Two‐dimensional metric spheres from gluing hemispheres
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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-NC 4.0 https://creativecommons.org/licenses/by-nc/4.0/ Two‐dimensional metric spheres from gluing hemispheres © 2022 the Authors Published version Ikonen, Toni Ikonen, T. (2022). Two‐dimensional metric spheres from gluing hemispheres. Journal of the London Mathematical Society, 106(4), 3069-3102. https://doi.org/10.1112/jlms.12656 2022
Received: 2 November 2021 Accepted: 27 April 2022 DOI: 10.1112/jlms.12656 Journal of the London Mathematical Society RESEARCH ARTICLE Two-dimensional metric spheres from gluing hemispheres Toni Ikonen University of Jyvaskyla, Department of Mathematics and Statistics, Jyvaskyla, Finland Correspondence Toni Ikonen, University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, Jyvaskyla, Finland. Email: toni.m.h.ik[email protected]i Funding information Academy of Finland, Grant/Award Number: 308659; Vilho, Yrjö and Kalle Väisälä Foundation Abstract We study metric spheres (𝑍, 𝑑𝑍)obtained by gluing two hemispheres of 𝕊2along an orientation-preserving homeomorphism g∶𝕊1→𝕊1, where 𝑑𝑍is the canonical distance that is locally isometric to 𝕊2off the seam. We show that if (𝑍, 𝑑𝑍)is quasiconformally equivalent to 𝕊2, in the geometric sense, then gis a welding homeomorphism with conformally removable welding curves. We also show that gis bi-Lipschitz if and only if (𝑍, 𝑑𝑍) has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping ℎ∶ 𝕊2→𝕊2. Furthermore, we show that if g−1 is absolutely continuous and gadmits a homeomorphic extension with exponentially integrable distortion, then (𝑍, 𝑑𝑍)is quasiconformally equivalent to 𝕊2. MSC 2020 30L10 (primary), 30C65, 28A75, 51F99, 52A38 (secondary) 1 INTRODUCTION In this paper, we work in the unit sphere 𝕊2⊂ℝ3. We denote the equator 𝕊2∩(ℝ2×{0})by 𝕊1and endow 𝕊2with the length distance 𝜎induced by the Euclidean distance of ℝ3. The open southern and northern hemispheres are denoted by 𝑍1and 𝑍2, respectively. Here (0,0,1)∈𝑍 2. Consider an orientation-preserving homeomorphism g∶𝕊1→𝕊1, mapping the boundary of 𝑍1 to the boundary of 𝑍2. We identify each 𝑧∈𝕊1with its image g(𝑧) ∈ 𝕊1. With this identification, we obtain a set 𝑍and inclusion maps 𝜄1∶ 𝑍1→𝑍and 𝜄2∶ 𝑍2→𝑍.Wecall𝑆𝑍=𝜄 1(𝕊1)=𝜄 2(𝕊1) the seam of 𝑍. © 2022 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. J. London Math. Soc. (2) 2022;1–34. wileyonlinelibrary.com/journal/jlms 1
2IKONEN We construct a pseudodistance 𝑑𝑍on 𝑍, see Section 3, making the inclusion maps local isometries off the seam and 1-Lipschitz everywhere. We consider the quotient map 𝑄∶ 𝑍 → ˜ 𝑍 identifying points 𝑥,𝑦 ∈ 𝑍 whenever 𝑑𝑍(𝑥, 𝑦) = 0, and endow ˜ 𝑍with the associated quotient distance. We are interested in this construction for the following reason: whenever the metric space ˜ 𝑍is quasiconformally equivalent to 𝕊2, there exist Riemann maps 𝜙1∶𝑍 1→Ω 1,𝜙2∶𝑍 2→Ω 2onto the complementary components of a Jordan curve with g=𝜙 −1 2◦𝜙1|𝕊1; with the Carathéodory theorem we can make sense of the composition 𝜙−1 2◦𝜙1|𝕊1[16]. Any such gis called a welding homeomorphism and awelding curve. A long-standing problem is to understand which homeomorphisms gsatisfy g=𝜙 −1 2◦𝜙1|𝕊1for some Riemann maps. We refer to the survey articles [18, 37] for further background information. We also investigate the properties of ˜ 𝑍, given an arbitrary welding homeomorphism g.Weshow in Section 4that the 1D Hausdorff measures on the seam 𝑄(𝑆𝑍)and on (the tangents of) are closely connected, using results from classical complex analysis [16]. For example, our results show that a given subarc of the welding curve has tangents only in a set negligible to the 1D Hausdorff measure if and only if the quotient map 𝑄collapses the corresponding part of the seam to a point. We present in Sections 7.1 and 7.2 examples illustrating that for some homeomorphisms g, after removing a portion 𝐸′of the seam 𝑄(𝑆𝑍), one can find a 1-quasiconformal embedding 𝜓∶ ˜ 𝑍⧵𝐸 ′→𝕊2, but not necessarily a quasiconformal homeomorphism Ψ∶ ˜ 𝑍→𝕊2. A similar phenomenon was investigated in [17]and[7] in more detail. We now state our first result. Theorem 1.1. Let g∶𝕊1→𝕊1be an orientation-preserving homeomorphism. The following are quantitatively equivalent. (1) gis 𝐿-bi-Lipschitz; (2) there exists an 𝐿′-bi-Lipschitz homeomorphism Ψ∶ ˜ 𝑍→𝕊2; (3) there exists 𝐶′⩾0such that for every 𝑦∈𝑄(𝑆 𝑍), lim inf 𝑟→0+ 2 ˜ 𝑍(𝐵˜ 𝑍(𝑦, 𝑟)) 𝜋𝑟2⩽𝐶′. In the implications “(1) ⇒(2)” we may take 𝐿′=𝐿,in“(2)⇒(3)” 𝐶′=(𝐿 ′)4, and in “(3) ⇒(1)” 𝐿=𝜋𝐶 ′. We prove “(1) ⇒(2)” by observing that if g∶𝕊1→𝕊1admits an 𝐿′-bi-Lipschitz extension 𝜙∶ 𝑍2→ 𝑍2, the space ˜ 𝑍has an 𝐿′-bi-Lipschitz parametrization. That we may take 𝐿′=𝐿in “(1) ⇒(2),” follows by applying a known planar extension result [24] and stereographic projection. The claim “(2) ⇒(3)” is a straightforward consequence of the properties of Hausdorff measures. The implication “(3) ⇒(1)” is proved by carefully analysing the behaviour of the inclusion mappings 𝜄𝑖∶ 𝑍𝑖→˜ 𝑍at the equator 𝕊1. Notice that the 𝜄𝑖are 1-Lipschitz everywhere and local isometries outside the equator. This implies 𝐶′⩾1in (3). Remark 5.9 shows two ways to improve the bi-Lipschitz constant 𝜋𝐶′. The improvements imply that as 𝐶′→1 +in (3), the bi-Lipschitz constant of gconverges to one. In particular, (3) holds with 𝐶′=1ifandonlyifgis an isometry. Theorem 1.1 is closely related to the following result.
