λ={fn:M→M | f uniformly quasiregular% , n∈N}. (1.3)

The Fatou set of is then defined as

 F(f)={p∈M | there is an open U,p∈U,Λ|U normal}, (1.4)

where normal means that every sequence of contains a locally uniformly convergent subsequence. The Julia set of is then defined as .

We prove:

###### Theorem 1.3.

Every lens space, with its natural sub-Riemannian structure, admits a uniformly quasiregular self-mapping with nonempty branch set and a Cantor set type Julia set.

We proceed to focus on an invariant of UQR mappings. In the setting of Riemannian spheres, Iwaniec and Martin [IM96] proved the existence of an invariant measurable structure for any abelian uniformly quasiregular semigroup. Tukia established earlier the corresponding result for groups of quasiconformal mappings, and [IM96] is an adapatation of this approach to the non-injective setting. Beyond the Riemannian setting, the existence of an invariant conformal structure for groups of quasiconformal mappings on compactified Heisenberg groups , , was first proved by Chow in [Cho96]. In [BFP12], the case of non-injective quasiregular mappings was studied independently for the first Heisenberg group , where invariant conformal structures can be related to invariant CR structures. It is not clear how this relation would generalize to higher dimensions.

In the present paper, generalizing [Cho96], we define a conformal structure on a sub-Riemannian manifold as an inner product on the horizontal bundle , up to rescaling (cf.  also the ‘sub-conformal structures’ on contact manifolds in [FV07, Definition 2.1. (2)]). That is, two inner products and represent the same conformal structure if we have for some positive function on . We say that a conformal class is measurable if it is represented by a measurable choice of an inner product (that is possibly undefined on a set of measure 0).

If the sub-Riemannian manifold is equipped with an inner product , one can in a natural way interpret a new inner product as function to a higher-rank symmetric space (see the proof of Theorem 6.1). The conformal class of is said to be bounded (with respect to ) if is a bounded map.

A conformal class represented by is invariant under a quasiregular mapping if we have

 ⟨f∗u,f∗v⟩=λ(x)⟨u,v⟩ (1.5)

for some positive function and any choice of vectors . We formalize the meaning of in §5 using the Margulis–Mostow derivative (which is equivalent to the Pansu derivative when a Carnot group is concerned). The discussion of differentiability leads to the following question:

###### Question 1.4.

Let be a mapping between sub-Riemannian manifolds with a continuous Margulis–Mostow derivative. Is then continuously differentiable in the classical sense? (See §6.2, where we prove the measurability of the Margulis–Mostow derivative.)

After addressing the differentiability questions, we prove in §6:

###### Theorem 1.5.

Every uniformly quasiregular self-mapping of a sub-Riemannian manifold admits an invariant measurable conformal structure that is bounded with respect to the given inner product.

The structure of the paper is as follows. After discussing basic properties of QR mappings in §2, we provide an explicit QR mapping on lens spaces in §3. We apply a “conformal trap” construction in §4 to transform it into a UQR mapping and thus prove Theorem 1.3. Following a review on differentiability in §5, we prove Theorem 1.5 in §6.

## 2. Quasiregular mappings

The theory of quasiregular mappings in developed as a generalization of the study of complex-analytic mappings of one variable. A crucial result is the following Picard-type theorem (see also [DP13]):

###### Theorem 2.1 (Rickman).

Let be a non-constant -quasiregular mapping. Then misses at most finitely many points, with the number of points bounded above in terms of and .

In the classical theory, one starts with the assumption that the quasiregular mapping is in the Sobolev class , with the differential satisfying the condition almost everywhere. One then proves that the analytic assumptions are equivalent to the topological and metric properties that we take in this paper as the definition of quasiregularity.

Recall that Carnot groups (with the Heisenberg group as the primary example) provide a generalization of Euclidean space admitting a sub-Riemannian metric. Namely, these are simply connected nilpotent Lie groups that admit a left-invariant sub-Riemannian metric and a one-parameter family of homotheties.

Heinonen and Holopainen first generalized the notion of quasiregularity and proved basic properties of quasiregular mappings on Carnot groups in [HH97]. Their assumptions were later relaxed by Dairbekov in [Dai99]. Quasiregular mappings satisfying these weaker conditions were then studied in general Carnot groups and the restricted setting of the Heisenberg group and other two-step Carnot groups in [Dai00b, MV06, Vod07] and other works. As in the classical case, these authors started with the analytic definition of quasiregularity. This definition does not easily generalize to arbitrary sub-Riemannian manifolds.

