On Locally Quasiconformal Teichmuller Spaces
aa r X i v : . [ m a t h . C V ] J u l ON LOCALLY QUASICONFORMAL TEICHMÜLLER SPACES
ALASTAIR FLETCHER AND ZHOU ZEMIN
Abstract.
We define a universal Teichmüller space for locally quasiconformal mappingswhose dilatation grows not faster than a certain rate. Paralleling the classical Teichmüllertheory, we prove results of existence and uniqueness for extremal mappings in the generalizedTeichmüller class. Further, we analyze the circle maps that arise. Introduction
Teichmüller theory is a major area of research in modern mathematics, bringing togetheranalysis, geometry, topology and dynamics. The Teichmüller space of a topological surfaceparameterizes the set of complex structures that can be equipped on the surface. For exam-ple, not all tori are conformally equivalent and the space of complex structures of a genus onesurface can be parameterized by the upper half-plane. A fundamental object in Teichmüllertheory is the universal Teichmüller space of the disk, denoted T (∆) . Via the Uniformiza-tion Theorem, every Teichmüller space of a hyperbolic surface is embedded in T (∆) , andso it is an important object to understand. We refer to [10, 11, 18, 19] for introductions toTeichmüller theory.There are various ways of modelling points of universal Teichmüller space. One can con-sider equivalence classes of quasiconformal maps f : ∆ → ∆ under the Teichmüller equiva-lence relation or, equivalently via solving the Beltrami differential equation f z = µf z , equiv-alence classes of Beltrami differentials. We recall that Beltrami differentials are elements µ ∈ L ∞ (∆) with || µ || ∞ < . Since every quasiconformal map f : ∆ → ∆ extends to aquasisymmetric homeomorphism e f : ∂ ∆ → ∂ ∆ , points of Teichmüller space can also bemodelled as quasisymmetric maps of the circle which fix the three points , − , i .The defining property of a quasiconformal map f : ∆ → ∆ is that it has uniformly boundeddistortion. Every quasiconformal map has a complex dilatation µ f = f z /f z which is definedalmost everywhere. The quasiconformality condition implies that there exists ≤ k < suchthat || µ f || ∞ ≤ k almost everywhere. Solving the Beltrami equation provides the converse tothis statement. Recently, there has been interest in the consequences of allowing || µ || ∞ = 1 in the Beltrami equation and investigating properties of the solutions that occur.In the literature, various classes of such mappings have been studied, for example Davidmappings [8, 27], µ -homeomorphisms [3, 4, 5, 6, 7, 13, 14, 17, 25] and locally quasiconformalmappings [20, 28] have all been studied in dimension two. These all sit in the larger frame-work of mappings of finite distortion in Euclidean spaces, see for example [12, 15, 16]. Such Mathematics Subject Classification.
Primary 30C62, Secondary 30C75.
Key words and phrases.
Quasiconformal mapping, Locally Quasiconformal mapping, generalizedTeichm ¨ uller space, Generalized maximum dilatation.AF is supported by a grant from the Simons Foundation ( appings are far from novelties: Petersen and Zakeri used David mappings in [23] to studySiegel disks in complex dynamics. Moreover, in the theory of length spectrum Teichmüllerspaces, it is known that it certain circumstances it differs from the quasiconformal Teich-müller space, see for example the work of Shiga [24]. In particular, in this paper a certainmap is constructed which has a uniform bound on the distortion of hyperbolic lengths ofessential curves, but is (in our language) locally quasiconformal. It is therefore conceivablethat locally quasiconformal mappings could play a role in the study of length spectrumTeichmüller spaces.In the current paper, we will work in the setting of locally quasiconformal mappings ofthe disk and initiate the study of a universal Teichmüller space of such mappings. Our aimis to set up a workable definition, solve an extremality problem in this setting and study theboundary mappings that arise.The paper is organized as follows: we recall some preliminary material on quasiconformaland locally quasiconformal mappings in section 2. In section 3, we define locally quasiconfor-mal Teichmüller spaces and state our main results on them. In section 4, we provide proofsof our results. Finally in section 5, we give some concluding remarks indicating directions offuture research. Acknowledgements:
The second named author would like to thank Professor Chen Jixiufor his many useful suggestions and help.2.
