Uniformization with Infinitesimally Metric Measures
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Uniformization with Infinitesimally Metric Measures © The Author(s) 2021 Published version Rajala, Kai; Rasimus, Martti; Romney, Matthew Rajala, K., Rasimus, M., & Romney, M. (2021). Uniformization with Infinitesimally Metric Measures. Journal of Geometric Analysis, 31(11), 11445-11470. https://doi.org/10.1007/s12220-021-00689-y 2021
The Journal of Geometric Analysis https://doi.org/10.1007/s12220-021-00689-y Uniformization with Infinitesimally Metric Measures Kai Rajala1·Martti Rasimus1·Matthew Romney1 Received: 18 July 2019 / Accepted: 27 April 2021 © The Author(s) 2021 Abstract We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces Xhomeomorphic to R2. Given a measure μon such a space, we introduce μ-quasiconformal maps f :X→R2, whose definition involves deforming lengths of curves by μ. We show that if μis an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a μ-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization. Keywords Metric doubling measure ·Quasiconformal mapping ·Quasisymmetric mapping ·Conformal modulus Mathematics Subject Classification Primary 30L10 ·Secondary 30C65 ·28A75 · 51F99 1 Introduction The quasisymmetric uniformization problem asks one to characterize, as meaningfully as possible, those metric spaces which may be mapped onto a domain in the Euclidean plane, or the 2-sphere, by a quasisymmetric homeomorphism. Informally, a mapping All authors were supported by the Academy of Finland, project number 308659. BMartti Rasimus [email protected] Kai Rajala [email protected] Matthew Romney matthew[email protected] 1Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), 40014 University of Jyvaskyla, Finland 123
K. Rajala et al. is quasisymmetric if it roughly preserves the relative distance between triples of points. See Sect. 4for the precise definition. Significant results on the uniformization problem, such as the Bonk–Kleiner theorem [4] and its extensions in [21] and [22], have been obtained for surfaces that are non-fractal, i.e., their 2-dimensional Hausdorff measure is locally finite. These spaces carry enough rectifiable paths for classical methods such as conformal modulus to be applicable. By surface, we mean a 2-manifold equipped with a continuous metric. In contrast, the class of fractal surfaces is too general for the standard methods. Consequently, understanding the quasisymmetric uniformization of fractal surfaces has proved extremely difficult. Any progress is desirable, especially due to applications to geometric group theory (cf. [3,12]) and complex dynamics (cf. [5]). The usual method for constructing quasisymmetric maps is to first show the existence of some conformal or quasiconformal map in the spirit of the classical uniformization theorem. Then, if the underlying surface has good geometric properties, one can use quasiconformal invariants to show that such a map is actually quasisymmetric. A fundamental difficulty in extending this method to fractal surfaces is the lack of a suitable definition of quasiconformality. The classical metric definition (see Sect. 4) is too weak to lead to a satisfactory theory in this generality. The geometric definition (see Sect. 2) requires the existence of many rectifiable paths, which need not be the case for fractal surfaces. In Sect. 2, we propose the definition of μ-quasiconformality for homeomorphisms f:X→R2, depending on a measure μon X. This is a modification of the geometric definition: we deform the metric on Xusing μto obtain the μ-length of a curve, and we define the corresponding μ-modulus of a family of curves in X. A homeomorphism fis μ-quasiconformal if the μ-modulus of every family of curves in Xis comparable to the conformal modulus of its image under fin R2. A quasisymmetric map f:X→R2is μ-quasiconformal when μis the pullback of the Lebesgue measure on R2. Our goal is to find measures μon a given space X for which the existence of μ-quasiconformal maps can be shown. In Sect. 3, we introduce the notion of infinitesimally metric measure on X. These correspond to the metric doubling measures of David and Semmes [6,13], the correspondence being similar to the one between metrically quasiconformal (MQC) maps and quasisymmetric (QS) maps, where the former is an infinitesimal condition and the latter is a global condition. Metric doubling measures can be used to produce quasisymmetric maps via deformation of the metric on X. Our first main result shows that aμ-quasiconformal map exists if μis an infinitesimally metric measure. Theorem 1.1 Let X be a metric space homeomorphic to R2which supports an infinitesimally metric measure μ. Then there exists a μ-quasiconformal map f :X→, where =D⊂R2or =R2. To prove Theorem 1.1, we first show that the metric don Xcan be deformed using μto yield a “quasiconformally equivalent” metric qthat has locally finite Hausdorff 2measure. Then, we apply the uniformization theorem in [14] to obtain a quasiconformal map (X,q)→R2. Composing, we then get the desired μ-quasiconformal map. 123