METRIC SPHERES FROM GLUING HEMISPHERES 3 Theorem 1.2. If an orientation-preserving homeomorphism g∶𝕊1→𝕊1is 𝐿-bi-Lipschitz, there exists a 1-quasiconformal homeomorphism 𝜑∶ 𝕊2→˜ 𝑍and a 𝐾-quasiconformal homeomorphism ℎ∶ 𝕊2→𝕊2such that the Jacobians satisfy 𝐶−1𝐽ℎ(𝑥) ⩽𝐽𝜑(𝑥) ⩽𝐶𝐽ℎ(𝑥) for 2 𝕊2-almost everywhere, 𝑥∈𝕊2(1.1) for 𝐾=𝐿 4and 𝐶=𝐿 2. Conversely, if there exists 𝐾,𝐶,andℎfor which Equation (1.1) holds, then g is 𝜋(𝐾𝐶)2-bi-Lipschitz. The Jacobians are defined in Section 2.3.Wenotethatifℎ∶ 𝕊2→𝕊2is an orientationpreserving quasiconformal homeomorphism, the 𝐽ℎcoincides with the usual distributional Jacobian; see for example [2, Section 3.8]. If gis 𝐿-bi-Lipschitz, the existence of 𝜑and ℎis a straightforward consequence of the implication “(1) ⇒(2)” in Theorem 1.1.Ifℎand 𝜑exist, we first check that Ψ=ℎ◦𝜑−1 is bi-Lipschitz, the study of the seam requiring a careful argument, and use the implications “(2) ⇒(3) ⇒(1)” from Theorem 1.1 to verify that gis bi-Lipschitz. Theorem 1.2 is a special case of the quasiconformal Jacobian problem: which weights 𝜔∶ 𝕊2→ [0, ∞] are comparable to the Jacobians of quasiconformal homeomorphisms ℎ∶ 𝕊2→𝕊2;see[6, 10], and references therein for further reading. Given that (1) and (3) are equivalent in Theorem 1.1, it is not entirely clear for which classes of homeomorphisms one can expect ˜ 𝑍to be quasiconformally equivalent to 𝕊2,orwhatkindof geometric properties one can expect from such a ˜ 𝑍. Question 1.3. Let ˜ 𝑍be the metric space obtained from a homeomorphism g∶𝕊1→𝕊1. When can we find a quasiconformal homeomorphism 𝜓∶ ˜ 𝑍→𝕊2? What kind of restrictions does this impose on g? As an example, if gis a welding homeomorphism corresponding to the von Koch snowflake, then 𝑑𝑍(𝑥, 𝑦) = 0 for every pair of points in the seam, see Remark 4.2. Hence ˜ 𝑍can fail to be quasiconformally equivalent, or homeomorphic, to 𝕊2when gis in quasisymmetry. We show that a simple measure-theoretic assumption removes this obstruction. Proposition 1.4. Let g∶𝕊1→𝕊1be a quasisymmetry whose inverse is absolutely continuous. Then ˜ 𝑍is quasiconformally equivalent to 𝕊2. The absolute continuity of g−1 is used in two ways. First, it guarantees that ˜ 𝑍=(𝑍,𝑑 𝑍). Second, if 𝜓∶ 𝑍2→ 𝑍2is a quasisymmetric extension of g, we show that the homeomorphism 𝐻∶ 𝕊2→˜ 𝑍 satisfying 𝐻|𝑍1=𝜄 1and 𝐻|𝑍2=𝜄 2◦𝜓|𝑍2is quasiconformal. A key step in the proof is showing the Sobolev regularity 𝐻−1 ∈𝑁 1,2(˜ 𝑍,𝕊2); the absolute continuity of g−1 is applied here. Proposition 1.4 is a special case of the following stronger result. Theorem 1.5. Let g∶𝕊1→𝕊1be an orientation-preserving homeomorphism whose inverse is absolutely continuous. If gextends to a homeomorphism 𝜓∶ 𝑍2→ 𝑍2for which 𝜓|𝑍2has exponentially integrable distortion, then ˜ 𝑍is quasiconformally equivalent to 𝕊2. We now explain the main steps of the proof of Theorem 1.5. We first show that there exists a homeomorphism 𝐻∶ 𝕊2→˜ 𝑍with exponentially integrable distortion. We also have
4IKONEN 𝐻−1 ∈𝑁 1,2(˜ 𝑍,𝕊2); see Remark 6.8. The exponential integrability of the distortion of 𝐻is used to verify the reciprocality condition of ˜ 𝑍, see Definition 2.5. Then [31, Theorem 1.4] shows that ˜ 𝑍is quasiconformally equivalent to 𝕊2. The key ingredients in the proof are the condenser estimates for mappings of exponentially distortion [27], applicable because 𝐻−1 ∈𝑁 1,2(˜ 𝑍,𝕊2),and the Stoilow factorization theorem [2, Chapter 20]. There are some known criteria which guarantee that gadmits an extension as in Theorem 1.5;see[25, 39]. In Section 7.1, we present an example of g∶𝕊1→𝕊1that is locally bi-Lipschitz outside a single point, but for which ˜ 𝑍is not quasiconformally equivalent to 𝕊2. This illustrates that the absolute continuity of g−1 is not enough to guarantee that ˜ 𝑍is quasiconformally equivalent to 𝕊2.Thisfact is a consequence of the following result, partially answering Question 1.3. Theorem 1.6. Suppose that g∶𝕊1→𝕊1is an orientation-preserving homeomorphism for which there exists a quasiconformal homeomorphism ℎ∶ 𝕊2→˜ 𝑍.Then˜ 𝑍=(𝑍,𝑑 𝑍)and there exists a 1quasiconformal homeomorphism 𝜋∶ 𝕊2→˜ 𝑍.Furthermore,gis a welding homeomorphism whose welding curves are conformally removable. The first step in the proof of Theorem 1.6 is showing that ℎcanbeassumedtobe1quasiconformal. Then, up to an orientation-reversing Möbius transformation, 𝜙𝑖=ℎ −1 ◦𝜄𝑖∶𝑍 𝑖→ 𝕊2are Riemann maps with welding curve =ℎ −1(𝑄(𝑆𝑍)) and welding homeomorphism 𝜙−1 2◦𝜙1|𝕊1. The equality ˜ 𝑍=(𝑍,𝑑 𝑍)and the conformal removability of follow from a connection we show between the tangents of the welding curve and the Hausdorff 1-measure on the seam 𝑄(𝑆𝑍); see Section 4. The equality ˜ 𝑍=(𝑍,𝑑 𝑍)implies g=𝜙 −1 2◦𝜙1|𝕊1. We recall that a compact proper subset 𝐾⊂𝕊2is conformally removable if every homeomorphism 𝑀∶ 𝕊2→𝕊2conformal in 𝕊2⧵𝐾is Möbius. The von Koch snowflake example illustrates that conformal removability of a welding curve is not enough to guarantee even that ˜ 𝑍is homeomorphic to 𝕊2. We refer the reader to [37]and[38] for further reading on conformal weldings and the connections to conformal removability. See [20] for some results in the context of Theorem 1.5. The paper is structured as follows. In Section 2, we introduce our notations and some preliminary results. In Section 3, we analyse the distance 𝑑𝑍induced by any given homeomorphism g∶𝕊1→𝕊1. When gis a welding homeomorphim, we establish in Section 4a connection between the geometry of the seam 𝑆𝑍and the tangents of the corresponding welding curves . We also prove Theorem 1.6 in this section. In Section 5, we prove Theorems 1.1 and 1.2.Proposition1.4 and Theorem 1.5 are proved in Section 6. In Section 7, we give some concluding remarks. 2 PRELIMINARIES 2.1 Notation Let (𝑌, 𝑑𝑌)be a metric space. We sometimes drop the subscript from 𝑑𝑌when there is no chance for confusion. For all 𝑄⩾0, the 𝑄-dimensional Hausdorff measure,oraHausdorff 𝑄-measure,is defined by 𝑄 𝑌(𝐵) = 𝛼(𝑄) 2𝑄sup 𝛿>0 inf {∞ ∑ 𝑖=1 (diam 𝐵𝑖)𝑄∶𝐵⊂ ∞ ⋃ 𝑖=1 𝐵𝑖,diam𝐵 𝑖<𝛿 }
METRIC SPHERES FROM GLUING HEMISPHERES 5 for all sets 𝐵⊂𝑌, where 𝛼(𝑄) is chosen so that 𝑛 ℝ𝑛coincides with the Lebesgue measure 𝑛for all positive integers. The length of a path 𝛾∶ [𝑎,𝑏]→ 𝑌is defined as 𝓁𝑑(𝛾) = sup 𝑛 ∑ 𝑖=1 𝑑(𝛾(𝑡𝑖), 𝛾(𝑡𝑖+1)), the supremum taken over all finite partitions 𝑎=𝑡 1⩽𝑡2⩽⋯⩽𝑡𝑛+1 =𝑏. A path is rectifiable if it has finite length. The metric speed of a path 𝛾∶ [𝑎,𝑏] →𝑌at the point 𝑡∈[𝑎,𝑏]is defined as 𝑣𝛾(𝑡) = lim 𝑡≠𝑠→𝑡 𝑑(𝛾(𝑠),𝛾(𝑡)) |𝑠−𝑡| whenever this limit exists. The limit exists 1-almost everywhere for every rectifiable path [12, Theorem 2.1]. A rectifiable path 𝛾∶ [𝑎,𝑏] →𝑌 is absolutely continuous if for all 𝑎⩽𝑠⩽𝑡⩽𝑏, 𝑑(𝛾(𝑡),𝛾(𝑠)) ⩽∫𝑡 𝑠 𝑣𝛾(𝑢) 𝑑1(𝑢) with 𝑣𝛾∈𝐿 1([𝑎, 𝑏]) and 1the Lebesgue measure on the real line. Equivalently, the rectifiable path 𝛾is absolutely continuous if it maps sets of 1-measure zero to sets of 1 𝑌-measure zero [12, Section 3]. Let 𝛾∶ [𝑎,𝑏]→𝑋be an absolutely continuous path. Then the (path) integral of a Borel function 𝜌∶ 𝑋 → [0,∞]over 𝛾is ∫𝛾 𝜌𝑑𝑠=∫𝑏 𝑎 (𝜌 ◦𝛾)𝑣𝛾𝑑1.(2.1) If 𝛾is rectifiable, then the path integral of 𝜌over 𝛾is defined to be the path integral of 𝜌over the arc length parametrization 𝛾𝑠of 𝛾;see[19, Chapter 5] for further details. Given a Borel set 𝐴⊂𝑌, the length of a path 𝛾∶ [𝑎,𝑏] →𝑌 in 𝐴is defined as ∫𝑌𝜒𝐴(𝑦)#(𝛾−1(𝑦)) 𝑑1 𝑌(𝑦), where #(𝛾−1(𝑥)) is the counting measure of 𝛾−1(𝑥).For𝐴=𝑌,[15, Theorem 2.10.13] states 𝓁(𝛾) = ∫𝑌 #(𝛾−1(𝑦)) 𝑑1 𝑌(𝑦). (2.2) When 𝛾is rectifiable, for every Borel function 𝜌∶ 𝑌 →[0,∞], ∫𝛾 𝜌𝑑𝑠=∫𝑌 𝜌(𝑦)#(𝛾−1(𝑦)) 𝑑1 𝑌(𝑦). (2.3) The equality (2.3) follows from [15, Theorem 2.10.13] via a standard approximation argument using simple functions.