Following the Heinonen–Koskela approach (see [HK95]) of rephrasing quasiconformality in terms of strictly metric properties, we start with a metric definition of quasiregularity. Note that our definition is akin to the characterization of Euclidean quasiregular mappings in [MRV69], see also [Ric93], and to the definition of quasiregularity in [OR09].

So far, important analytic properties – such as Sobolev regularity and the -inequality – of metrically defined quasiregular mappings are only known in Carnot groups, see the discussion in [UV10, 4], or for Euclidean source spaces, which is the case considered in [OR09]. We conjecture that analogous properties hold for mappings between equiregular sub-Riemannian manifolds, but do not pursue this further in the present paper.

### 2.1. Rectifiable curves

We first recall some terminology and notation for curves, spheres, and balls in metric spaces (see also for instance [BR96]).

Let be a sub-Riemannian manifold and a continuous curve. Recall that the length of is defined as

 length(γ)=sup∑d(γ(ti),γ(ti−1))≤∞,

where the supremum is taken over all partitions of . A curve is a geodesic if . By the Hopf-Rinow theorem, any pair of sufficiently nearby points in can be joined by a geodesic. If is complete, then in fact any two points can be joined by a geodesic.

Recall further that is said to be rectifiable if , in which case it can be given a Lipschitz reparametrization. An absolutely continuous curve in a sub-Riemannian manifold, in particular a Lipschitzly parametrized curve, is rectifiable if and only if it is horizontal, that is, for a.e. . In this case the length of can be computed using the sub-Riemannian inner product as

 length(γ)=∫ba|˙γ(t)|dt.

We will denote balls and spheres (respectively) in a metric space as follows:

 B(x,r)={y∈M:d(x,y)

### 2.2. Definitions of dilatation

In this subsection, we clarify the different definitions of dilatation in the introduction and show that a quasiregular map is locally quasiconformal away from the branch set. We then provide a way to compute the maximum dilatation of a quasiregular map under some smoothness conditions.

###### Lemma 2.2.

Let be a locally geodesic space and a homeomorphism. Then for every and small enough (depending continuously on ),

 ℓ:=inf|x−y|≥r|fx−fy|≥inf|x−y|=r|fx−fy|=ℓ∗. (2.1)
###### Proof.

Without loss of generality, we may assume that is geodesic. To see this, let be a geodesic open neighborhood of . Since both and approach as , we have that for sufficiently small , the infima in (2.1) remain unchanged if solely points in are considered. We may therefore replace with and with .

Fix , and let minimize the expression (perhaps up to a small error term). Let be a geodesic joining to , and its preimage in . By continuity of the metric on , there is an on with . Since is a geodesic, we have . ∎

###### Proposition 2.3.

Let be a quasiregular map between equiregular sub-Riemannian manifolds. Then almost every point in possesses a neighborhood such that is quasiconformal.

###### Proof.

The statement follows from the definition of as the set of points where does not define a local homeomorphism. Since the branch set is closed, each point possesses a neighborhood restricted to which is a homeomorphism and, in fact quasiconformal by the local and essential boundedness of . The only point worth noting here is that the definition of quasiconformality on metric spaces requires the boundedness of the distortion function (1.1), whereas quasiregularity yields only that the a priori smaller function as defined in (1.2) is bounded. Lemma 2.2 ensures that we obtain the same class of quasiconformal mappings for both distortion functions.

Now if we consider an open subset in a sub-Riemannian manifold , there are two natural distance functions on : the restriction of the cc-distance and the cc-distance which is given by the restriction of the sub-Riemannian metric to . Since is locally geodesic, the two metrics agree on small balls inside . By Lemma 2.2 the uniform boundedness of , where the distortion is computed with respect to and , then implies the uniform boundedness of the ‘quasiconformal distortion’ (1.1) in terms of and . ∎

In the study of uniformly quasiregular mappings later on, the precise value of a mapping’s distortion will be relevant. For this purpose we record the following result.

###### Proposition 2.4.

Let be a quasiregular map between sub-Riemannian manifolds such that each point in possesses an open neighbourhood restricted to which is a diffeomorphism and in which the functions

 λ−(y)=infv∈HyM|f∗v|N|v|M λ+(y)=supv∈HyM|f∗v|N|v|M

(with always nonzero, and denoting the usual derivative) are continuous and satisfy

 λ+(y)λ−(y)≤K.