Infinitesimally Metric Measures In view of the correspondence between infinitesimally metric measures and metric doubling measures, it is natural to attempt to characterize the class of metric spaces Xthat admit metrically quasiconformal maps f:X→R2in terms of infinitesimally metric measures. However, it turns out that the existence of such maps can be rather arbitrary unless strong conditions are imposed on X. Instead, we consider the notion of infinitesimally quasisymmetric (I-QS) mapping (Definition 4.1). Such maps form an intermediate class between those of MQC and QS maps. In our second main result, we characterize the metric spaces which admit such maps into R2as the spaces that carry infinitesimally metric measures with suitable properties. Theorem 1.2 Let X be a metric space homeomorphic to R2. There exists an infinitesimally quasisymmetric map f :X→, where =Dor =R2, if and only if X is infinitesimally linearly locally connected and supports an infinitesimally metric measure μsuch that (X,μ)is infinitesimally Loewner. See Sect. 4for definitions. The proof combines Theorem 1.1 with estimates for the μ-modulus that generalize the modulus estimates in [11]. One motivation for our work is to understand the conformal geometry of metric surfaces in the absence of strong geometric assumptions such as Ahlfors regularity, linear local connectedness, and the Loewner condition (see Sect. 4). In Sect. 5,we present four examples to illustrate possible behaviors of metric surfaces under weaker geometric assumptions. We remark that, while the main theorems of this paper are applicable to any metric space homeomorphic to R2, including fractal spaces, all of these examples have locally finite Hausdorff 2-measure. The four examples are summarized here, listed by section in which they appear. 5.1. A surface that admits an MQC parametrization by R2but not an I-QS parametrization. This surface is linearly locally connected (LLC) but not Loewner. This example also illustrates how metric quasiconformality is not preserved under taking inverses or precomposing with a QS map. 5.2. A surface that admits a geometrically quasiconformal (QC) parametrization by R2 but not a MQC parametrization. This surface is upper Ahlfors 2-regular but not infinitesimally LLC. 5.3. A surface that admits an I-QS parametrization by R2but not a quasisymmetric parametrization. This surface is upper Ahlfors 2-regular but not LLC. 5.4. A surface that, despite being a geodesic space of locally finite Hausdorff 2measure, violates infinitesimal upper Ahlfors 2-regularity at every point along a non-degenerate continuum. This surface is LLC, and it admits a QC parametrization by R2but not a MQC parametrization. In particular, these examples show that the class of I-QS maps from R2onto a metric space differs from both the class of QS maps and the class of MQC maps. 2-Quasiconformal Maps We assume throughout the paper that (X,d)is a metric space homeomorphic to the Euclidean plane R2. We denote B(x,r)={y∈X:d(x,y)<r},B(x,r)={y∈ 123
K. Rajala et al. X:d(x,y)≤r}, and S(x,r)={y∈X:d(x,y)=r}.IfBis a ball of radius r,we denote by λBthe ball with the same center and radius λr.Apath in Xis a continuous map γ:I→X, where Iis an interval. The image of such a path is called a curve in X. We recall the Carathéodory construction of measures, cf. [7, 2.10]. Let Fbe a family of subsets of X, and ϕ:F→[0,∞].ForA⊂Xand δ>0, the δ-content φδ(A)is φδ(A)=inf S∈G ϕ(S), where the infimum is taken over all countable G⊂{S∈F:diam(S)≤δ}such that A⊂ S∈G S. Then, since φδ(A)is decreasing with respect to δ, the limit ψ(A)=lim δ→0+φδ(A)∈[0,∞] exists. Moreover, if every S∈Fis a Borel set, then ψis a Borel regular measure in X. Applying the Carathéodory construction with Fall the non-empty subsets of X and φ(S)=α(m)2−mdiam(S)mgives the m-dimensional Hausdorff measure Hmin X, where α(1)=2 and α(2)=π. Before defining μ-quasiconformal maps, we review the classical geometric definition of quasiconformality. However, we replace the standard modulus of path families with the modulus of curve families, which lead to equivalent definitions but are easier to work with in our setting. Let be a family of curves (i.e., images of paths) in X. A Borel function ρ:X→ [0,∞] is admissible for if CρdH1≥1 for all C∈with locally finite H1measure. The (conformal) modulus of is defined as mod =inf X ρ2dH2,(1) where the infimum is taken over all admissible functions ρ. Let X,Ybe metric spaces homeomorphic to R2and f:X→Ya homeomorphism. Then fis geometrically quasiconformal (QC), if there exists K≥1 such that K−1mod ≤mod f≤Kmod for all curve families in X. In this case, we also say that fis geometrically K - quasiconformal (K-QC). 123