Preliminaries
We recall some basic facts about quasiconformal mappings which can be found in manytexts, for example [1, 10, 11, 16, 18, 19]. Let U ⊂ C be a domain. The distortion of ahomeomorphism f : U → V ⊂ C is defined by D f ( z ) = | f z ( z ) | + | f z ( z ) || f z ( z ) − | f z ( z ) | . A homeomorphism f : U → V is called K -quasiconformal if f is absolutely continuous onalmost every horizontal and vertical line in U and moreover sup z ∈ U D f ( z ) ≤ K . The smallestsuch K that holds here is called the maximal dilatation of f and denoted K f . If we do notneed to specify the K , then we just call the map quasiconformal . The complex dilatation of f is µ f = f z /f z and satisfies the equation D f ( z ) = 1 + | µ ( z ) | − | µ ( z ) | . If f is quasiconformal, then there exists ≤ k < such that || µ f || ∞ ≤ k and K f = 1 + || µ f || ∞ − || µ f || ∞ . Every quasiconformal map f : ∆ → ∆ extends to a homeomorphism of ∆ and, moreover,the boundary map e f : ∂ ∆ → ∂ ∆ is quasisymmetric, that is, there exists M ≥ so that M ≤ | e f ( e i ( θ + t ) ) − e f ( e iθ ) || e f ( e iθ ) − e f ( e i ( θ − t ) ) | ≤ M holds for all θ ∈ [0 , π ) and t > .We now define the class of mappings that will form the basis of our study. efinition 2.1. A homeomorphism f : ∆ → ∆ is called locally quasiconformal if and onlyif for every compact set E ⊂ ∆ , f | E is quasiconformal.This definition means we allow that distortion of our map to blow up as we head outtowards the boundary. Moreover, the complex dilatation of a locally quasiconformal mappingis defined almost everywhere in the disk and we allow || µ f || ∞ = 1 . However, we have torestrict the types of locally quasiconformal mappings we study. Example 2.2. (i) The map f ( z ) = z (1 − | z | ) − from [20] is a locally quasiconformalmap from ∆ onto C . In this paper, we want to consider only locally quasiconformalself-mappings of ∆ .(ii) The spiral map f ( re iθ ) = r exp( i ( θ +ln − r )) is a locally quasiconformal map f : ∆ → ∆ but it does not extend continuously to the boundary. In this paper, we want to considerlocally quasiconformal mappings which extend homeomorphically to the boundary. Definition 2.3.
We say that a continuous increasing function ρ : [0 , → [1 , ∞ ) is allowable if the following conditions hold:(i) ρ (0) = 1 ,(ii) Z ρ ( r ) dr < ∞ , (iii) for some constant R > and every ξ ∈ ∂ ∆ , lim t → + Z Rt drrρ ∗ ( | z | ) = + ∞ , where ρ ∗ ( r ) is defined by(2.1) ρ ∗ ( r ) = Z S ( ξ,r ) ∩ ∆ ρ ( | z | ) dθ,z = ξ + re iθ and S ( ξ, r ) is the circle centred at ξ of radius r .Each allowable ρ yields a family of locally quasiconformal mappings. Definition 2.4.
Suppose ρ is allowable. The family QC ρ (∆) consists of locally quasiconfor-mal mappings f : ∆ → ∆ such that there exists C > with D f ( z ) ≤ Cρ ( | z | ) , for all z ∈ ∆ .Condition (i) in Definition 2.3 is a convenient normalization condition which implies that D f ( z ) /ρ ( | z | ) ≤ for every conformal map f : ∆ → ∆ with equality at the origin. Condition(ii) implies by [20, Theorem 1] that if µ ∈ L ∞ (∆) with | µ ( z ) | < then there exists a locallyquasiconformal map f : ∆ → ∆ with complex dilatation µ . Finally, condition (iii) impliesby, for example, [5, Theorem 1] that f extends continuously to ∂ ∆ .Observe that if f is quasiconformal, then f ∈ QC ρ (∆) for every allowable ρ . Typicallyfor a locally quasiconformal mapping, the maximal dilatation is not finite, but there is amaximal dilatation with respect to ρ . Definition 2.5.