Then is -quasiregular.

###### Proof.

Since the branch set of a quasiregular map is of zero measure, it suffices to show that

 H(x,f)≤K,for all x∈M∖Bf.

Let be a point in . Then there exists an such that is a diffeomorphism onto its image and , are continuous functions on whose quotient satisfies the desired bound. We may further assume that has been chosen small enough so that can be joined to any point in by a geodesic . Then

 dN(f(x),f(y)) ≤∫|f∗˙γ(t)|Ndt ≤∫λ+(γ(t))|˙γ(t)|Mdt ≤(supt∈[0,ℓ]λ+(γ(t)))(∫|˙γ(t)|Mdt) =supB(x,r)λ+⋅dM(x,y).

For the reverse estimate, we notice that there is such that for , the points and can be joined by a geodesic which lies entirely inside . Since is a local horizontal diffeomorphism, is the image of the horizontal curve and therefore

 dN(f(x),f(z)) =length(γ′) ≥inft∈[0,ℓ′]λ−(f−1(γ′(t)))⋅length(f−1∘γ′) ≥infB(x,r)λ−⋅dM(x,y)

It follows

 dN(f(x),f(y))dN(f(x),f(z)))≤supB(x,r)λ+infB(x,r)λ−

and thus by letting ,

 H(x,f)≤λ+(x)λ−(x),

as desired. ∎

### 2.3. Lusin’s conditions

A quasiconformal map between equiregular sub-Riemannian manifolds is absolutely continuous in measure, see Theorem 7.1 in [MM95]. Since the inverse of such a map is again quasiconformal by Corollary 6.4 in [MM95], we have both Lusin’s condition and . That is, the image, respectively preimage, under a quasiconformal map of each set of measure zero is a set of measure zero. The same holds true for non-constant quasiregular mappings.

###### Lemma 2.5.

Let be a non-constant quasiregular mapping on an equiregular sub-Riemannian manifold and let be a set with . Then .

###### Proof.

By Proposition 2.3, each point in possesses a neighbourhood restricted to which is quasiconformal. Since is strongly Lindelöf, that is, every open cover of an arbitrary open subset of has a countable subcover, we may cover the open set by a countable union of sets so that is quasiconformal. By the absolute continuity of quasiconformal maps, we have

 |f(E∩Bi)|=|(f|Bi)−1(E)|=0,for all i∈N

and thus, by the assumptions on the branch set of a quasiregular map and its image,

 |f(E)|≤|f(Bf)|+∞∑i=1|f(E∩Bi)|

and

 |f−1(E)|≤|Bf|+∑i∈N|f−1(E)∩Bi|=|Bf|+∑i∈N|(f|Bi)−1(E∩f(Bi))|,

which concludes the proof. ∎

Although very natural, Definition 1.1 is not easy to work with: it can be difficult to verify directly that a given map is quasiregular, and it is not clear in general whether the composition of two quasiregular maps is again quasiregular. For that reason we introduce a sub-class of quasiregular mappings that are easier to handle: the radially bilipschitz mappings with . Examples of such mappings are those of bounded length distortion with finite multiplicity, as well as open, discrete and sense-preserving locally -bilipschitz maps. Both the quasiregular and uniformly quasiregular mappings we construct below will be radially bilipschitz.

###### Definition 2.6.

Let and be two sub-Riemannian manifolds endowed with Carnot-Carathéodory distances. We say that a branched cover is radially bilipschitz (RBL for short) if there exist constants and for each a number such that for all with , one has

 c1d(x,y)≤d(f(x),f(y))≤c2d(x,y). (2.2)

We emphasize that the condition (2.2) is formulated with respect to the base point , in particular we do not require that the restriction of to any curve emanating from , or even to a small neighborhood around , would be bilipschitz.

###### Example 2.7.

The prototypical example of RBL maps is the planar map

 f(z)=z2/|z|.

In polar coordinates, the map is given by , so that away from the origin the map is locally 2-bilipschitz. At the origin, one sees branching but no dilatation: a circle of any radius is mapped onto itself.

Immediately from the definition of RBL maps, we obtain the following criterion for quasiregularity.

###### Proposition 2.8.

Any RBL map with and between sub-Riemannian manifolds is quasiregular.

We state several useful properties of RBL maps.