Infinitesimally Metric Measures We now define μ-quasiconformal maps. Let μbe a Radon measure in Xwith no atoms such that μ(B)>0 for every open ball B⊂X. Recall that a Borel regular measure μis Radon if it is finite on compact sets. We associate with μa collection Bof open balls in Xsuch that for every point x∈Xthere is rx>0 such that B(x,r)∈Bfor every r<rx.Wealsomakethe requirement that B(x,rx)is compact for all x. We refer to such a collection Bas an admissible cover. From now on we use the convention that every measure μcomes equipped with an admissible cover B. Definition 2.1 The μ-length measure μin Xis defined by the Carathéodory construction with F=Band ϕ:B→[0,∞],ϕ(B)=2π−1/2μ(B)1/2. The μis normalized so that if X=R2and μthe Lebesgue measure, then μ=H1 (for any choice of B). Definition 2.2 Let be a family of curves in X. We say that a Borel function ρ:X→ [0,∞] is μ-admissible for if Cρdμ≥1 for all C∈with locally finite μmeasure. We denote the set of such functions by μ().Theμ-modulus of is modμ=inf ρ∈μ() X ρ2dμ. Notice that if μ(C)=0forsomeC∈, then there are no μ-admissible functions for and thus modμ=∞. On the other hand, if μis not locally finite on any C∈, then modμ=0. Definition 2.2 coincides with (1) when X=R2and μthe Lebesgue measure. Definition 2.3 Let f:X→be a homeomorphism, where is a domain in R2.We say that fand f−1are μ-quasiconformal, if there exists K≥1 such that K−1modμ≤mod f≤Kmodμ for every curve family in X. Definition 2.3 naturally leads to the following questions: (1) How to decide if a given metric space Xcarries a measure μfor which there exists aμ-quasiconformal map into R2? (2) How to decide if there exists a μ-quasiconformal map for a given (X,μ)? Concerning Question (2), it is reasonable to ask if the reciprocality condition (Definition 3.7 below) can be modified to yield a characterization similar to the one obtained in [14] for the 2-dimensional Hausdorff measure. In the next section, we introduce infinitesimally metric measures and show that they lead to the existence of μ-quasiconformal maps. 123
K. Rajala et al. 3 Infinitesimally Metric Measures We now define the infinitesimally metric measures. Let X,μ,B, and μbe as above. Moreover, for x,y∈Xlet q(x,y)=inf μ(C(x,y)), where the infimum is taken over all curves C(x,y)that join xand yin X. Thus qdefines a pseudometric on X. In the following, we use the subscripts dand qto indicate which (pseudo)metric is being used in our notation for balls, spheres, and diameter. Definition 3.1 The measure μis infinitesimally metric (I-MM) if there exist >1, Ci≥1 such that C−1 iq(y,z)≤μ(Bd(x,r))1/2≤Ciq(y,z)(2) for every Bd(x,r)∈B,y∈Bd(x,r/), and z∈Sd(x,r). It follows immediately from the definition that if μis I-MM, then qis a metric on X. Recall that a metric space Xis (Ahlfors) 2-regular if there exists C≥1 such that C−1r2≤H2(B(x,r)) ≤Cr2for all x∈X,r∈(0,diam X). We say that X is lower or upper 2-regular if, respectively, the first or second of these inequalities holds. Definition 3.1 imposes a similar infinitesimal condition on the measure μ.In fact, we show in Lemmas 3.4 and 3.5 that (X,q)is infinitesimally Ahlfors 2-regular. The remainder of this section is dedicated to the proof of Theorem 1.1.Wefirst restate the theorem. Theorem 3.2 Let X be a metric space homeomorphic to R2which supports an I-MM μ. Then there exists a μ-quasiconformal map f :X→, where =D⊂R2or =R2. As groundwork, we require several lemmas to estimate the 1and 2-dimensional Hausdorff measures corresponding to the metric q. We fix an I-MM μ.LetB={Bd(x,r):x∈X,r<rx}be the admissible cover associated with μ. The assumption that μhas no atoms implies that limr→0μ(Bd(x,r)) =0 for all x∈X. Definition 3.1 then implies that the metrics dand qare topologically equivalent. Lemma 3.3 We have μ(Bd(x,r)) ≤C2 iμ(Bd(x,r)) for every Bd(x,r)∈B, where Ciis the constant in Definition 3.1. 123