Let ρ be allowable and let f ∈ QC ρ (∆) . Then the maximal dilatation withrespect to ρ is defined by K ρf := sup z ∈ ∆ D f ( z ) ρ ( | z | ) . s remarked above, condition (i) in Definition 2.3 implies that K ρf = 1 for every conformalmap f and every allowable ρ . Example 2.6. By [20, Theorem 4] , if ρ ( r ) = log − r , then any locally quasiconformal map f : ∆ → ∆ with D f ( z ) ≤ Cρ ( | z | ) extends homeomorphically to ∆ . It is a short computationto show that, if σ denotes Lebesgue measure, then any such f satisfies σ { z ∈ ∆ : D f ( z ) > K } < πe − K/C . In particular, this means that any such f is a David mapping. We don’t know how QC ρ (∆) and David mappings are related in general. Since ρ (0) = 1 , we have K ρf ≥ , and so K ρf gives a quantity the describes how far f isfrom a conformal map, except that here K ρf = 1 does not imply that f is conformal. To seethis, we only need D f ( z ) to grow slower than ρ ( | z | ) .3. Locally quasiconformal Teichmüller spaces
In this section we define locally quasiconformal Teichmüller spaces with respect to anallowable ρ and state our results. Definition 3.1.
Let ρ be allowable. Then the set L ρ (∆) consists of locally quasiconformalmappings f : ∆ → ∆ such that f − ∈ QC ρ (∆) .We need to control the growth of f − for our results. Note that such a condition on f does not imply the same condition holds for f − , in contrast to the fact that the inverse ofa K -quasiconformal map is also a K -quasiconformal map. Example 3.2.
Consider radial maps given in polar coodinates by f a ( re iθ ) = [1 − (1 − r ) a ] e iθ for a > . A computation shows that D f ( re iθ ) = 1 − (1 − r ) a ar (1 − r ) a − , and the right hand side is the appropriate ρ to consider for such a map. However, condition(ii) in Definition 2.3 is only satisfied when < a < . Since f − a = f /a and f a ◦ f b = f ab ,we see that condition (ii) is not closed under inverses and compositions. Lemma 3.3. If f ∈ L ρ (∆) , then f can be extended homeomorphically to a map ∆ → ∆ .Proof. Since ρ is allowable and f ∈ L ρ (∆) , then D f − ( z ) ≤ Cρ ( | z | ) , where ρ satisfies theconditions in Definition 2.3. It immediately follows from [28, Theorem 1.1] that µ f − can beintegrated to give a locally quasiconformal map g which extends to a homeomorphism of ∆ .Now, for z ∈ ∆ , the complex dilatation of g ◦ f satisfies µ g ◦ f ( f − ( z )) = ω ( z ) · µ g ( z ) − µ f − ( z )1 − µ f − ( z ) µ g ( z ) ≡ , since µ g = µ f − , and where | ω ( z ) | = 1 for all z ∈ ∆ . Hence g ◦ f is a conformal map from ∆ to itself which extends to the boundary. Hence f , and also f − , extend homeomorphicallyto ∆ . (cid:3) ecall that A (∆) is the Bergman space of integrable holomorphic functions on ∆ , that is, A (∆) = { ϕ : Z ∆ | ϕ | < ∞} . Complex dilatations µ ( z ) = kϕ/ | ϕ | for < k < and ϕ ∈ A (∆) are said to be of Teichmüller-type and play an important role in extremal problems in Teichmüller theory. We show thatan analogue exists for locally quasiconformal mappings. Lemma 3.4.
Let ϕ ∈ A (∆) be not identically zero, K ≥ be a constant and let ρ beallowable. Then there exists a locally quasiconformal mapping f ∈ L ρ (∆) with Teichmüller-type complex dilatation (3.1) µ f ( z ) = ρ ( | f ( z ) | ) K − ρ ( | f ( z ) | ) K + 1 · ϕ ( z ) | ϕ ( z ) | . We prove this lemma in the next section. Since elements of L ρ (∆) extend homeomorphi-cally to the boundary, we may always assume that we have post-composed by a Möbius mapso that elements of L ρ (∆) fix , − and i . Definition 3.5.