###### Proposition 2.9.

Let and be RBL maps between sub-Riemannian manifolds. Then is RBL.

###### Proposition 2.10.

If is RBL and is a domain, then is RBL (with respect to the restriction of either the distance function or sub-Riemannian inner product on ).

###### Proposition 2.11.

Let and be sub-Riemannian manifolds of equal dimension. Assume that can be written as a finite union of open sets, and suppose further that we are given RBL maps so that . Then the map , defined by is RBL.

###### Proof.

The condition on ensures that is well-defined and continuous. By [HR02, Remark 3.2], every discrete and open map between oriented generalized -manifolds is sense-preserving or sense-reversing (so that if each is sense-preserving, so is ). Thus, regarding the topological properties, it is sufficient to verify openness and discreteness. For that purpose, let be an open set in , then

 f(U)=f(k⋃i=1U∩Bi)=k⋃i=1fi(U∩Bi)

is open as a union of open sets. The discreteness is also immediate since for all , we have

 f−1({y})=k⋃i=1f−1i({y}),

which is discrete as a finite union of discrete sets. The metric condition in the definition of RBL is easy to verify, as it is a local condition and the sets , on which is RBL, are open. ∎

### 2.5. Examples of RBL mappings

An important class of RBL maps are those of bounded length distortion introduced in Definition 2.12 below, which is analogous to [HR02, Definition 0.1].

###### Definition 2.12.

Let and be two sub-Riemannian manifolds endowed with Carnot-Carathéodory distances. We say that a mapping is of bounded length distortion (BLD) if it is continuous, open, discrete, sense-preserving and there exists such that

 1Llength(γ)≤length(f∘γ)≤Llength(γ)

for all continuous paths in .

###### Proposition 2.13.

Let and be oriented sub-Riemannian -manifolds and assume that is complete. Let be a surjective -BLD map of finite multiplicity. Then is RBL.

###### Proof.

The -Lipschitz continuity is immediate: Since is a geodesic space, for every , there exists such that

 dM(x,y)=length(γ).

Then

 dN(f(x),f(y))≤length(f∘γ)≤Llength(γ)=LdM(x,y).

We now prove the local lower bound for at an arbitrarily chosen point . Since has finite multiplicity, consists of finitely many points . By [HR02, Proposition 4.13], we have

 B(x,ρC)∪N⋃i=1B(xi,ρC) ⊆f−1(B(f(x),ρ)) (2.3) ⊆B(x,Cρ)∪N⋃i=1B(xi,Cρ)

for all and some constant which depends on the data associated to (including multiplicity). There exists such that the balls and are pairwise disjoint for . Set

 r:=2Cρ1.

Now let be an arbitrary point in with . If , there is nothing to prove. Otherwise, we have and for by (2.3) and so

 dN(f(x),f(y))>ρ=r′2C=dM(x,y)2C,

which concludes the proof. ∎

From Proposition 2.8 and Proposition 2.13, we obtain the following result.

###### Corollary 2.14.

Let and be oriented sub-Riemannian -manifolds and assume that is complete. Let be a surjective BLD map of finite multiplicity with . Then is quasiregular.

A second class of RBL mappings is provided by locally bilipschitz maps.

###### Lemma 2.15.

Let and be compact sub-Riemannian manifolds and let be a smooth map which is horizontal, that is, for all , then is Lipschitz.

###### Proof.

Since is smooth, we can consider the pull-back metric . This is a semidefinite form (on the subbundle ). As is compact, the same holds true for the unit horizontal subbundle and the map defined thereon is continuous and finite. Thus there exists a constant such that

 |f∗v|N≤L|v|Mfor all v∈HM. (2.4)

The claim follows since is horizontal. ∎

###### Corollary 2.16.

Let be a compact sub-Riemannian manifold whose sub-Riemannian structure is given by a contact form. Then every sense-preserving contactomorphism is BLD, RBL and quasiconformal.

###### Proof.

We apply Lemma 2.15 to both and its inverse in order to see that is bilipschitz. It follows that is distorting the length of curves only by a controlled amount. Thus, is BLD, RBL and quasiconformal. ∎

### 2.6. Remarks on QR mappings

There is a well-established analytic definition available for quasiregularity on the Heisenberg group and a rather complete theory has been developed for mappings which are quasiregular according to this definition [Dai00a]. A continuous mapping between equiregular sub-Riemannian manifolds which are modelled conformally and locally bilipschitzly on (‘Heisenberg manifolds’) could now be called -quasiregular if it is -quasiregular in charts according to the analytic definition. Similarly, one could study quasiregular mappings on manifolds which are modelled on other two-step Carnot groups, based on the notions and results in [Dai00b].