6IKONEN 2.2 Metric Sobolev spaces In this section we give an overview of Sobolev theory in the metric surface setting, and refer to [19] for a comprehensive introduction. Let Γbe a family of paths in 𝑌. A Borel function 𝜌∶ 𝑌 → [0,∞]is admissible for Γif the path integral ∫𝛾𝜌𝑑𝑠⩾1for all rectifiable paths 𝛾∈Γ.Given1⩽𝑝<∞, the 𝑝-modulus of Γis mod𝑝Γ=inf∫𝑌 𝜌𝑝𝑑2 𝑌, where the infimum is taken over all admissible functions 𝜌. Observe that if Γ1and Γ2are path families and every path 𝛾1∈Γ 1contains a subpath 𝛾2∈Γ 2, then mod𝑝Γ1⩽mod𝑝Γ2. In particular, this holds if Γ1⊂Γ 2. When 𝑝=2, and there is no chance for confusion, we omit the subscript from mod2. If 𝜌is admissible for a path family Γ⧵Γ 0, where mod𝑝Γ0=0, we say that 𝜌is 𝑝-weakly admissible for Γ. If a property holds for every path 𝛾∈Γexcept in a subfamily of 𝑝-modulus zero, the property is said to hold on 𝑝-almost every path in Γ. We also refer to 2-almost every path as almost every path. We recall the following lemma [19, Lemma 5.2.8]. Lemma 2.1. Let 1⩽𝑝<∞. A family of nonconstant paths Γsatisfies mod𝑝Γ=0if and only if there exists 𝜌∶ 𝑌 → [0,∞],𝜌∈𝐿 𝑝(𝑌) with ∞=∫𝛾 𝜌𝑑𝑠 for every 𝛾∈Γ. Let 𝜓∶ (𝑌,𝑑𝑌)→(𝑍,𝑑 𝑍)be a mapping between metric spaces 𝑌and 𝑍. A Borel function 𝜌∶ 𝑌 → [0,∞]is an upper gradient of 𝜓if 𝑑𝑌(𝜓(𝑥), 𝜓(𝑦)) ⩽∫𝛾 𝜌𝑑𝑠 for every rectifiable path 𝛾∶ [𝑎,𝑏]→ 𝑌 connecting 𝑥to 𝑦. The function 𝜌is a 𝑝-weak upper gradient of 𝜓if the same holds for 𝑝-almost every rectifiable path. A𝑝-weak upper gradient 𝜌∈𝐿 𝑝 loc(𝑌) of 𝜓is minimal if it satisfies 𝜌⩽˜ 𝜌almost everywhere for all 𝑝-weak upper gradients ˜ 𝜌∈𝐿 𝑝 loc(𝑌) of 𝜓.If𝜓has a 𝑝-weak upper gradient 𝜌∈𝐿 𝑝 loc(𝑌), then 𝜓has a minimal 𝑝-weak upper gradient, which we denote by 𝜌𝜓. We refer to Section 6of [19] and Section 3of [36] for further details. Minimal 2-weak upper gradients are also referred to as minimal weak upper gradients. Fix a point 𝑧∈𝑍,andlet𝑑𝑧=𝑑 𝑍(⋅,𝑧).Thespace𝐿𝑝(𝑌, 𝑍) is defined as the collection of measurable maps 𝜓∶ 𝑌 →𝑍 such that 𝑑𝑧◦𝜓is in 𝐿𝑝(𝑌).Moreover,𝐿𝑝 loc(𝑌, 𝑍) is defined as those measurable maps 𝜓∶ 𝑌 →𝑍 for which, for all 𝑦∈𝑌, there is an open set 𝑈⊂𝑌containing 𝑦 such that 𝜓|𝑈is in 𝐿𝑝(𝑈, 𝑍). The metric Sobolev space 𝑁1,𝑝 loc (𝑌, 𝑍) consists of those maps 𝜓∶ 𝑌 →𝑍 in 𝐿𝑝 loc(𝑌, 𝑍) that have a minimal 𝑝-weak upper gradient 𝜌𝜓∈𝐿 𝑝 loc(𝑌).
METRIC SPHERES FROM GLUING HEMISPHERES 7 For subsets ∅≠𝑈⊂𝑌, we say that 𝜓∈𝑁 1,𝑝(𝑈, 𝑍) if 𝜓|𝑈∈𝑁 1,𝑝 loc (𝑈, 𝑍),𝜌𝜓|𝑈∈𝐿 𝑝(𝑈) and 𝜓|𝑈∈𝐿 𝑝(𝑈, 𝑍).If𝑍=ℝ, we denote 𝑁1,𝑝(𝑈,𝑍)=𝑁 1,𝑝(𝑈), and in the case 𝑝=2, 𝐸(𝜓) ∶=2 −1‖‖‖𝜌𝜓‖‖‖ 2 𝐿2(𝑈). We refer to 𝐸(𝜓) as the Dirichlet energy of 𝜓. We repeatedly use the following technical lemma in later sections. Lemma 2.2. Let 𝜓∶ 𝑌 →𝑍 be continuous, 𝜌∶ 𝑌 → [0,∞] a Borel function and 𝛾∶ [0,1]→𝑌 absolutely continuous with ∫𝛾𝜌𝑑𝑠<∞. Suppose that 𝐸⊂𝑌is compact, 1 𝑍(𝜓(𝐸)) = 0,and𝓁(𝜓 ◦𝛾|𝐼)⩽∫𝛾|𝐼𝜌𝑑𝑠for each closed interval 𝐼⊂[0,1]⧵𝛾 −1(𝐸).Then𝓁(𝜓 ◦𝛾) ⩽∫𝛾𝜌𝑑𝑠. Proof. First, for every closed interval 𝐽⊂[0,1]⧵𝛾 −1(𝐸),𝜓◦𝛾|𝐽is absolutely continuous with 𝑣𝜓◦𝛾(𝑠) ⩽(𝜌 ◦𝛾)(𝑠)𝑣𝛾(𝑠) for 1-almost every 𝑠∈𝐽. This follows from [19, Proposition 6.3.2]. Second, consider the connected components {𝐼𝑖}∞ 𝑖=1 of [0, 1] ⧵ (𝜓 ◦𝛾)−1(𝜓(𝐸)). Notice that 𝐼𝑖⊂ [0, 1] ⧵ 𝛾−1(𝐸) for every 𝑖. Let 𝐽𝑖= 𝐼𝑖. Then 𝑣𝜓◦𝛾(𝑠) ⩽(𝜌 ◦𝛾|𝐽𝑖)(𝑠) 1-almost everywhere on 𝐽𝑖(on 𝐼𝑖). This fact, the continuity of 𝜓◦𝛾and ∫𝛾𝜌𝑑𝑠<∞imply 𝓁(𝜓 ◦𝛾|𝐽𝑖)⩽∫𝐼𝑖 (𝜌 ◦𝛾)𝑣𝛾𝑑𝑠 < ∞. By summing over 𝑖, we conclude ∞ ∑ 𝑖=1 𝓁(𝜓 ◦𝛾|𝐽𝑖)⩽∫⋃∞ 𝑖=1 𝐼𝑖 (𝜌 ◦𝛾)𝑣𝛾𝑑𝑠 ⩽∫𝛾 𝜌𝑑𝑠. Given 1 𝑍(𝜓(𝐸)) = 0,(2.2)and(2.3)imply 𝓁(𝜓 ◦𝛾) = ∫𝑍⧵𝜓(𝐸) #((𝜓 ◦𝛾)−1(𝑥)) 𝑑1 𝑍(𝑥) ⩽ ∞ ∑ 𝑖=1 ∫𝑍⧵𝜓(𝐸) #((𝜓 ◦𝛾|𝐽𝑖)−1(𝑥)) 𝑑1 𝑍(𝑥) = ∞ ∑ 𝑖=1 𝓁(𝜓 ◦𝛾|𝐽𝑖)⩽∫𝛾 𝜌𝑑𝑠. Hence 𝓁(𝜓 ◦𝛾) ⩽∫𝛾𝜌𝑑𝑠.□ 2.3 Measure theory Let 𝑌be a Borel subset of a complete and separable metric space. A Borel measure 𝜇on 𝑌is 𝜎-finite if there exists a Borel decomposition {𝐵𝑖}∞ 𝑖=1 of 𝑌for which 𝜇(𝐵𝑖)<∞for every 𝑖.