Let ρ be allowable. Then f, g ∈ L ρ (∆) are Teichmüller related with respectto ρ , denoted by f ∼ g , if and only if the boundary extensions of f and g , normalized to fix , − , i , agree. We then define the generalized Teichmüller space with respect to ρ by T ρ (∆) = L ρ (∆) / ∼ . Elements of T ρ (∆) are Teichmüller equivalence classes denoted by [ f ] ρ , or simply [ f ] if thecontext is clear. Given [ f ] ∈ T ρ (∆) , every representative of [ f ] is a locally quasiconformalmap f whose boundary values agree with f and so that K ρf − < ∞ . Definition 3.6.
Let [ f ] ∈ T ρ (∆) .(i) We say that f ∈ [ f ] is extremal if K ρf − ≤ K ρg − for all g ∈ [ f ] .(ii) We say that f ∈ [ f ] is uniquely extremal if K ρf − < K ρg − for all g ∈ [ f ] \ { f } .Our first result on extremal maps is that extremal representatives always exist. Theorem 3.7.
Let ρ be allowable and let [ f ] ∈ T ρ (∆) . Then there exists an extremalrepresentative f ∈ [ f ] . We show next that uniquely extremal representatives exist.
Theorem 3.8.
Let ρ be allowable, K > , ϕ ∈ A (∆) and suppose f ∈ T ρ (∆) hasTeichmüller-type complex dilatation µ f ( z ) = ρ ( | f ( z ) | ) K − ρ ( | f ( z ) | ) K + 1 · ϕ ( z ) | ϕ ( z ) | . Then f is uniquely extremal in [ f ] . We next turn to the boundary mapping induced by an element [ f ] ∈ T ρ (∆) . It is well-known that every quasiconformal self-map of ∆ extends to a quasisymmetric map of the unitcircle and, conversely, every quasisymmetric map of the unit circle extends to a quasiconfor-mal map of ∆ . For h : ∂ ∆ → ∂ ∆ , we define the quasisymmetric function λ h ( ξ, t ) = | h ( ξe it ) − h ( ξ ) || h ( ξ ) − h ( ξe − it ) | . his is the circle version of the standard quasisymmetric function considered by many au-thors, for example [5, 27]. Theorem 3.9.
Let ρ be allowable and [ f ] ∈ T ρ (∆) with associated circle map h . Then thereexists a function λ depending only on ρ such that λ ( t ) ≤ λ h − ( θ, t ) ≤ λ ( t ) for all θ ∈ [0 , π ) . In proving this result we will give an explicit formula for λ ( t ) . Note that we may have λ ( t ) → ∞ as t → . Theorem 3.9 is a circle version of a result proved for the extended realline in [5]. However, the extended real line has a special point at infinity, whereas the circlehas no special point. In particular, it is not a trivial task to obtain our result from [5].4. Proofs of results
Generalized Teichmüller-type dilatations.
Here we prove Lemma 3.4. First, werecall some results on non-linear elliptic systems from [2, Chapter 8]. Assume that h : C × C × C → C satisfies the following conditions:(i) the homogeneity condition that f z = 0 whenever f z = 0 , or equivalently, H ( z, w, ≡ , for almost every ( z, w ) ∈ C × C ; (ii) the uniform ellipticity condition that for almost every z, w ∈ C and all ζ , ξ ∈ C , | H ( z, w, ζ ) − H ( z, w, ξ ) | ≤ k | ζ − ξ | , for some ≤ k < ;(iii) H is Lusin measurable (see [2, p.238] for further details).A solution f ∈ W , loc ( C ) to(4.1) ∂f∂z = H ( z, f, ∂f∂z ) , for z ∈ C and normalized by the condition f ( z ) = z + a z − + a z − + · · · outside a compact set will be called a principal solution . A homeomorphic solution f ∈ W , loc ( C ) to (4.1) is called normalized if f (0) = 0 and f (1) = 1 . Naturally any such solutionfixes the point at infinity too. Lemma 4.1 (Theorem 8.2.1, [2]) . Under the hypotheses above, equation (4.1) admits anormalized solution. If, in addition, H ( z, w, ζ ) is compactly supported in the z -variable,then the equation admits a principal solution. We now prove Lemma 3.4.