###### Question 2.17.

Does this definition of ‘quasiregularity in charts’ on Heisenberg manifolds yield the same class of mappings as Definition 1.1?

Mappings on the Heisenberg group which are quasiregular according to the analytic definition stated in [Dai00a] behave analogously as their Euclidean counterparts: non-constant quasiregular mappings satisfy Lusin’s condition and their branch set is of vanishing measure, it is also known that the composition of two such mappings is again quasiregular. If we start with a metric distortion condition on a sub-Riemannian manifold, our knowledge is less advanced.

###### Question 2.18.

What are geometric or analytic characterizations of mappings which are quasiregular in the sense of Definition 1.1?

An answer to Question 2.18 would most likely yield that quasiregular mappings are closed under compositions, and provide us with other useful properties.

As we show in §3, nontrivial quasiregular mappings exist between any two lens spaces (see Theorem 3.5). On the other hand, for certain Carnot groups any quasiregular mapping is in fact conformal [HH97]. More generally, one can ask:

###### Question 2.19.

For which sub-Riemannian manifolds does there exist a non-constant quasiregular mapping ? Does a cohomological obstruction exist for non-constant QR mappings , where is a Carnot group, as for (Riemannian) QR-elliptic manifolds in [BH01]?

If a compact Riemannian manifold supports a nontrivial UQR map, then is elliptic [IM01] (that is, admits a quasiregular mapping from ). A detailed proof based on applying Zalcman’s rescaling principle on the Julia set can be found in [Kan08]. The corresponding statement in the sub-Riemannian setting is an open question.

###### Question 2.20.

Suppose is an equiregular sub-Riemannian manifold that admits a nontrivial UQR map and that has at almost every point a fixed Carnot group as a tangent space in the sense of Margulis and Mostow. Does then admit a quasiregular mapping from ?

## 3. QR mappings on spheres and lens spaces

In this section we introduce the setting of sub-Riemannian spheres and lens spaces and provide nontrivial examples for quasiregular mappings on such manifolds. For Riemannian lens spaces, the existence of nontrivial UQR maps follows from [Pel99] and various properties of such mappings have been studied. For instance, G. Martin and the third author have shown that the -torus cannot appear as a Julia set either on the standard or on a three-dimensional lens space equipped with the quotient metric [MP]. In the present paper, we initiate the study of quasiregular mappings on sub-Riemannian lens spaces. The UQR maps constructed here provide also new Riemannian counterparts. All these mappings have Cantor set type Julia sets.

Recall that a differentiable map between sub-Riemannian manifolds of the same dimension is locally isometric (in the metric sense) if one has and also . More generally, the mapping is said to be conformal if one has and , that is, there exists a positive function such that for all and one has .

Notice that a conformal diffeomorphism is -quasiconformal with respect to the Carnot-Carathéodory distances defined by the corresponding sub-Riemannian metrics. The assumed smoothness might seem a too strong regularity condition. It is, however, natural, for instance in the context of rigidity results for conformal mappings on Carnot groups [CC06] and for isometries on equiregular sub-Riemannian manifolds [CLD13]. Later in the paper, we consider mappings for which the conformality condition above is formulated in terms of the so-called Margulis–Mostow differential and required to hold only almost everywhere.

### 3.1. Sub-Riemannian spheres and lens spaces

The sphere has a natural sub-Riemannian structure. Namely, one obtains the horizontal subbundle by taking a maximal complex subspace of : . The Euclidean inner product on then restricts to as a sub-Riemannian metric and corresponding norm .

We will next take a quotient of . Let be an integer and relatively prime to . Set , and define

 Rp,q(z1,z2,…,zn+1)=(e2π\mathbbmiq1/pz1,e2π\mathbbmiq2/pz2,…,e2π\mathbbmiqn+1/pzn+1). (3.1)

Note that the transformation preserves the unit sphere , and that its restriction to is conformal with respect to the standard sub-Riemannian structure. Furthermore, it has finite order and no fixed points on . The associated quotient space is called a lens space. Lens spaces are well-studied, especially in dimension 3, see [Hat02].