8IKONEN Apairof𝜎-finite Borel measures 𝜇and 𝜈on 𝑌are said to be mutually singular if there exists a Borel set 𝐵⊂𝑌such that 𝜇(𝐵) = 0 and 𝜈(𝑌 ⧵ 𝐵) = 0. The measure 𝜇admits a Lebesgue decomposition (with respect to 𝜈), where 𝜇=𝑓⋅𝜈+𝜇 ⟂,with𝜇⟂and 𝜈mutually singular and 𝑓 Borel measurable [9, Sections 3.1–3.2 in Volume I]. We say that 𝜇and 𝜈are mutually absolutely continuous if 𝜇=𝑓⋅𝜈with density 𝑓>0𝜈-almost everywhere. Given a homeomorphism 𝜓∶ 𝑌 →𝑍and measures 𝜈on 𝑌and 𝜇on 𝑍, the measure 𝜓∗𝜇(𝐵) = 𝜇(𝜓(𝐵)) is called the pullback measure. Such a measure admits a decomposition 𝜓∗𝜇=𝑓⋅𝜈+𝜇 ⟂ with 𝜈and 𝜇⟂mutually singular. If 𝜈=2 𝑌and 𝜇=2 𝑍, the density 𝑓is called the Jacobian of 𝜓 and denoted by 𝐽𝜓. 2.4 Quasiconformal mappings Here we define quasiconformal maps and recall some basic facts. Definition 2.3. Let (𝑌, 𝑑𝑌)and (𝑍, 𝑑𝑍)be metric spaces with locally finite Hausdorff 2-measures. A homeomorphism 𝜓∶ (𝑌,𝑑𝑌)→(𝑍,𝑑 𝑍)is quasiconformal if there exists 𝐾⩾1such that for all path families Γin 𝑌 𝐾−1 mod Γ ⩽mod 𝜓Γ ⩽𝐾modΓ, (2.4) where 𝜓Γ = {𝜓 ◦𝛾∶ 𝛾 ∈ Γ}.IfEquation(2.4) holds with a constant 𝐾⩾1, we say that 𝜓is 𝐾quasiconformal. A special case of [36, Theorem 1.1] yields the following. Theorem 2.4. Let 𝑌and 𝑍be locally compact separable metric spaces with locally finite Hausdorff 2-measure and 𝜓∶ 𝑌 →𝑍 a homeomorphism. The following are equivalent for the same constant 𝐾>0: (i) mod Γ ⩽𝐾mod𝜓Γfor all path families Γin 𝑌. (ii) 𝜓∈𝑁 1,2 loc(𝑌, 𝑍) and satisfies 𝜌2 𝜓(𝑦) ⩽𝐾𝐽𝜓(𝑦) for 2 𝑌-almost every 𝑦∈𝑌. The outer dilatation of 𝜓is the smallest constant 𝐾𝑂⩾0for which the modulus inequality mod Γ ⩽𝐾𝑂mod 𝜓Γ holds for all Γin 𝑌.Theinner dilatation of 𝜓is the smallest constant 𝐾𝐼⩾0 for which mod 𝜓Γ ⩽𝐾𝐼mod Γ holds for all Γin 𝑌. The number 𝐾(𝜓) = max{𝐾𝐼(𝜓), 𝐾𝑂(𝜓)} is the maximal dilatation of 𝜓. For a set 𝐺⊂𝑌and disjoint sets 𝐹1,𝐹 2⊂𝐺,letΓ(𝐹1,𝐹 2;𝐺)denote the family of paths with each path starting at 𝐹1,endingat𝐹2and whose images are contained in 𝐺.Aquadrilateral is a set 𝑄homeomorphic to [0, 1]2with boundary 𝜕𝑄 consisting of four boundary arcs, overlapping only at the end points, labelled 𝜉1,𝜉 2,𝜉 3,𝜉 4in cyclic order. Ametric surface is a separable metric space 𝑌with locally finite Hausdorff 2-measure that is homeomorphic to a (connected) 2-manifold without boundary.
METRIC SPHERES FROM GLUING HEMISPHERES 15 negligible 1 ˜ 𝑍-measure. Either way, the multiplicity is negligible in Equation (4.2), so the second equality is justified. Proposition 4.1. Let gbe a welding homeomorphism with a welding circle and 𝐼⊂asubarc. Then 𝑑˜ 𝑍(˜ 𝜋(𝑥), ˜ 𝜋(𝑦))=0for all 𝑥,𝑦 ∈ 𝐼 if and only if 𝜔1|𝐼and 𝜔2|𝐼are mutually singular. If such an interval exists, then ˜ 𝑍is not quasiconformally equivalent to 𝕊2. Remark 4.2. If gis a welding homeomorphism obtained from Remark 3.9 or any welding gcorresponding to the von Koch snowflake [16, Example 4.3], Proposition 4.1 implies that 𝑄(𝑆𝑍)is a singleton. In particular, ˜ 𝑍is not even homeomorphic to the sphere. For a given g, this happens if and only if g∗1 𝕊1and 1 𝕊1are mutually singular. A key step in the proof of the conformal removability in Theorem 1.6 is the following. Proposition 4.3. Let gbe a welding homeomorphism and ˜ 𝜋as in Equation (4.1). Then ˜ 𝜋is continuous, monotone, and surjective. Moreover, for all path families Γon 𝕊2,mod Γ ⩽mod ˜ 𝜋Γ.The metric space ˜ 𝑍is quasiconformally equivalent to 𝕊2if and only if ˜ 𝜋is a homeomorphism for which mod Γ = mod ˜ 𝜋Γ for all path families. The proof of Proposition 4.3 requires some preparatory work. Given the curve , we say that 𝑥0∈is a tangent point if there exists a homeomorphism 𝛾∶ (−𝜖,𝜖) → ′⊂with 𝛾(0) = 𝑥0, and a tangent vector 𝑣0∈𝑇 𝑥0𝕊2with unit length such that for every smooth 𝑓∶ 𝕊2→ℝ,its differential 𝑑𝑓 satisfies 𝑑𝑓(𝑣0)= lim 𝑡→0+ 𝑓(𝛾(𝑡)) − 𝑓(𝑥0) 𝜎(𝛾(𝑡),𝑥0)and 𝑑𝑓(−𝑣0)= lim 𝑡→0− 𝑓(𝛾(𝑡)) − 𝑓(𝑥0) 𝜎(𝛾(𝑡),𝑥0). If 𝑣0exists, the tangent vector 𝑣0is independent of the parametrization 𝛾and ′up to multiplication by −1;see[16, Chapter II, Section 4]. The collection of tangents points of is denoted by Tn(). The key properties of Tn()are self-contained in the following statement. Lemma 4.4. The Borel set Tn()has 𝜎-finite Hausdorff 1-measure. Moreover, on the set Tn(),the measures 𝜔1,𝜔2,and1 are mutually absolutely continuous. Given any Borel set 𝐸⊂with 𝜔1(𝐸) ⋅𝜔2(𝐸) > 0, the restrictions 𝜔1|𝐸and 𝜔2|𝐸are mutually singular on 𝐸if and only if 1 (Tn()∩𝐸)=0. Proof. The Borel measurability of Tn()follows from [16, Chapter II, Theorem 4.2] which connects the tangents of and the angular derivatives of any given Riemann map 𝜙′ 1∶𝑍 1→Ω 1, where 𝜕Ω1=. The fact that Tn()has 𝜎-finite Hausdorff 1-measure follows from [16, Chapter VI, Theorem 4.2]. Theorem 6.3 of [16, Chapter VI] states that if a Borel set 𝐸⊂is such that 𝜔1(𝐸) ⋅𝜔2(𝐸) > 0, then 𝜔1|𝐸and 𝜔2|𝐸are mutually singular on 𝐸if and only if 1 (Tn()∩𝐸)=0. The fact that on the set Tn()the measures 𝜔1,𝜔2,and1 are mutually absolutely continuous follows from [16, Chapter VI, Theorem 4.2 and the following discussion on p. 211]. □ Lemma 4.5. The measures 𝜒˜ 𝜋∗1 ˜ 𝑍,𝜒Tn()𝜔1,𝜒Tn()𝜔2and 𝜒Tn()1 are mutually absolutely continuous.