Proof of Lemma 3.4.
Recalling (3.1), the equation we want to solve is ∂f ∂z = ρ ( | f ( z ) | ) K − ρ ( | f ( z ) | ) K + 1 · ϕ | ϕ | · ∂f ∂z . o that end, let H ( z, w, ζ ) = H ( z, f , ∂f ∂z ) = ρ ( | f | ) K − ρ ( | f | ) K + 1 · ϕ ( z ) | ϕ ( z ) | · ∂f ∂z . For n = 2 , , . . . , set ∆ n = { z : | z | < − n } and D n = { z : 1 − n < | z | < − n } .By interpolating in D n with a continuous function F n satisfying | F n ( z, w, ζ ) | ≤ | H ( z, w, ζ ) | ,there exists a continuous function H n ( z, w, ζ ) := H ( z, w, ζ ) when z ∈ ∆ n F n ( z, w, ζ ) when z ∈ D n when z ∈ C − ∆ n − D n . Applying Lemma 4.1, we see that the equation ∂f∂z = H n ( z, w, ∂f∂z ) admits a principal and normalized solution f n : C → C .For every fixed j , the family { f n | ∆ j } consists of quasiconformal mappings with a uniformbound on the distortion. Since f n (0) = 0 and f n (1) = 1 for all n , the family is uniformlybounded in ∆ j . It follows from the quasiconformal version of Montel’s Theorem (see [22]) thatthe family { f n | ∆ j } is normal. By a standard diagonal argument, we can find a subsequence ( f n p ) ∞ p =1 which converges locally uniformly in ∆ . Suppose f is the limit function. Theimage f (∆) may not be ∆ , but it must be a simply connected proper subset of C . Bypost-composing by a suitable conformal map, via the Riemann Mapping Theorem, we canassume that f : ∆ → ∆ and f still fixes and . Then f is either a locally quasiconformalmapping (since it is quasiconformal on each ∆ j ) with complex dilatation (3.1) or a constant.However, since f (0) = f (1) , f cannot be constant.Finally, we need to show that f ∈ L ρ (∆) . Set g = f − . Then, by the formula for thecomplex dilatation of an inverse (see [10, p.6]), for z ∈ ∆ we have | µ g ( z ) | = | µ f ( f − ( z )) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ( | z | ) K − ρ ( | z | ) K + 1 · ϕ ( f − ( z )) | ϕ ( f − ( z )) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ρ ( | z | ) K − ρ ( | z | ) K + 1 . We therefore obtain | µ g ( z ) | − | µ g ( z ) | ≤ K ρ ( | z | ) and conclude that g ∈ QC ρ (∆) and hence f ∈ L ρ (∆) . The proof is complete. (cid:3) Extremal Mappings.
We will show that extremal mappings always exist, but first weneed to prove a normal family result which generalizes [6, Theorem 3.1].
Lemma 4.2.
Let ρ be allowable, suppose F ⊂ QC ρ (∆) and there exists a constant C > sothat D f ( z ) ≤ Cρ ( | z | ) for all f ∈ F . Then F is a normal family and relatively compact in QC ρ (∆) viewed as asubset of continuous functions from the disk to itself. roof. Let r < . Then on ∆ r = { z : | z | < r } , every f ∈ F is Cρ ( r ) -quasiconformalwith image contained in ∆ . By Montel’s Theorem for quasiconformal mappings (see [22]),if ( f n ) ∞ n =1 is any sequence in F , then ( f n | ∆ r ) ∞ n =1 contains a subsequence ( f n k | ∆ r ) ∞ k =1 whichconverges to either a Cρ ( r ) -quasiconformal map or a constant.By a standard diagonal sequence argument, we find a subsequence ( f n p ) ∞ p =1 with f n p con-verging to f locally uniformly on ∆ . Then f is either a constant or a locally quasiconformalmap with D f ( z ) ≤ Cρ ( | z | ) . Hence f ∈ QC ρ (∆) . (cid:3) Proof of Theorem 3.7.