Infinitesimally Metric Measures Proof Since Bd(x,r)is compact and Xhomeomorphic to R2, there exists a point z∈∂(X\Bd(x,r)). Observe that z∈Sd(x,r).Letε>0, and let w∈Bq(z,ε)such that r<d(x,w)<rx.Now, μ(Bd(x,r))1/2≤μ(Bd(x,d(x,w)) 1/2 ≤Ciq(x,w)≤Ciq(x,z)+Ciε ≤C2 iμ(Bd(x,r))1/2+Ciε. Letting ε→0 proves the claim. Lemma 3.4 We have C−2 ir2≤μ(Bq(x,r)) ≤C3 ir2 for every ball Bq(x,r)contained in Bd(x,rx/2), where Ciis the constant in Definition 3.1. Proof Let s=inf y∈X\Bq(x,r)d(x,y)and t=sup z∈Bq(x,r) d(x,z). Clearly Bd(x,s)⊂Bq(x,r). We claim that there exists y∈Sd(x,s)such that q(x,y)≥r. If not, then X\Bq(x,r)and Bd(x,s)are disjoint closed sets, with Bd(x,s)compact. This implies that dist(X\Bq(x,r), Bd(x,s)) > 0, contradicting the definition of s. Since μisassumedtobeI-MM,wehaveμ(Bq(x,r)) ≥ μ(Bd(x,s)) ≥C−2 ir2. Likewise, Bq(x,r)⊂Bd(x,t). Similarly to the first part of the proof, we note that (X\Bd(x,t)) ∩Bq(x,r)=∅. Thus, there exists z∈Sd(x,t)such that q(x,z)≤r. Since μis I-MM, Lemma 3.3 gives μ(Bq(x,r)) ≤μ(Bd(x,t)) ≤C2 iμ(Bd(x,t)) ≤C3 ir2. For s,δ > 0, let Hs qand Hs q,δ denote the s-dimensional Hausdorff measure and Hausdorff δ-content on (X,q), respectively. Lemma 3.5 We have π 4C2 i μ(A)≤H2 q(A)≤100πC2 iμ(A) for any Borel set A ⊂X, where Ciis the constant in Definition 3.1. 123
K. Rajala et al. Proof Let δ>0, and let U⊂Xbe an open set with A⊂Uand μ(U)≤μ(A)+δ. Using the basic covering theorem (see [9, Thm. 1.2]), choose a sequence of pairwise disjoint balls Bj=Bq(xj,rj)with Bj⊂U,Bj⊂Bd(xj,rxj/2)and 10rj<δfor all j, such that U⊂∪ ∞ j=15Bj. Then H2 q,δ(A)≤π∞ j=1 (10rj)2≤Cπ∞ j=1 μ(Bj)≤Cπμ(U)≤Cπ(μ(A)+δ), where C=100C2 i(the πcomes from the normalization of H2). The upper bound for H2 q(A)follows. For the lower bound, fix nand define the Borel set An= {x∈A:Bq(x,1/n)⊂Bd(x,rx/2)}∩A. Let {Ej}be a cover for Anwith diamq(Ej)< 1 2nfor all j. Removing sets from the cover if necessary, we may assume that for every jthere exists xj∈Ansuch that Ej⊂Bq(xj,2diam qEj)and Bq(xj,1/n)⊂Bd(xj,rxj/2). Since μ(An)≤∞ j=1 μ(Bq(xj,2diam qEj)) ≤4C2 i ∞ j=1 diamq(Ej)2, we get π 4C2 i μ(An)≤H2 q,1/2n(An)≤H2 q(A). Since μ(A)=limn→∞ μ(An), the claim follows. Lemma 3.6 We have 2 Ci√πH1 q(A)≤μ(A)≤4C3 i √πH1 q(A) for any Borel set A ⊂X, where Ciis the constant in Definition 3.1. Proof Since Xis homeomorphic to R2, it is locally compact and can be exhausted by compact sets Xj. We can also approximate both μ(A)and H1 q(A)from below with the measures of the sets Aj=A∩Xj, and by considering some compact neighborhood Xj+kof Ajwe can assume that sup x∈X diamq(Bd(x,r)), sup x∈X diamd(Bq(x,r)) →0asr→0. We first consider Borel sets An= {x∈A:1/n<rx}∩A,n∈N. 123
Infinitesimally Metric Measures Proposition 4.7 Let f , μ, and Bbe as in Lemma 4.5. Then μis I-MM and satisfies the I-Loewner condition. Proof Let >1 be large enough so that η(1/) ≤1 2.Fixx∈Xand 0 <r<rx/2 so that B(f(x), diam f(B(x,r))) ⊂. In order to prove the I-MM condition (2), fix y∈B(x,r/) and z∈S(x,r). Then the segment [f(y), f(z)]is contained in . Let C=f−1([f(y), f(z)]), which is a curve connecting yand z. Now let T=sup{t>0:B(f(x), t)⊂f(B(x,r))}. Using Lemma 4.5 and infinitesimal quasisymmetry, we have μ(C)≤4η(5)H1(f(C)) =4η(5)|f(y)−f(z)|≤4η(5)diam fB(x,r) ≤8η(1)η(5)T≤8η(1)η(5) √πL2(f(B(x,r)))1/2 =8η(1)η(5) √πμ(B(x,r))1/2, so the first inequality in (2) holds. For the reverse inequality, notice first that our choice of implies that |f(x)− f(y)|≤1 2|f(x)−f(z)|and thus |f(y)−f(z)|≥1 2|f(x)−f(z)|.LetCbe any curve connecting yand z. Now by Lemma 4.5 μ(C)≥η(1)−1H1(f(C)) ≥η(1)−1|f(y)−f(z)|≥ 1 2η(1)|f(x)−f(z)| ≥1 2√πη(1)2L2(f(B(x,r)))1/2=1 2√πη(1)2μ(B(x,r))1/2, since f(B(x,r)) ⊂B(f(x), η(1)|f(x)−f(z)|). Hence also the second inequality in (2) holds. We conclude that μis I-MM. Finally, we show the I-Loewner condition. Fix x∈Xand disjoint continua Eand Fas in Definition 4.3, so that there are y∈F∩S(x,s)and z∈E∩S(x,t).By infinitesimal quasisymmetry, dist(fE,fF) diam E≤|f(y)−f(x)| |f(z)−f(x)|≤η(s/t). By definition, Fcontains S(x,rx). In particular, fS(x,rx)surrounds f(x), and we have dist(fE,fF)≤diam fF. Combining the estimates yields dist(fE,fF) min{diam E,diam F}≤max{η(s/t), 1}. 123