16 IKONEN More precisely, a given Borel set 𝐵⊂Tn()has positive 1D Hausdorff measure if and only if 1 ˜ 𝑍(˜ 𝜋(𝐵)) > 0.Furthermore,if𝐵⊂⧵Tn(),then1 ˜ 𝑍(˜ 𝜋(𝐵)) = 0. Proof. We write g∗1 𝕊1=𝑣 g1 𝕊1+2𝜋⋅𝜇⟂with 1 𝕊1and 𝜇⟂being mutually singular. We recall from Proposition 3.6 that for every Borel set 𝐵⊂, 1 ˜ 𝑍(˜ 𝜋(𝐵)) = ∫𝜙−1 1(𝐵) min {1, 𝑣g}𝑑1 𝕊1.(4.3) We denote ℎ=𝑣 g◦𝜙−1 1and observe the equality 𝜔2=ℎ𝜔 1+(𝜙 1)∗𝜇⟂, where (𝜙1)∗𝜇⟂(𝐸) = 𝜇⟂(𝜙−1 1(𝐸)) for each 𝐸⊂𝕊2. Then Equation (4.3) is equivalent to (2𝜋)−11 ˜ 𝑍(˜ 𝜋(𝐵)) = ∫𝐵 min {1, ℎ}𝑑𝜔1.(4.4) Lemma 4.4 implies that the measures 𝜒⧵Tn()𝜔1and 𝜒⧵Tn()𝜔2are mutually singular. Consequently, ℎ=0𝜔 1-almost everywhere in ⧵Tn(). In particular, if 𝐵=⧵Tn(), the left-hand side equals zero in Equation (4.4). Lemma 4.4 yields that the measures 𝜒Tn()𝜔1,𝜒Tn()𝜔2and 𝜒Tn()1 are mutually absolutely continuous. Hence ∞>ℎ>0𝜔 1-almost everywhere in Tn(). This implies that the measure in Equation (4.4) is mutually absolutely continuous with the measures 𝜒Tn()𝜔1,𝜒Tn()𝜔2and 𝜒Tn()1 . The claim follows from the equalities (4.2)for𝛼=1.□ Proof of Proposition 4.1. Consider a subarc 𝐼⊂.Proposition3.6 implies that ˜ 𝜋(𝐼) has zero 1 ˜ 𝑍measure if and only if for every 𝑥,𝑦 ∈ 𝐼,𝑑˜ 𝑍(˜ 𝜋(𝑥), ˜ 𝜋(𝑦)) = 0 ifandonlyif𝑣g=01 𝕊1-almost everywhere on 𝜙−1 1(𝐼). Equivalently, 𝜔1|𝐼and 𝜔2|𝐼are mutually singular. Lemma 3.3 shows that ˜ 𝑍≠(𝑍, 𝑑𝑍)if and only if there exists a closed arc 𝐼⊂𝕊1such that 𝑦= ˜ 𝜄1(𝐼). Assume that such an 𝐼exists. Having fixed 𝑥0∈𝑍 1and 0<𝑠<𝜎(𝑥 0,𝕊1), there exists 𝑐= 𝑐(𝑥0,𝐼,𝑠)for which mod Γ(𝐼, 𝐵𝕊2(𝑥0,𝑠);𝐼∪𝑍 1)⩾𝑐>0; a positive lower bound can be shown, for example, by estimating the modulus of all geodesics joining 𝐼to 𝐵𝕊2(𝑥0,𝑠)in 𝐼∪𝑍 1. When 𝑅>0is small enough, for every 𝑅>𝑟>0and every path in Γ(𝐼, 𝐵𝕊2(𝑥0, 𝑠)); 𝐼 ∪ 𝑍1),we find a subpath 𝛾′∶ [0, 1] → 𝑍1so that ˜ 𝜄1◦𝛾′joins 𝐵˜ 𝑍(𝑦, 𝑟) to ˜ 𝑍⧵𝐵 ˜ 𝑍(𝑦, 𝑅) within 𝐵˜ 𝑍(𝑦, 𝑅).Since ˜ 𝜄1is a local isometry off the seam, this implies lim inf 𝑟→0+mod Γ(𝐵˜ 𝑍(𝑦, 𝑟), ˜ 𝑍⧵𝐵 ˜ 𝑍(𝑦, 𝑅); 𝐵˜ 𝑍(𝑦, 𝑅))⩾𝑐. Recalling Theorem 2.6, we see that ˜ 𝑍is not quasiconformally equivalent to 𝕊2.□ Lemma 4.6. For 𝑖=1,2,let𝜌𝑖∶Ω 𝑖→ [0, ∞] denote the operator norm of the differential of 𝐷(𝜙−1 𝑖). Then 𝐺=𝜒 Ω1𝜌1+𝜒 Ω2𝜌2+∞⋅𝜒Tn()∈𝐿 2(𝕊2)(4.5) is a weak upper gradient of ˜ 𝜋.
METRIC SPHERES FROM GLUING HEMISPHERES 17 Proof. The 𝐿2-integrability of 𝐺follows from the change of variables formulas of the Riemann maps 𝜙1and 𝜙2and the fact that Tn()has negligible area. Hence, as a consequence of Lemma 2.1, 𝐺is integrable along almost every absolutely continuous path 𝛾∶ [0,1]→𝕊2. Given such a 𝛾,we claim that 𝑑˜ 𝑍(˜ 𝜋(𝛾(0)), ˜ 𝜋(𝛾(1))) ⩽∫𝛾 𝐺𝑑𝑠, (4.6) implying that 𝐺is a weak upper gradient of ˜ 𝜋. Since 𝐺is integrable along 𝛾,𝛾has negligible length in Tn(). Then Equation (2.3)implies 1 𝕊2(Tn()∩|𝛾|)=0. We conclude 1 ˜ 𝑍(˜ 𝜋()∩|˜ 𝜋◦𝛾|)=0from Lemma 4.5. The assumptions of Lemma 2.2 are satisfied and the conclusion 𝓁(˜ 𝜋◦𝛾) ⩽∫𝛾𝐺𝑑𝑠follows. The inequality (4.6)isa consequence. □ We define the Jacobian of ˜ 𝜋to be the density of ˜ 𝜋∗2 ˜ 𝑍, defined in Equation (4.2), with respect to 2 𝕊2. Lemma 4.7. The mapping ˜ 𝜋satisfies Lusin’s condition (𝑁) and the Jacobian 𝐽˜ 𝜋coincides with 𝐺2 2 𝕊2-almost everywhere, with 𝐺being from Equation (4.5). Proof. The Lusin’s condition (𝑁)of˜ 𝜋follows from the fact that ˜ 𝜋()has negligible 2 ˜ 𝑍-measure, the fact that 𝜄𝑖∶𝑍 𝑖→˜ 𝑍𝑖is a local isometry, and as 𝜙−1 𝑖∶Ω 𝑖→𝑍 𝑖satisfies condition (𝑁). Here 𝐽˜ 𝜋=02 𝕊2-almost everywhere on , so the equality 𝐽˜ 𝜋=𝐺 2follows from the fact that 𝜙1and 𝜙2 are Riemann maps. □ Proof of Proposition 4.3. The claimed topological properties of ˜ 𝜋were already verified at the beginning of this section. Lemmas 4.6 and 4.7 prove that 𝐽˜ 𝜋=𝐺 2∈𝐿 1(𝕊2)with 𝐺being a weak upper gradient of ˜ 𝜋. This fact and the fact that the multiplicity of ˜ 𝜋is negligible for ˜ 𝜋∗2 ˜ 𝑍imply mod Γ ⩽mod ˜ 𝜋Γ for all path families Γ. Lastly, we argue that a 𝐾-quasiconformal map 𝜓∶ ˜ 𝑍→𝕊2exists (for some 𝐾⩾1)ifandonly if ˜ 𝜋is a 1-quasiconformal homeomorphism. The “if”-direction is obvious. In the “only if”-direction, the fact that ˜ 𝜋is a homeomorphism follows from Proposition 4.1.So ℎ=𝜓◦˜ 𝜋∶ 𝕊2→𝕊2is a homeomorphism satisfying mod Γ ⩽𝐾modℎΓfor all path families Γ. Theorem 2.4 and [2, Definition 3.1.1 and Theorem 3.7.7] prove that ℎis 𝐾-quasiconformal. Consequently, ˜ 𝜋is 𝐾′-quasiconformal for some 𝐾′⩽𝐾2. This self-improves to 𝐾′=1due to Lemma 4.8 below. This yields mod ˜ 𝜋Γ = mod Γ for all path families. □ Lemma 4.8. Suppose that ˜ 𝜋∶ 𝕊2→˜ 𝑍from Equation (4.1) is a homeomorphism. Then ˜ 𝜋∶ 𝕊2→˜ 𝑍 is 1-quasiconformal if and only if for every 1-Lipschitz ℎ∶ 𝕊2→ℝ,ℎ◦˜ 𝜋−1 ∈𝑁 1,2(˜ 𝑍). Proof. The “only if”-claim is clear, given Theorem 2.4 (ii). In the “if”-direction, fix a 1-Lipschitz ℎ∶ 𝕊2→ℝfor now. Consider the Borel function 𝐺∶ 𝕊2→ [0, ∞] defined on Lemma 4.6. Then 𝜌=1∕𝐺◦˜ 𝜋−1 is such that 𝜌2is the Jacobian of ˜ 𝜋−1, as a consequence of Lemma 4.7. Hence 𝜌∈𝐿 2(˜ 𝑍). Given that ℎ◦˜ 𝜋−1 ∈𝑁 1,2(˜ 𝑍) and 2 ˜ 𝑍(𝑄(𝑆𝑍)) = 0, for almost every 𝛾∶ [0,1]→ ˜ 𝑍, the composition (ℎ ◦˜ 𝜋−1)◦𝛾is absolutely continuous, 𝛾has negligible length on the seam 𝑄(𝑆𝑍),and