Let [ f ] ∈ T ρ (∆) . Then for each g ∈ [ f ] we have ≤ K ρg − < ∞ . Let K = inf g ∈ [ f ] K ρg − , and find f n ∈ [ f ] with K ρf − n → K . Without loss of generality, we may assume that K ρf − n isdecreasing. Set C = K ρf − . Then for all n ∈ N , we have D f − n ( z ) ≤ Cρ ( | z | ) . By Lemma 4.2, the family { f − n : n ∈ N } is normal and hence there exists a subsequence ( f − n k ) ∞ k =1 which converges locally uniformly on ∆ to a continuous map h . For each k , f − n k agrees with f − on ∂ ∆ and hence h cannot be a constant. We conclude that h ∈ QC ρ (∆) and K ρh = K . Setting f = h − we see that f is extremal in [ f ] . (cid:3) Uniquely Extremal Mappings.
Proof of Theorem 3.8.
By Lemma 3.4, there exists a locally quasiconformal map f withcomplex dilatation of Teichmüller type given by (3.1). Consequently, the Teichmüller class [ f ] is well-defined and at least contains the representative f .Let f ∈ [ f ] and set g = f − ◦ f . As we observed in Example 3.2, elements of QC ρ (∆) are not necessarily preserved by taking inverses or compositions. However, g is locallyquasiconformal in ∆ , extends to a homeomorphism from ∆ to itself, has generalized partialderivatives on ∆ and is the identity on ∂ ∆ . We can therefore apply the generalized versionof the Reich-Strebel Main Inequality proved by Marković and Mateljević [21, Theorem 1] tosee that for any ϕ ∈ A (∆) , we have(4.2) Z ∆ | ϕ | ≤ Z ∆ | ϕ | | µ g ϕ | ϕ || − | µ g | . For convenience, we write µ ( z ) = µ f − ( f ( z )) , µ = µ f and τ = ( f ) z / ( f ) z . Since µ g = µ + µ τ µµ τ we obtain(4.3) | µ g ϕ/ | ϕ || − | µ g | ≤ | µϕ/ | ϕ || (1 − | µ | ) | µ τ ϕ | ϕ | (1 + µ ϕ | ϕ | ) / (1 + µ ϕ | ϕ | ) | (1 − | µ | ) . Consequently(4.4) Z ∆ | ϕ | ≤ Z ∆ | ϕ | | µϕ/ | ϕ || (1 − | µ | ) | µ τ ϕ | ϕ | (1 + µ ϕ | ϕ | ) / (1 + µ ϕ | ϕ | ) | (1 − | µ | ) . etting ϕ = − ϕ in (4.4) and using (3.1), we obtain Z ∆ | ϕ | ≤ Z ∆ | ϕ | K ρ ( | f ( z ) | ) | µ τ | − | µ | ≤ K Z ∆ | ϕ | ρ ( | f ( z ) | ) 1 + | µ | − | µ |≤ K Z ∆ | ϕ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | µ | − | µ | ρ ( | f ( z ) | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ = K ρf − K Z ∆ | ϕ | . We conclude that K ≤ K ρf − . Since K ρf − = K , we see that f is extremal.To show that f is uniquely extremal, suppose f ∈ [ f ] is extremal and keep the samenotation as above. Therefore K ρf − = K ρf − . We obtain from (3.1), (4.4) and setting ϕ = − ϕ that Z ∆ | ϕ | ≤ Z ∆ | ϕ | ρ ( | f ( z ) | ) K | − µ τ ϕ / | ϕ || − | µ | . Since | − µ τ ϕ / | ϕ || − | µ | ≤ (1 + | µ f − ( f ( z )) | ) − | µ f − ( f ( z )) | ≤ ρ ( | f ( z ) | ) K ρf − , it follows that Z ∆ | ϕ | ≤ Z ∆ | ϕ | ρ ( | f ( z ) | ) K ρ ( | f ( z ) | ) K ρf − = Z ∆ | ϕ | . Since there must be equality everywhere, in particular we must have µ ( z ) = µ f − ( f ( z )) = − τ K ρ ( | f ( z ) | ) − K ρ ( | f ( z ) | ) + 1 ϕ | ϕ | . However, we also have from the definition of f that µ f − ( f ( z )) = − τ µ f ( z ) = − τ K ρ ( | f ( z ) | ) − K ρ ( | f ( z ) | ) + 1 ϕ | ϕ | . Since f − and f − have the same complex dilatation, then the same argument as in theproof of Lemma 3.3 shows that they are related via post-composition by a conformal map.However, since both f − and f − agree on ∂ ∆ , the conformal map must be the identity. Weconclude that f = f and so f is uniquely extremal. (cid:3) Quasisymmetry functions.