K. Rajala et al. Since R2is Loewner, there is φsuch that mod ( fE,fF)≥φ(max{η(s/t), 1}). On the other hand fis μ-quasiconformal by Theorem 1.1,so modμ(E,F)≥K−1mod ( fE,fF) for some K≥1. We conclude that the I-Loewner condition holds with φ(T)= K−1φ(max{η(T), 1}). Proposition 4.8 Let μbe an I-MM on X, and f :X→aμ-quasiconformal homeomorphism. Suppose that X is I-LLC and μsatisfies the I-Loewner condition. Then f is I-QS. Proof Let and λbe the constants in Definitions 3.1 and 4.2 of I-MM and I-LLC, respectively. We will prove the equivalent statement that g=f−1is I-QS. In this proof, for a point a∈and set A⊂,leta=g(a)and A=g(A). Fix x∈and r>0 so that B(x,3r)⊂∩g−1B(x,rx/(10λ44)), and y,z∈B(x,r). By our choice of r, we can choose w∈g−1S(x,rx)so that the segment [x,w]contains z. Moreover, taking rto be sufficiently small, we can ensure that the segment [x,w]lies in . Notice that w/∈B(x,3r).Letm=d(x,y) and =d(x,z).Lett>0. We must find an upper bound η(t)on m/ that holds whenever |x−y|/|x−z|≤t, such that η(t)→0ast→0. Assume then that y,z satisfy |x−y|/|x−z|≤t. Suppose first that m/ ≥λ2. Then, by the I-LLC property, we can connect xto zby a continuum Econtained in B(x,λ), and yto wby a continuum Fcontained in X\B(x,m/λ).Letk=log(m/(λ2)), Bj=B(x,j/λ), and Aj=B(x,j/λ) \B(x,j−1/λ). Then, by the definition of I-MM, ρ=1 k k j=1 CiχAj μ(Bj)1/2 is μ-admissible for (E,F). Thus modμ(E,F)≤X ρ2dμ≤1 k2 k j=1 C2 iμ(Aj) μ(Bj)≤C2 i k≤C2 i log(m/(λ2)). Hence modμ(E,F)becomes arbitrarily small as m/ increases to infinity. 123
Infinitesimally Metric Measures Since gis μ-quasiconformal, mod (E,F)is also small, where E=g−1(E)and F=g−1(F). But these sets connect xto zand yto w, respectively, and have relative distance (E,F)=dist(E,F) min{diam E,diam F}≤|x−y| |x−z|. Thus, by the Loewner property of R2,wehave|x−y|/|x−z|→∞as m/ →∞, establishing the distortion inequality in this case. Suppose then that 0 <m/ < λ2. In this case we choose E=[x,y]and F=[z,w]∪g−1S(x,rx). We may assume that 2|x−y|<|x−z|, since otherwise there is nothing to prove. Applying the I-Loewner condition to Eand F,wehave modμ(E,F)≥φ(/m). Combining with the μ-quasiconformality of g, we get mod (E,F)≥K−1φ(/m). On the other hand, by our choice of wwe can estimate mod (E,F)from above as follows: mod (E,F)≤mod (S(x,|x−z|), S(x,|x−y|)) =2πlog |x−z| |x−y|−1 . Combining the estimates, we see that φ(/m)≤2πK(log(1/t))−1. Observe that this bound becomes arbitrarily small as t→0. Since φis decreasing, this yields an upper bound η(t)on m/ that goes to zero as t→0. 5 Examples In this section, we work out in detail a number of specific examples of metric spaces homeomorphic to the plane. All of our examples have locally finite Hausdorff 2measure, and we assume throughout this section that a given metric space is equipped with the Hausdorff 2-measure. We write a point xin coordinates as x=(x1,x2)if x∈R2or x=(x1,x2,x3)if x∈R3. In addition to the examples of this section, we refer the reader to Example 4.7 of [10] for a family of uniformly LLC surfaces in R3, equipped with the ambient Euclidean metric, that are conformally equivalent but not uniformly QS equivalent to the Euclidean plane. We also refer to Example 2.1 of [14] for an example of a non-reciprocal metric on the plane, and to Example 17.1 of [14] for a non-rectifiable surface in R3that is QC equivalent to the Euclidean plane. Finally, see [17]forthe construction of a surface of locally finite Hausdorff 2-measure that is QS equivalent to the plane but not QC equivalent. 123
K. Rajala et al. 5.1 Conformal Weight that Decreases Rapidly Near the Origin Define a metric don the Riemann sphere R2=R2∪{∞}via the conformal weight ω(x)=e−1/|x|/|x|2if x= 0 0ifx=0,∞. That is, for all x,y∈ R2, the metric dis given by d(x,y)=infγγωds, where the infimum is taken over all absolutely continuous paths γ:[0,1]→ R2such that γ(0)=xand γ(1)=y. It is easy to check that d(0,x)=e−1/|x|for all x∈ R2\{0}. In particular, d(0,∞)=1 and we see that dis finite. Next, let x,y∈ R2\{0}and assume that |x|≤|y|. By considering the concatenation of the straight-line path from xto (|y|/|x|)xand a circular arc from (|y|/|x|)xto y, we obtain the estimate d(x,y)≤e−1/|y|−e−1/|x|+2πe−1/|y| |y|. As a consequence, if (xj)and (yj)are sequences in R2such that xj→∞and yj→ ∞, then d(xj,yj)→0. This is sufficient to conclude that ( R2,d)is homeomorphic to the Riemann sphere. In fact, by considering the pushforward of ωunder the inversion map x→ x/|x|2, we see that ( R2,d)is isometric to the metric space ( R2, d), where dis the metric generated by the conformal weight ω(x)=e−|x|. In particular, any ball in ( R2,d) centered at ∞not containing the origin is bi-Lipschitz equivalent to a Euclidean disk. In Fig. 1, a number of geodesics emanating from the point p=(.3,0)are plotted. Observe that the length-minimizing path from pto a point qin the upper left region of the plot is the concatenation of the straight-line path from pto the origin and the straight-line path from the origin to q. This example illustrates how metric quasiconformality is not preserved in general under taking inverses