18 IKONEN ∫𝛾𝜌𝑑𝑠<∞. Indeed, the absolute continuity of (ℎ ◦˜ 𝜋−1)◦𝛾for almost every path follows from [19, Proposition 6.3.2]. The fact that almost every path has negligible length on 𝑄(𝑆𝑍)follows from Lemma 2.1 and the 𝐿2-integrability of ∞⋅𝜒𝑄(𝑆𝑍). Similarly, the conclusion ∫𝛾𝜌𝑑𝑠<∞follows from Lemma 2.1 and the 𝐿2-integrability of 𝜌. If we denote 𝐸=(ℎ◦˜ 𝜋−1)(|𝛾|∩𝑄(𝑆 𝑍)), the absolute continuity of (ℎ ◦˜ 𝜋−1)◦𝛾implies 1 ℝ(𝐸) = 0. Then Lemma 2.2 yields 𝓁((ℎ ◦˜ 𝜋−1)◦𝛾) ⩽∫𝛾𝜌𝑑𝑠. We conclude that 𝜌is a weak upper gradient of ℎ◦˜ 𝜋−1. Since 𝜌is independent of ℎand ℎis an arbitrary 1-Lipschitz function, Theorem 7.1.20 [19]shows that 𝜌is a weak upper gradient of ˜ 𝜋−1.Since𝜌2is the Jacobian of ˜ 𝜋−1, we conclude 𝐾𝑂(˜ 𝜋−1)=1. Recall 𝐾𝑂(˜ 𝜋) = 1 from Proposition 4.3.□ Remark 4.9. If the welding curve happens to be rectifiable, the Hausdorff 1-measure on and 𝜒Tn()1 are mutually absolutely continuous [16, Chapter VI, Theorem 1.2 (F. and M. Riesz)]. With this fact at hand, Lemma 4.5 implies that ˜ 𝜋is a homeomorphism. Moreover, one can show that ℎ◦˜ 𝜋−1 ∈𝑁 1,2(˜ 𝑍) for every 1-Lipschitz ℎ∶ 𝕊2→ℝ. Hence ˜ 𝜋is 1-quasiconformal by Lemma 4.8. Proof of Theorem 1.6. Suppose the existence of a quasiconformal homeomorphism 𝜓∶ ˜ 𝑍→𝕊2. Up to postcomposing 𝜓by an orientation-reversing Möbius transformation of 𝕊2, we may assume that ˜ 𝜙𝑖∶=𝜓◦˜ 𝜄𝑖|𝑍𝑖∶𝑍 𝑖→𝕊2is orientation-preserving for 𝑖=1,2.Let= 𝜓(𝑄(𝑆𝑍)). The set 𝕊2⧵is the disjoint union of Jordan domains Ω1and Ω2, where Ω𝑖is the image of ˜ 𝜙𝑖 for 𝑖=1,2. Next, since 𝜓∶ ˜ 𝑍→𝕊2is a quasiconformal homeomorphism, 𝜓satisfies Lusin’s Condition (𝑁) [31, Section 17]. Consequently, has zero 2D Hausdorff measure. We consider the Beltrami differential 𝜇=𝜒 Ω1𝜇1+𝜒 Ω2𝜇2, where 𝜇𝑖is the Beltrami differential of ˜ 𝜙−1 𝑖.Ifℎ∶ 𝕊2→𝕊2is a normalized solution to the Beltrami equation induced by 𝜇[2, Measurable Riemann mapping theorem], the mapping ˜ 𝜓=ℎ◦𝜓is 1-quasiconformal. Since has zero measure, this is readily verified by hand or by applying [22, Theorem 4.12]. We have verified that (𝑍, 𝑑𝑍)=˜ 𝑍andwemayassumethat𝜓∶ (𝑍,𝑑𝑍)→𝕊2is 1-quasiconformal with 𝜙𝑖=𝜓◦˜ 𝜄𝑖|𝑍𝑖being Riemann maps [2, Weyl’s lemma]. Proposition 4.1 implies (𝑍, 𝑑𝑍)=˜ 𝑍. The definition of 𝑍implies that g=𝜙 −1 2◦𝜙1|𝕊1. Consequently, gis a welding homeomorphism. In order to show the removability of ∶=𝜓(𝑆 𝑍), we are given an orientation-preserving homeomorphism 𝑀∶ 𝕊2→𝕊2conformal in the complement of . Then 𝜋′∶=𝜓 −1 ◦𝑀−1 defines a mapping as in Equation (4.1) for the curve ′=𝑀().Proposition4.3 implies that 𝜋′is 1-quasiconformal. Consequently, 𝑀−1 =𝜓◦𝜋′is 1-quasiconformal, that is, a Möbius transformation. □ 5MASS UPPER BOUND In this section, we prove Theorems 1.1 and 1.2. We first consider the implication “(3) ⇒(1).” Recall that we are given an orientation-preserving homeomorphism g∶𝕊1→𝕊1and the canonical quotient map 𝑄∶ 𝑍 → ˜ 𝑍. We are assuming the existence of a constant 𝐶>0for which lim inf 𝑟→0+ 2 ˜ 𝑍(𝐵˜ 𝑍(𝑦, 𝑟)) 𝜋𝑟2⩽𝐶for every 𝑦∈𝑄(𝑆 𝑍). (5.1)
METRIC SPHERES FROM GLUING HEMISPHERES 19 In order to make transparent how the Lipschitz constant of g(respectively, g−1) is related to 𝐶 in Equation (5.1), we define 𝐶1,𝐶 2⩾0to be the smallest constants for which lim inf 𝑟→0+ 2 ˜ 𝑍(˜ 𝜄1(𝑍1)∩𝐵˜ 𝑍(𝑦, 𝑟)) 𝜋𝑟2⩽𝐶1for every 𝑦∈𝑄(𝑆 𝑍)(5.2) lim inf 𝑟→0+ 2 ˜ 𝑍(˜ 𝜄2(𝑍2)∩𝐵˜ 𝑍(𝑦, 𝑟)) 𝜋𝑟2⩽𝐶2for every 𝑦∈𝑄(𝑆 𝑍). (5.3) Recalling from Lemma 3.3 the fact that the inclusion maps are 1-Lipschitz and local isometries outside the seam, the limit infimum in Equations (5.2)and(5.3) are bounded from below by 1∕2. Since the seam is negligible, we also have 𝐶1,𝐶 2⩾1∕2 and max{𝐶1,𝐶 2}⩽𝐶−1∕2. We show that the constant 𝐶1in Equation (5.2) and the Lipschitz constant 𝐿1of g−1 are connected via the following function 𝑓(𝜖) ∶=(sin |(0,𝜋∕2])−1(𝜖) 𝜋+√1−𝜖 2 𝜋𝜖 for 0<𝜖⩽1. (5.4) Definition 5.1. For every 𝐶⩾1∕2,𝐿=𝐿(𝐶)⩾1denotes the unique positive number such that for every 0<𝜖⩽𝐿−1,𝑓(𝜖) ⩾𝐶. Equivalently, 𝐿=1∕𝑓 −1(𝐶). Remark 5.2. We note that for every 0<𝜖⩽1,wehave𝑓(𝜖) ⩾(𝜋𝜖)−1. We use this fact during the proof of Theorem 1.1. Proposition 5.3. If Equation (5.2)holds with constant 𝐶1and 𝐿1=𝐿(𝐶 1)is as in Definition 5.1, then g−1 is 𝐿1-Lipschitz and ˜ 𝜄1∶ 𝑍1→˜ 𝑍satisfies for every 𝑥,𝑦 ∈ 𝑍1,𝜎(𝑥,𝑦) ⩾𝑑𝑍(˜ 𝜄1(𝑥),˜ 𝜄1(𝑦)) ⩾ 𝜎(𝑥,𝑦)∕𝐿1. The symmetry in the argument yields the following result. Proposition 5.4. If Equation (5.3) holds with constant 𝐶2and 𝐿2=𝐿(𝐶 2)is as in Definition 5.1, then gis 𝐿2-Lipschitz and ˜ 𝜄2∶𝑍 2→˜ 𝑍satisfies for every 𝑥,𝑦 ∈ 𝑍2,𝜎(𝑥, 𝑦) ⩾𝑑𝑍(˜ 𝜄2(𝑥),˜ 𝜄2(𝑦)) ⩾ 𝜎(𝑥,𝑦)∕𝐿2. We start the proof of Proposition 5.3. We consider the decomposition g∗1 𝕊1=𝑣 g1 𝕊1+𝜇 ⟂with 𝜇⟂and 1 𝕊1being singular. We fix a Borel representative of 𝑣g.Let𝑓be as in Equation (5.4). The following statement holds for every ˜ 𝑍. Proposition 5.5. Given 1>𝜖>0and a 1 𝕊1-density point 𝑥0∈𝕊1of 𝐸∶={𝑣 g⩽𝜖},wehave 𝑓(𝜖) ⩽lim inf 𝑟→0+ 2 ˜ 𝑍(˜ 𝜄1(𝑍1)∩𝐵˜ 𝑍(𝑥0,𝑟)) 𝜋𝑟2.(5.5) Proof. For the duration of the proof, we fix normal coordinates 𝐹∶ 𝐵(0,𝜋∕2)→𝕊2centred at 𝑥0in such a way that the preimage of 𝕊1∩𝐵(𝑥 0,𝜋∕2)is (−𝜋∕2, 𝜋∕2) × {0} [28, Section 5]. Recall that this means that 𝐹is an isometry along radial geodesics and the metric has the expansion