Before we prove Theorem 3.9, we need to recall someresults. If Γ is a curve family, then let M (Γ) denote its modulus. We refer to [26, ChapterII] for the precise definition. In particular, if E, F ⊂ C are disjoint continua, then ∆( E, F ) denotes the family of curves starting in E and terminating in F and M (∆( E, F )) is thecorresponding modulus.For ξ ∈ ∂ ∆ and < r < R , let Q ( ξ, r, R ) be the quadrilateral Q ( ξ, r, R ) = { z : r ≤ | z − ξ | ≤ R, z ∈ ∆ } with vertices taken in order as the intersections of the circle {| z − ξ | = r } and {| z − ξ | = R } with the unit circle. The modulus mod Q ( ξ, r, R ) is then defined as the modulus of the curve amily joining the two components of Q ( ξ, r, R ) ∩ ∂ ∆ . If f : ∆ → ∆ is a homeomorphism,then we define mod f ( Q ( ξ, r, R )) analogously. Lemma 4.3 (Lemma 2.1, [28]) . Let ρ be allowable and let f ∈ QC ρ (∆) . Then Z Rr dttρ ∗ ( t ) ≤ mod f ( Q ( ξ, r, R )) , where ρ ∗ ( t ) = Z S ( ξ,t ) ∩ ∆ ρ ( | z | ) dθ, and z = ξ + te iθ . Let τ : (0 , ∞ ) → (0 , ∞ ) be the Teichmüller capacity function (see [26, p.66]). Observethat τ is decreasing. Lemma 4.4 (Lemma 7.34, [26]) . If Ω ⊂ C is an open ring with complementary components E, F and a, b ∈ E , c, ∞ ∈ F , then M (∆( E, F )) ≥ τ (cid:18) | a − c || a − b | (cid:19) . Proof of Theorem 3.9.
Since [ f ] ∈ T ρ (∆) , if g = f − , then g ∈ QC ρ (∆) and extends to aboundary map that, by abuse of notation, we will also call g .Let ξ ∈ ∂ ∆ , t > and w = ξe it/ ∈ ∂ ∆ be the midpoint of the arc of ∂ ∆ from ξ to ξe it . Let Q be the quadrilateral Q = Q ( w, s, S ) , where s = | ξ − w | = | e it/ − | and S = | ξe − it − w | = | e it/ − | . Then since g ∈ QC ρ (∆) , by Lemma 4.3,(4.5) Z Ss drrρ ∗ ( r ) ≤ mod f ( Q ) . Let I ( z ) = z/ | z | be inversion in the unit circle. Then Ω := f ( Q ) ∪ I ( f ( Q )) is a ring domain.If Γ is the curve family separating boundary components of Ω , by symmetry we have(4.6) M (Γ) = mod f ( Q )2 . Now, if Γ ′ is the curve family connecting the complementary components of Ω , then(4.7) M (Γ ′ ) = 1 /M (Γ) . Applying Lemma 4.4 with a = g ( ξ ) , b = g ( ξe it ) and c = g ( ξe − it ) , we have(4.8) M (Γ ′ ) ≥ τ (cid:18) | g ( ξ ) − g ( ξe − it ) || g ( ξ ) − g ( ξe it ) | (cid:19) = τ (cid:18) λ g ( ξ, t ) (cid:19) . Combining (4.5), (4.6), (4.7) and (4.8), we conclude that Z Ss drrρ ∗ ( r ) ≤ (cid:18) τ (cid:18) λ g ( ξ, t ) (cid:19)(cid:19) − . Rearranging in terms of λ g ( ξ, t ) and using the fact that τ is decreasing, we obtain λ g ( ξ, t ) ≤ " τ − R Ss drrρ ∗ ( r ) ! − . or the reverse inequality, we apply the same argument as above, except this time we let w ′ = ξe − it/ be the midpoint of the arc of ∂ ∆ between ξ and ξe − it/ and we let Q ′ be thequadrilateral Q ( w ′ , s, S ) . Using the same notation as above, the argument is the same untilwe reach (4.8) and we obtain M (Γ ′ ) ≥ τ (cid:18) | g ( ξ ) − g ( ξe it ) || g ( ξ ) − g ( ξe − it ) | (cid:19) = τ ( λ g ( ξ, t )) . This yields λ g ( ξ, t ) ≥ τ − R Ss drrρ ∗ ( r ) ! . Consequently, we obtain the desired quasisymmetry estimate with λ ( t ) = " τ − R Ss drrρ ∗ ( r ) ! − , recalling that s = | e it/ − | and S = | e it/ − | . (cid:3) Concluding remarks