We can give a natural sub-Riemannian structure using the following construction.

###### Proposition 3.1.

Let be an equiregular (simply connected) sub-Riemannian manifold and a discrete group of isometries of , acting freely and properly discontinuously on .

Let be the usual quotient map. Then the following holds:

1. can be given the structure of a sub-Riemannian manifold which can be endowed with a natural metric so that becomes a locally isometric covering.

2. Every -quasiregular map with

 π(p)=π(q)⇒F(p)=F(q), (3.2)

induces a -quasiregular map , defined by .

###### Proof.

Under the given assumptions it is possible to endow with a sub-Riemannian metric by setting

 g[p](v∗,v′∗):=gp(v,v′),for p∈M,v,v′∈HpM

with , , , where denotes the standard projection given by the quotient map , which in this case becomes a locally isometric covering.

Let now be a -quasiregular map satisfying condition (3.2). Then is well-defined by setting . For small enough and , we have

 dM/Γ([p],[q])=dM(p,q)

for and in the same fundamental region. Then the -quasiregularity of follows from the corresponding property of . ∎

###### Example 3.2.

Let be an integer, and with each relatively prime to . Let be the associated rotation of and be the associated lens space. For any , the rotation

 Rθ1,…,θn+1(z1,…,zn+1)=(e\mathbbmiθ1z1,…,e\mathbbmiθn+1zn+1) (3.3)

of induces a rotation of that commutes with and is conformal with respect to the standard metric on (it is, in fact, an isometry). We thus have an induced “rotation” isometry on the lens space .

The sphere admits a larger family of conformal “rotations”. Namely, the group of unitary matrices acts transitively on by isometries. However, this action does not descend to because it does not commute with .

###### Example 3.3.

The sphere admits a one-parameter family of “loxodromic” conformal maps defined, for , by

 Td(z1,…,zn+1)=(cosh(d)z1+sinh(d),z2,z3,…,zn+1)sinh(d)z1+cosh(d).

For , the transformation maps the “right hemisphere” to a smaller neighborhood of the point . Because is complex-analytic (as a map on ), it preserves the horizontal distribution of . Because it is linear-fractional, is, in fact, conformal with respect to the sub-Riemannian metric. See [KR85] for the case .

We remark that inside the unit ball, the transformation is an isometry of complex hyperbolic space with translation length (see [Gol99, §6.2] or [Par12]). Furthermore, under stereographic projection, corresponds to a dilation map of the Heisenberg group.

### 3.2. Multi-twist mappings

We define a multi-twist map of as follows:

###### Definition 3.4.

For , the multi-twist map is given by

 Fa(r1e\mathbbmiθ1,…,rn+1e\mathbbmiθn+1)=(r1ea\mathbbmiθ1,…,rn+1e\mathbbmiaθn+1). (3.4)

The goal of this section is to prove (see Corollary 3.10 and Lemma 3.11):

###### Theorem 3.5.

The mapping is quasiregular for each . For each dividing and any choice of vector of integers relatively prime to , the map induces quasiregular mappings and .

To describe the properties of , it is convenient to work on the domain

 U:={z=(z1,…,zn+1)∈Cn+1:|zi|≠0for% i=1,…,n+1},

in polar coordinates . Clearly, is everywhere differentiable on and for all ,

 (Fa)∗,z(HzS2n+1)⊆HFa(z)S2n+1. (3.5)

This can be seen most easily from the following characterization of the horizontal distribution:

###### Lemma 3.6.

The horizontal bundle is given by

 HS2n+1=kerαwith α=\mathbbmi2(n+1∑i=1zid¯zi−¯zidzi)=n+1∑i=1r2idθi.
###### Proof.

One can write , where and the mapping is the restriction to the sphere of the multiplication by . Since and on , we find , and the expression of in Cartesian coordinates follows. The expression in polar coordinates is obtained by noting that . ∎

We observe . Moreover, a direct computation yields

 gE((Fa)∗v,(Fa)∗v)=n+1∑i=1((cosθivxi+sinθivyi)2+a2(sinθivxi−cosθivyi)2),

where

 v=n+1∑i=1vxi∂xi+vyi∂yi.