20 IKONEN g𝑖𝑗(𝑥) = 𝛿𝑖𝑗 +𝑂(‖𝑥‖2 2)in these coordinates. In particular, as 𝑟→0 +, the bi-Lipschitz constant of 𝐹|𝐵(0,𝑟) is of the form 1+𝑂(𝑟 2). We denote Γ(𝑠) ∶=𝐹(𝑠,0)for |𝑠|⩽𝜋∕2. We fix 0<𝜂<1∕𝜖−1.Since𝑥0is a density point of 𝐸, there exists 𝑠0<𝜋∕2such that for every 0<𝑠⩽𝑠0, 1 𝕊1(Γ([−𝑠, 𝑠]) ⧵𝐸 )⩽𝜖𝜂𝑠. (5.6) We fix 0<𝑟⩽𝜖𝑠0. Then, for every 0<𝑠<𝑟∕𝜖,Proposition3.6 yields, for both 𝐼=[0,𝑠]and 𝐼= [−𝑠, 0], 𝓁(𝐸 ∩ (˜ 𝜄1◦Γ|𝐼)) ⩽𝜖𝑠. (5.7) Since˜ 𝜄1is 1-Lipschitz, according to Lemma 3.3,Equations(5.6)and(5.7)imply 𝓁(˜ 𝜄1◦Γ|𝐼)⩽𝑠𝜖+𝜖𝜂𝑠=𝜖(1+𝜂)𝑠<𝑠. (5.8) We denote for every |𝑠|<𝑟∕(𝜖(1+𝜂)),𝜌𝑠∶=𝑟−𝜖(1+𝜂)|𝑠|. For each 𝑧∈𝑍 1∩𝐵 𝕊2(𝐹(𝑠, 0), 𝜌𝑠), the inequality (5.8)implies˜ 𝜄1(𝑧) ∈ 𝐵˜ 𝑍(˜ 𝜄1(𝑥0), 𝑟). We estimate 𝐴𝑟∶=2 ˜ 𝑍(˜ 𝜄1(𝑍1)∩𝐵˜ 𝑍(˜ 𝜄1(𝑥0), 𝑟)) as 𝑟→0 +. In estimating 𝐴𝑟, we use the fact that the seam 𝑄(𝑆𝑍)has negligible 2 ˜ 𝑍-measure and that ˜ 𝜄1is a local isometry outside the seam. We claim that for each 0<𝜃<𝜋∕2the following holds: 𝐴𝑟⩾(1 + 𝑂((𝑟∕𝜖)2))−2(𝜃𝑟2+ cos(𝜃) 𝑟2 (1 + 𝑂((𝑟∕𝜖)2))𝜖(1 + 𝜂)).(5.9) The term (1 + 𝑂((𝑟∕𝜖)2))−2 comes from estimating the Jacobian of ˜ 𝜄1◦𝐹. The first term in the brackets comes from the fact that 𝐹preserves the speed of radial geodesics, so (˜ 𝜄1◦𝐹)({(𝑠, 𝑡)∶ √𝑠2+𝑡 2<𝑟,0<𝑡 })⊂˜ 𝜄1(𝑍1)∩𝐵˜ 𝑍(˜ 𝜄1(𝑥0), 𝑟). We use this inclusion in a circular sector 𝐶𝜃(𝑟) which has a total angle 2𝜃 and an angle bisector {0} × ℝ. The second term in the brackets is twice the area of a suitable triangle. The factor of two comes from the symmetry of the estimate in Equation (5.8) with respect to the parameter 𝑠=0. We consider a triangle 𝑇𝜃(𝑟) ⊂ ℝ2foliated by line segments 𝓁(𝑠), where 0⩽𝑠 < 𝑟∕((1 + 𝜂)𝜖), with 𝓁(𝑠) having the start point (𝑠, 0), tangent in the direction (sin(𝜃), cos(𝜃)), and has length 𝜌𝑎∕(1 + 𝑂((𝑟∕𝜖)2)).The˜ 𝜄1◦𝐹image of such a triangle 𝑇𝜃(𝑟) contributes to 𝐴𝑟. The inequality (5.9) follows. We choose the angle 𝜃to satisfy sin(𝜃)=𝜖(1+𝜂). We divide Equation (5.9)by𝜋𝑟2, pass to the limit 𝑟→0 +, and then to 𝜂→0 +, and conclude lim inf 𝑟→0+ 𝐴𝑟 𝜋𝑟2⩾(sin |(0,𝜋∕2])−1(𝜖) 𝜋+√1−𝜖 2 𝜋𝜖 = 𝑓(𝜖). (5.10) The inequality (5.5) is the same as Equation (5.10). □
METRIC SPHERES FROM GLUING HEMISPHERES 21 Remark 5.6. Given 0<𝜖<1, the lower bound in Equation (5.10) is sharp. This can be shown by considering a bi-Lipschitz g∶𝕊1→𝕊1with metric speed 𝑣g≡𝜖everywhere in an open neighbourhood of 𝑥0∈𝕊1. If the circular sector 𝐶𝜃(𝑟) and triangle 𝑇𝜃(𝑟) are defined as in the proof of the lower bound Equation (5.10), with 𝜂=0,and𝜃=(sin|(0,𝜋∕2])−1(𝜖),wehave lim inf 𝑟→0+ 𝐴𝑟 𝜋𝑟2=2 ℝ2(𝐶𝜃(1)) + 22 ℝ2(𝑇𝜃(1)) 𝜋= 𝑓(𝜖). This can be showed using Lemma 3.2 and Proposition 3.6. The key property of the angle 𝜃is that the line on ℝ2containing (𝑟∕𝜖, 0) with tangent vector (− cos(𝜃), sin(𝜃)) intersects every ball 𝐵ℝ2((𝑠, 0), 𝜌𝑠)tangentially when 0⩽𝑠<1∕𝜖and 𝜌𝑠=1−𝜖𝑠. Proof of Proposition 5.3. Given Equation (5.2) and Proposition 5.5,wehave𝑣g(𝑥) ⩾𝐿−1 1for 1 𝕊1almost every 𝑥∈𝕊1. This implies that g−1 is absolutely continuous and 𝑣g−1 (𝑥) ⩽𝐿1for 1 𝕊1almost every 𝑥∈𝕊1. Therefore g−1 is 𝐿1-Lipschitz. Thefactthat˜ 𝜄11-Lipschitz follows from Lemma 3.3.Proposition3.6 implies that 𝑑𝑍(˜ 𝜄1(𝑥),˜ 𝜄1(𝑦)) ⩾𝜎(𝑥,𝑦)∕𝐿1,for every 𝑥,𝑦 ∈ 𝕊1. Given this inequality, the equality (3.1) in Lemma 3.2 implies the corresponding inequality for every pair 𝑥,𝑦 ∈ 𝑍1. Hence˜ 𝜄−1 1is 𝐿1-Lipschitz. □ Next, we verify a lemma about radial extensions of bi-Lipschitz maps, which we need during the proof of Theorem 1.1. For the south pole 𝑃1∈𝑍 1, we consider the stereographic projection 𝑃∶ 𝕊2⧵{𝑃 1}→ℝ2×{0} fixing the equator and mapping the north pole 𝑃2= (0, 0, 1) to the origin. We identify ℝ2×{0} with ℝ2. We note that 𝑃−1 has the explicit definition 𝑃−1(𝑥, 𝑦) = (2𝑥 1+𝑥 2+𝑦 2,2𝑦 1+𝑥 2+𝑦 2,1−𝑥 2−𝑦 2 1+𝑥 2+𝑦 2). The Riemannian tensor of 𝕊2in these coordinates is 𝐼=(4∕(1+𝑟 2)2)gE, where 𝑟is the distance to the origin and g𝐸the Euclidean inner product. In polar coordinates, gE=𝑑𝑟 2+𝑟 2𝑑𝜃2.Wesee from the form of 𝐼that the bi-Lipschitz constants of ˜g=𝑃◦g◦(𝑃|𝕊1)−1 and g∶𝕊1→𝕊1coincide. We represent the polar coordinates using the complex notation 𝑟𝑒𝑖𝜃. We note that there exists a homeomorphism ˜ 𝐺∶ ℝ→ℝwith ˜g(𝑒𝑖𝜃)=𝑒 𝑖˜ 𝐺(𝜃) for every 𝜃∈ℝ. For every 0⩽𝑟⩽1and 𝜃∈ℝ, we set ˜ 𝜓(𝑟𝑒𝑖𝜃)∶=𝑟𝑒 𝑖˜ 𝐺(𝜃) and refer to ˜ 𝜓as the radial extension of ˜g. We recall from [24, Theorem 2.2] that the bi-Lipschitz constants of ˜gand ˜ 𝜓coincide. Let 𝜓=𝑃 −1 ◦˜ 𝜓◦𝑃|𝑍2∶𝑍 2→𝑍 2. We use the following fact during the proof of Lemma 5.8; see for example [11, 13]. Lemma 5.7. For every 𝑥,𝑦 ∈ 𝕊2,0<𝜖<1,and0<4𝑟<𝜎(𝑥,𝑦), the modulus of the family of paths joining 𝐵𝕊2(𝑥, 𝑟) to 𝐵𝕊2(𝑦, 𝑟) with length (1 + 𝜖)𝜎(𝑥, 𝑦) is positive. Lemma 5.8. The map 𝜓∶ 𝑍2→𝑍 2is 𝐿-bi-Lipschitz if gis 𝐿-bi-Lipschitz. Proof. We refer the interested reader to [24, Section 2] for the proof of the fact that ˜ 𝜓is bi-Lipschitz if ˜g(equivalently g) is bi-Lipschitz. We take this as given.