Boundary maps.
In Theorem 3.9, we showed that if [ f ] ∈ T ρ (∆) and f is any rep-resentative, then f − extends to the boundary and the boundary map has controlled qua-sisymmetry function λ depending on ρ which may, however, blow up. What is not clearis whether given a boundary map with quasisymmetry controlled by λ , there is a locallyquasiconformal extension contained in QC ρ (∆) . If so, then we would have an alternate pa-rameterization of T ρ (∆) through boundary maps of controlled quasisymmetry, in analogywith the quasisymmetric parameterization of universal Teichmüller space.There are various extensions available. The Douady-Earle extension [9] extends a home-omorphism of the circle to a diffeomorphism of the (open) disk and hence this extensionwill be locally quasiconformal. It would be interesting to know how the distortion of theextension is controlled by λ .Another extension is obtained through the Beurling-Ahlfors extension of homeomorphismsof the real line to homeomorphisms of the upper half-plane. In [5], it is shown that controlof the quasisymmetry function of the boundary map leads to control of the distortion of theBeurling-Ahlfors extension, but again we don’t know whether we can obtain ρ from λ .5.2. Pseudo-metrics and metrics.
It is well-known that universal Teichmüller space car-ries the Teichmüller metric, and this has been well studied. For T ρ (∆) , it is not clear howto make it into a metric space. One immediate barrier is that different classes can haveextremal representatives f , f which both have K ρf − i = 1 for i = 1 , . Moreover, the factthat QC ρ (∆) need not be closed under compositions and inverses makes a direct analogy ofthe Teichmüller metric impossible.On the other hand, it is easy to turn T ρ (∆) into a pseudo-metric space by consideringmaps F : T ρ (∆) → ( X, d X ) , where X is a metric space with distance function d X . We thendefine d T,X on T ρ (∆) via d T,X ([ f ] , [ g ]) = d X ( F ([ f ]) , F ([ g ])) . or example F ([ f ]) = inf f ∈ [ f ] K ρf mapping T ρ (∆) into R + with the usual Euclidean metricyields a pseudo-metric. We can therefore ask how to construct a metric on T ρ (∆) or, slightlynebulously, how to construct as interesting a pseudo-metric as possible.5.3. Riemann surfaces.
Our construction of generalized universal Teichmüller spaces leadsto an obvious generalization to generalized Teichmüller spaces of hyperbolic Riemann sur-faces, that is, those surfaces covered by the unit disk. It seems plausible that a study of suchobjects could yield information about the interplay between length-spectrum and quasicon-formal Teichmüller spaces. In particular, does length spectrum Teichmüller space sit insidea generalized Teichmüller space for every infinite-type Riemann surface?
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