or under precomposition with a quasisymmetry, as the following proposition shows. Proposition 5.1 Let ι:(R2,|·|)→(R2,d)be the identity map, and let h :R2→R2 be the linear map defined by h(x1,x2)=(x1/2,x2). (a) ιis MQC with H =1, as is its inverse. (b) ιis 1-QC. (c) ιis not I-QS. (d) (ι ◦h)−1is MQC. (e) ι◦h is not MQC. Proof Claim (a) is immediate for all x= 0 by virtue of ωbeing a conformal weight, and it also holds for x=0 by the radial symmetry of ω. Claim (b) is also immediate if we exclude x=0. However, observe that reciprocality condition (4) holds for the metric dat the origin. Thus the geometric definition is unaffected by adding the origin back in, so the claim holds on all of R2. 123
Infinitesimally Metric Measures For claim (c), let (tj)be a sequence of positive numbers converging to zero, and let yj=(2tj,0),zj=(tj,0). Then |yj−0|=2tj,|zj−0|=tj,d(yj,0)=√e−1/tj, and d(zj,0)=e−1/tj. But then |yj−0|/|zj−0|=2 while d(yj,0)/d(zj,0)→∞, violating the I-QS condition. For claim (d), note that (ι ◦h)−1=h−1◦ι−1:(R2,d)→(R2,|·|)is the postcomposition of a MQC map by a QS map, which is always MQC. Claim (e) follows from a variation of the argument for (c). Let (tj)again be a sequence of positive numbers converging to zero, and let yj=(tj,0)and zj=(0,tj). Then h(yj)=(tj/2,0)and h(zj)=zj. This gives d(h(yj), 0)=√e−1/tjand d(zj,0)=e−1/tj, showing that ι◦his not MQC. Claim (c) of Proposition 5.1 can be strengthened to the following. Proposition 5.2 There is no I-QS map f :(R2,|·|)→(R2,d). Proof Suppose that such an I-QS map fexists. Then f−1is also I-QS. Since metric quasiconformality is preserved under postcomposition by an I-QS map, it follows that f−1◦ιis an MQC map of the Euclidean plane. By the equivalence of definitions of quasiconformality in the Euclidean setting (for example, see [20, Thm. 34.1]), we conclude that f−1◦ιis QS and thus that ιitself is I-QS. This contradicts claim (c) of Proposition 5.1. Note that the claims in Proposition 5.1 all hold if we replace R2with R2equipped with the spherical metric. We also observe that (R2,d)is not upper 2-regular: The Hausdorff 2-measure of the ball Br=B(0,r), where r∈[0,1], is given by H2(Br)=Br ω2dL2=2πR 0 e−2/t/t3dt, where R=−(log r)−1. This evaluates to H2(Br)=2πe−2/R1 4+1 2R=2πr21 4−log r 2. Since −log r→∞as r→0, we see that upper 2-regularity fails. Proposition 5.3 The space ( R2,d)is linearly locally connected. However, it is not a Loewner space. The proof of linear local connectedness uses the following lemma. Lemma 5.4 Let x ∈R2and r >0be such that B(x,r)⊂B(0,e−2). Then B(x,r)is simply connected. Proof The claim is obvious when x=0, so we assume that x= 0. We argue by contradiction. Suppose that B=B(x,r)is not simply connected. Since ( R2,d)is a geodesic space, all metric balls are connected. Hence the failure of simple connectivity implies that there exists a component Vof R2\Bnot containing ∞. 123
K. Rajala et al. Fig. 1 Geodesics emanating from the point (.3,0) Observe that B(0,e−2)coincides with the Euclidean ball B(0,1/2). In this region, ωis increasing as a function of the radius. Let Lbe the Euclidean straight line which contains xand the origin. The increasing property of ωimplies that L∩B(0,e−2)is a geodesic segment. Thus L∩B(x,r)is connected, and in particular V∩L=∅. It follows that Vis contained in one of the two open half-planes defined by the line L, denoted by W.Letz∈Vand let Sdenote the Euclidean circle of radius |z|centered at the origin. Let Ldenote the Euclidean straight line containing 0 and z. Then W\L consists of two disjoint open sets W1,W2, where x∈∂W1. We observe that there exists a point y∈S∩B∩W2. A length-minimizing curve from xto ymust cross L at some point v. However, the radial symmetry of ωimplies that d(v, z)≤d(v, y), and thus that d(x,z)≤d(x,y). This gives a contradiction, and we conclude that Bis simply connected. Proof of Proposition 5.3 That ( R2,d)is linearly locally connected can be shown from Lemma 5.4 as follows. By Lemma 2.5 in [4], it suffices to show that there exists r0>0 and λ≥1 such that every ball B(x,r)of radius r∈(0,r0)is contractible inside the ball B(x,λr). Let s=e−2/4 and let L≥1 be such that ( R2\B(0,s), d)is L-bi-Lipschitz equivalent to a Euclidean disk. Let r0=e−2/(4L2)and λ=L2. For any r∈ (0,r0)and x∈ R2, the ball B(x,λr)is contained in B(0,e−2)or it is contained in ( R2\B(0,s), d). In the first case, B(x,r)is simply connected by Lemma 5.4 and hence contractible. In the second case, the L-bi-Lispchitz equivalence of ( R2\B(0,s), d) 123