Hence

 |v|E≤|(Fa)∗v|E≤a|v|,v∈HpS2n+1,p∈U. (3.6)

If were smooth, (3.6) would give immediately that is -Lipschitz and a BLD map, but we cannot assume that the curves we are considering lie entirely in . We will use the following result, which is immediate for curves contained in the domain where is smooth. The situation is more subtle when a curve meets the set

 B:=S2n+1∖U={z∈S2n+1:zi=0 for % some i∈{1,…,n+1}},

but even if is not differentiable in such points, it is “differentiable along the considered curves”.

###### Proposition 3.7.

A curve is horizontal if and only if is horizontal. In this case,

 |˙γ(s)|E≤∣∣˙γ′(s)∣∣E≤a|˙γ(s)|E (3.7)

for almost every .

###### Proof.

Notice that and . Each point in possesses a neighborhood restricted to which is a diffeomorphism. This shows immediately that if is a curve contained entirely in , then it is horizontal if and only if is. So let us concentrate on the case where the curve intersects . We notice

 γ(s)∈B⇔γ′(s)∈B.

Even more, using standard coordinates from the ambient space , the -th component of vanishes at some point if and only if the -th component of vanishes. Denote now by the th component of , and by the th component of . The strategy is the following: Assume first that is horizontal. Then the set of points in where all the components of are differentiable is a full measure set . Let us look at . If , then also is nonvanishing and differentiable at with

 ˙γi(s)2≤˙γ′i(s)2≤a2˙γi(s)2 (3.8)

and

 a(γi(s)˙¯γi(s)−¯γi(s)˙γi(s))=γ′i(s)˙¯γ′i(s)−¯γ′i(s)˙γ′i(s). (3.9)

Let us now consider such that . We can neglect the case where is an isolated zero, because there can be at most countably many such in , and hence they form a set of measure zero. So assume that is not an isolated zero. Since is by assumption differentiable at and it vanishes at as well as on a sequence of points converging to , we must have . In order to prove that also is differentiable at with vanishing tangent (so that (3.8) and (3.9) remain valid in this case), we resort to the following auxiliary result.

###### Lemma 3.8.

Let be differentiable at with , and a smooth function. Then the function , defined (using any branch of the logarithm) by

 w(s):={r(s)e\mathbbmiη(φ(s))if z(s)=r(s)e\mathbbmiφ(s)0if z(s)=0

is differentiable in the point as well, and .

###### Proof.

This follows immediately from the definition of derivative. ∎

Applying this lemma to the non-isolated zeros of concludes the proof of (3.8) and (3.9) for almost every . Thus, if is horizontal, so is by (3.9), and (3.8) gives (3.7).

Second, one has to show that is horizontal whenever the image curve is. Lemma 3.8 has been formulated in a general setting, so that it can also be applied here, and the rest of the proof goes completely analogously, so we do not carry it out here. ∎

###### Corollary 3.9.

is -Lipschitz with

for every path .

###### Proof.

This is an immediate consequence of Proposition 3.7 and the discussion of rectifiable curves in §2.1. ∎

###### Corollary 3.10.

The multi-twist map is a BLD (and thus RBL) map on and -quasiregular.

###### Proof.

Apply Proposition 2.4 and Corollary 3.9. ∎

If for some positive integer , then induces a well-defined map on the lens space, namely

 fa:Lp,q→S2n+1,fa([z]):=Fa(z)for z∈S2n+1. (3.10)

Denoting by the usual projection, we obtain a multi-twist map of the lens space as

 π∘fa:Lp,q→Lp,q. (3.11)
###### Lemma 3.11.

Let . The multi-twist maps and are RBL and quasiregular.

###### Proof.

The first statement follows from Corollary 3.10 and Proposition 3.1. The quasiregularity of is an easy consequence of the fact that is quasiregular and the projection is isometric. Indeed, the Carnot-Carathéodory distance of and on equals the distance of and on if only and are close enough, which can be arranged by continuity. Since and the branches of its inverse are locally isometric, they are RBL, and so the RBL property of and follows from the corresponding property of . ∎

## 4. UQR mappings on spheres and lens spaces

There are essentially two methods known to produce UQR mappings: Lattès construction and the conformal trap method. The proof of Theorem 1.3 uses the conformal trap construction, which was first introduced by Iwaniec and Martin in [IM96] and [Mar97] to prove the following theorem.

###### Theorem 4.1 (Iwaniec, Martin).

Let and let be a quasiregular map of the Riemannian sphere. Then there exists a UQR mapping with the same branch set as .

###### Example 4.2.

Suppose we have a quasiregular planar map with the following properties:

1. The point has two preimages