Infinitesimally Metric Measures with a Euclidean disk implies that B(x,r)is contractible inside B(x,λr). We conclude that ( R2,d)is linearly locally connected. We now show that ( R2,d)is not Loewner. Let E=(−∞,0)×{0}and let Ft=[rt,Rt]×{0}for t∈(0,1), where rt=−1/log(t/2)and Rt=−1/log t. Then dist(E,Ft)=diam(Ft)=t, so that (E,Ft)=1 for all t. Observe that limt→0Rt/rt=1. Since the identity map ι:(R2,|·|)→(R2,d)is 1-QC, the modulus of (E,Ft) relative to the metric dis the same as the modulus of the same curve family relative to the Euclidean metric. These curve families arise classically in the Teichmüller ring problem [1, Chapter III]. One can give an upper bound on their modulus as follows. Let tdenote the family of curves which span the open Euclidean annulus At=A((rt,0);Rt−rt,rt), where tis sufficiently small so that Rt<2rt.For sufficiently small t, the annulus Atdoes not intersect E. The family tmajorizes (E,Ft)and has modulus 2π/log(rt/(Rt−rt)). As t→0, we have that mod (E,Ft)goes to zero. Hence (R2,d)is not Loewner. The Loewner condition and linear local connectedness are conceptually similar in that they both rule out the existence of cusps and sequences of bottlenecks that become arbitrarily thin. In fact, the two properties are equivalent for the class of Ahlfors 2-regular metric spheres. This follows from Theorem 1.1 and Theorem 1.2 in [4] together with the quasisymmetric invariance of the Loewner condition [19,Cor.1.6]. This example illustrates how, for metric spheres of finite Hausdorff 2-measure, linear local connectedness does not imply the Loewner condition without the assumption of Ahlfors regularity. 5.2 An Accumulation of Spikes, I The purpose of this example is to give a metric surface Xso that the Hausdorff 2measure on Xis upper 2-regular but Xfails to be I-LLC. Upper regularity implies, by Proposition 3.9, that there is a QC parametrization of Xby the Euclidean plane. However, Xdoes not admit an MQC parametrization by the Euclidean plane, as shown by the following simple lemma. Lemma 5.5 Suppose there is an MQC map g :→X, where is a domain in R2. Then X is I-LLC. Proof Let x∈Xand x=g−1(x).LetRx>0 be sufficiently small so that Hg(x,R)≤2Hfor all R<Rx. For small r>0, g−1(B(x,r)) ⊂B(x,Rx).Lety,z∈B(x,r),y=g−1(y), z=g−1(z), and R=sup{|x−w|:w∈g−1(B(x,r))}. Then there is a curve C from yto zwhich is contained in B(x,R). The metric quasiconformality implies that g(C)is a curve from yto zcontained in B(x,2Hr). Similarly, let y,z∈X\B(x,r), with y=g−1(y)and z=g−1(z).Now,let R=inf{|x−w|:w∈\g−1(B(x,r))}. Connect yto zby a curve Cin \B(x,R). Then metric quasiconformality implies that g(C)is a curve from yto zcontained in X\B(x,r/(2H)). This establishes that Xis I-LLC. 123
K. Rajala et al. We construct this example as a surface in R3containing a sequence of spikes that become progressively smaller and converge to a point. For all n∈N,lettn=2−n, hn=2−n/2, and rn=2−2·2−3n/2. The surface Xis constructed by removing each Euclidean disk B((tn,0), rn)from R2, identified here with R2×{0}, and replacing it with a cone Snof height hn. That is, Snhas vertex (tn,0,hn)and joins to R2along the circle S((tn,0), rn). We equip Xwith the ambient Euclidean metric from R3, though the example works just as well if we were to take the induced length metric. We check that Xis upper 2-regular. Let x∈Xand r>0. In the first case, assume that r≤|x|/20, where |·|is the Euclidean norm in R3. A computation shows that B(x,r)intersects at most one of the cones Sn. It is clear that H2(B(x,r)∩ (R2×{0})) ≤πr2. By the elementary geometry of cones in R3, it also holds that H2(B(x,r)∩Sn)≤πr2. We conclude that H2(B(x,r)≤2πr2. In the second case, assume that r>|x|/20. Then B(x,r)⊂B(0,21r), writing 0 to denote the origin in R3. For this, we compute H2(B(0,2−n)) ≤π2−2n+∞ k=n H2(Sn) ≤π2−2n+π∞ k=n 2−3n/22−n+2−3n ≤π2−2n+π∞ k=n 2−3n/2(2−n/2+2−3n/2)2−2n. We deduce that H2(B(0,21r)) r2, and therefore that Xis upper 2-regular. Finally, the point yn=(tn,0,hn)lies outside the ball Bn=B(0,|yn|/2).Any continuum connecting ynto the unbounded component of R2\Bnmust pass through the smaller ball X\B(0,2tn). However, limn→∞ tn/|yn|=0, violating the I-LLC property. 5.3 An Accumulation of Spikes, II By modifying the previous example, we construct a space which is I-QS equivalent to the plane but not QS equivalent. We carry out the same construction as above, now taking tn=2−n,hn=2−n, and rn=2−2·2−2n. Instead of cones, we replace the Euclidean disks B((tn,0), rn)with cylinders Cnof height hn. More precisely, Cn=En∪Fn, where En={(x1,x2,x3): (x1,x2)∈S((tn,0), rn), 0≤x3≤hn}and Fn=B((tn,0), rn)+(0,0,hn).Again, we equip the resulting space Xwith the restriction of the ambient Euclidean metric to Xto get (X,d). The space Xis not LLC because the cylinders get progressively narrower; thus X is not QS equivalent to the Euclidean plane. However, we claim that Xequipped with μ=H2satisfies the conditions of Theorem 1.2 and therefore admits an I-QS map from R2. 123
Infinitesimally Metric Measures First, notice that for every x∈X\{(0,0,0)}there is rx>0 so that B(x,rx)⊂X is 10-bi-Lipschitz equivalent to a planar disk. In particular, the conditions of Theorem 1.2 hold for all such points x. We still need to verify the conditions of Theorem 1.2 for x=0=(0,0,0).Take r0=1/2. The I-LLC condition follows from our choices of tn,hn, and rn.Also, calculating as in Sect. 5.2, we conclude that r2H2(B(0,r)) r2for all r>0. Therefore, the q-metric on Xis comparable to the metric d, and μis I-MM. Finally, we show that the I-Loewner condition is satisfied at 0.ForafixedT>0, let s,t>0 satisfy s/t≤T.Letn∈Nbe such that 2−n−1≤s<2−n. Consider two disjoint continua E,F⊂Xas in Definition 4.3. We make the observation that the cylinders Cnand Cn−1are separated by a distance of at least 2−n−1. Thus F∩(R2× {0})∩B(0,2−n+1)contains a continuum Fof diameter at least 2−n−1. Next, we split into two cases. If t≥s, then by similar reasoning E∩(R2×{0})∩B(0,2−n)contains a continuum Eof diameter at least 2−n−2.Ift≤s, then take Eto be a continuum in E∩(R2×{0})∩B(0,t)of diameter at least t/16. Then d(E,F)≤2−n+2, and Eand Fhave relative distance (E,F)≤2−n+2 min{2−n−2,t/16}≤max {128T,16}. Let T=max{128T,16}, so that (E,F)≤T. Consider the domain G=R2\∞ n=1 B((tn,0), rn)×{0}⊂X. The domain Gis Loewner; let ϕbe the associated Loewner function. We have then the inequality mod (E,F)≥mod (E,F;G)≥ϕ(T). We conclude that the I-Loewner condition is satisfied at 0. 5.4 Gluing a Grushin Half-Plane to a Euclidean Half-Plane The Grushin plane is a basic example of a sub-Riemannian manifold. See [2, Sect. 3.1] for an overview. One approach to the Grushin plane, studied in [16], is given by the following definition. For each β∈(0,1),theβ-Grushin plane is R2equipped with the metric dobtained from the singular conformal weight ω:R2→[0,∞]defined by ω(x)=|x1|−β. The standard Grushin plane is obtained by taking β=1/2. Note that the standard Grushin plane does not have locally finite Hausdorff 2-measure. However, in the case when β∈(0,1/2),itwasshownin[18] and [23] that the β-Grushin plane is bi-Lipschitz equivalent to the Euclidean plane. In particular, the β-Grushin plane is Ahlfors 2-regular. Moreover, the identity map R2→(R2, d)is QS. A proof of this can be found in [16,Thm.4.3]. 123
K. Rajala et al. Here, we present a modified version of the Grushin plane. Let β∈(0,1/2). Define the conformal weight ω:R2→[0,∞] by ω(x)=|x1|−βif x1>0 1ifx1≤0. Let ddenote the resulting metric. First, we establish a ball–box relationship. For all r≤1, let Dr=[−r,(1−β)r1/(1−β)]×[−r,r]. Note that, for all x2∈R, the straight-line curve from (0,x2)to ((1−β)r1/(1−β),x2) has length r. Observe further that ω≥1onDr. From this, it follows that d(x,0)≥r for all x∈∂Dr. Next, by considering the concatenation of the vertical line segment from 0 to (0,x2)with the horizontal line segment from (0,x2)to x, we see that d(x,0)≤2rfor all x∈∂Dr. We conclude that Bd((0,0), r)⊂Dr⊂Bd((0,0), 2r)(11) for all r≤1. Next, observe that H2(Bd(0,2r)) is bounded from below by Dr ω2dL2=2r2+2rR 0 t−2βdt =2r2+r(2−3β)/(1−β) 1−2β.(12) For β∈(0,1/2), the inequality (2−3β)/(1−β) < 2 holds, from which we conclude that lim inf r→0 H2(Bd(x,r)) r2=∞ for all xlying on the vertical axis. On the other hand, (12) is an upper bound on H2(Bd(x,r)), showing that (R2,d)has locally finite Hausdorff 2-measure. Since ωis constant on each vertical line, we see that metric balls are simply connected. In particular, (R2,d)is LLC. This example illustrates how a metric surface with locally finite 2-measure can violate infinitesimal upper 2-regularity at every point in a fairly large set, namely a non-degenerate continuum. Since any metric surface that is infinitesimally upper 2-regular is reciprocal, this suggests the following question. Question 5.6 Is there a metric surface for which reciprocality condition (4) fails at every point on a non-degenerate continuum? The space (R2,d)in this example is reciprocal and hence does not answer this question. In fact, the identity map onto the Euclidean plane is 1-QC. This can be shown by a change of variables argument; see also Proposition 3.5 in [8], where the corresponding fact is proved for the β-Grushin plane. In contrast, we have the following. 123