22 IKONEN Since ˜ 𝜓is bi-Lipschitz, it has a differential at 2-almost every point in 𝔻. Given this fact, the following computations are understood to hold at 2-almost every (𝑥, 𝑦) = 𝑟𝑒𝑖𝜃 in the unit disk. The pullback ˜ 𝜓∗𝐼is a diagonal matrix with respect to the basis (𝑑𝑟, 𝑑𝜃), with diagonal 4∕(1 + 𝑟2)2and 4|˜ 𝐺′(𝜃)|2𝑟2∕(1 + 𝑟2)2. Hence the maximum of the operator norms of 𝐷˜ 𝜓∶ (𝑇𝔻,𝐼)→ (𝑇𝔻,𝐼)and its inverse is equal to 𝐿(𝑟𝑒𝑖𝜃)=max{|˜ 𝐺′(𝜃)|,|˜ 𝐺′(𝜃)|−1}. Then, if 𝐿′denotes the essential supremum of 𝐿(𝑟𝑒𝑖𝜃), Lemma 5.7 implies that 𝜓is 𝐿′-bi-Lipschitz. On the other hand, 𝐿′is the bi-Lipschitz constant of g.□ Proof of Theorem 1.1. We first claim that “(1) ⇒(2).” Lemma 5.8 provides us with an 𝐿-bi-Lipschitz 𝜓∶ 𝑍2→𝑍 2extension of the given 𝐿-bi-Lipschitz g. We define 𝐻(𝑥) =˜ 𝜄1(𝑥) for each 𝑥∈𝑍1 and 𝐻(𝑥) =˜ 𝜄2◦𝜓(𝑥) otherwise. Proposition 3.6 implies that 𝐻is 𝐿-bi-Lipschitz at the seam, and Lemma 3.2 implies that 𝐻is 𝐿-bi-Lipschitz everywhere. Notice that if 𝐻∶ 𝕊2→˜ 𝑍is 𝐿′-bi-Lipschitz, we may choose 𝐶=(𝐿 ′)4as an upper bound for the 2D Hausdorff lower density. Hence “(2) ⇒(3)” follows, quantitatively. Lastly, “(3) ⇒(1)” follows from Propositions 5.3 and 5.4. In fact, given 𝐶⩾1for which the lower density bound of Equation (5.1) holds, gis 𝐿′-bi-Lipschitz for 𝐿′solving 𝐶=𝑓(1∕𝐿 ′).Since𝑓(𝜖) ⩾1∕𝜋𝜖 for every 0<𝜖⩽1,wehave𝐶𝜋 ⩾𝐿′. Hence gis 𝐶𝜋-bi-Lipschitz. □ Remark 5.9. The estimates between the constants in “(3) ⇒ (1)” in Theorem 1.1 canbeimprovedin two ways. First, the constants 𝐶1and 𝐶2in Equations (5.2)and(5.3)satisfymax{𝐶1,𝐶 2}⩽𝐶−1∕2, so gis (𝐶 − 1∕2)𝜋-bi-Lipschitz. The second improvement isobtained by using the constant 𝐿′=𝐿(𝐶−1∕2)from Definition 5.1. Then gis 𝐿′-bi-Lipschitz, where 𝐿′⩽(𝐶 − 1∕2)𝜋. These improvements imply that the bi-Lipschitz constant of gconverges to 1 as 𝐶→1 +. These facts also improve Theorem 1.2 and the following result, Proposition 5.10. Before proving Theorem 1.2, we investigate a related problem. To this end, suppose that we are given Riemann maps 𝜙𝑖∶𝑍 𝑖→Ω 𝑖with Ω1and Ω2denoting the complementary components of a welding curve ,andsetg=𝜙 −1 2◦𝜙1|𝕊1. Proposition 5.10. Let 𝐾,𝐶 ⩾1. The welding homeomorphism gis 𝜋(𝐾𝐶)2-bi-Lipschitz if there exists a 𝐾-quasiconformal homeomorphism ℎ∶ 𝕊2→𝕊2such that for both 𝑖=1,2, 𝐶−1𝐽ℎ(𝑥) ⩽𝐽𝜙−1 𝑖(𝑥) ⩽𝐶𝐽ℎ(𝑥) for 2 𝕊2-almost everywhere, 𝑥∈Ω 𝑖.(5.11) Conversely, if gis 𝐿-bi-Lipschitz, then there exists 𝐿4-quasiconformal homeomorphism ℎ∶ 𝕊2→𝕊2 such that Equation (5.11) holds for 𝐶=𝐿 2. Proof. We first assume that g∶𝕊1→𝕊1is 𝐿-bi-Lipschitz. Then Theorem 1.1 provides us with an 𝐿bi-Lipschitz homeomorphism Ψ∶ ˜ 𝑍→𝕊2.Proposition4.3 and Equation (4.1) imply that ˜ 𝜋∶ 𝕊2→ ˜ 𝑍defined via the formula ˜ 𝜋(𝑥) = {˜ 𝜄1◦𝜙−1 1(𝑥), 𝑥 ∈ Ω1, ˜ 𝜄2◦𝜙−1 2(𝑥), 𝑥 ∈ Ω2 (5.12)
METRIC SPHERES FROM GLUING HEMISPHERES 23 is a 1-quasiconformal homeomorphism. Therefore, ℎ∶=Ψ◦˜ 𝜋∶ 𝕊2→𝕊2is 𝐾-quasiconformal for 𝐾=𝐿 4,andasΨis 𝐿-bi-Lipschitz, the Jacobians of ℎand ˜ 𝜋are comparable with comparison constant 𝐶=𝐿 2. Next,wearegivenaJordancurve⊂𝕊2corresponding to a welding homeomorphism g= 𝜙−1 2◦𝜙1|𝕊1,a𝐾-quasiconformal homeomorphism ℎ∶ 𝕊2→𝕊2, and a constant 𝐶⩾1such that 𝐶−1𝐽ℎ(𝑥) ⩽𝐽˜ 𝜋(𝑥) ⩽𝐶𝐽ℎ(𝑥) 2 𝕊2-almost everywhere, 𝑥∈𝕊2⧵.(5.13) For 𝑖=1,2, the composition ℎ◦𝜙𝑖is 𝐾-quasiconformal with Jacobian bounded from above 𝐶 and below by 𝐶−1, respectively; here we apply Equation (5.13). Theorem 2.4 (ii) and Hadamard’s inequality imply that 𝐶−1 ⩽𝜌2 ℎ◦𝜙𝑖⩽𝐾𝐶 2 𝕊2-almost everywhere in 𝑍𝑖. Lemma 5.7 implies that the homeomorphism ℎ◦𝜙𝑖is locally 𝐿′-bi-Lipschitz for 𝐿′=√𝐾𝐶. Since, for both 𝑖=1,2,𝑍𝑖is geodesic, it is immediate that ℎ◦𝜙𝑖∶ 𝑍𝑖→𝕊2is 𝐿′-Lipschitz. Since this holds for both 𝑖=1,2, the construction of 𝑑𝑍implies that whenever 𝑥,𝑦 ∈ 𝕊1, 𝜎(ℎ ◦𝜙1(𝑥), ℎ ◦𝜙1(𝑦)) ⩽𝐿′𝑑˜ 𝑍(˜ 𝜄1(𝑥),˜ 𝜄1(𝑦)). Lemma 3.2 (3.1) establishes the same inequality for each 𝑥,𝑦 ∈ 𝑍1. Hence the mapping ˜ 𝜋defined by the expression (5.12) is a homeomorphism and Ψ∶=ℎ◦˜ 𝜋−1 is 𝐿′-Lipschitz on the southern hemisphere. A similar argument shows that Ψis 𝐿′-Lipschitz on both of the hemispheres. Then Lemma 3.2 (3.2) implies that Ψis 𝐿′-Lipschitz everywhere. Since mod Γ ⩽𝐾 mod Ψ−1Γfor all path families (recall Proposition 4.3), we have Ψ−1 ∈ 𝑁1,2(𝕊2,˜ 𝑍). On the other hand, Ψ(𝑄(𝑆𝑍)) has negligible 2 𝕊2-measure and Ψ−1 is locally 𝐿′-Lipschitz in the complement of that set. In particular, almost every absolutely continuous 𝛾∶ [0,1] →𝕊2has zero length in Ψ(𝑄(𝑆𝑍)) and Ψ−1 ◦𝛾is absolutely continuous. As a consequence, 1 ˜ 𝑍(𝑄(𝑆𝑍)∩|Ψ−1 ◦𝛾|)=0. Denoting 𝐸=𝑄(𝑆 𝑍)∩|Ψ−1 ◦𝛾|and 𝜌=𝐿 ′𝜒𝕊2, we conclude from Lemma 2.2 that 𝓁(Ψ−1 ◦𝛾) ⩽ ∫𝛾𝜌𝑑𝑠⩽𝐿′𝓁(𝛾). Lemma 5.7 implies that Ψ−1 is 𝐿′-Lipschitz. We have verified that Ψis 𝐿′-bi-Lipschitz. By applying the implications “(2) ⇒(3) ⇒(1)” in Theorem 1.1, we conclude that gis 𝐿-bi-Lipschitz for 𝐿=𝜋(𝐿 ′)4=𝜋(𝐾𝐶) 2.□ Next, we prove Theorem 1.2. This essentially follows from Proposition 5.10. Proof of Theorem 1.2. We claim that g∶𝕊1→𝕊1is bi-Lipschitz if and only if there exists a quasiconformal homeomorphism ℎ∶ 𝕊2→𝕊2and a 1-quasiconformal homeomorphism 𝜑∶ 𝕊2→˜ 𝑍 such that 𝐽𝜑and 𝐽ℎare comparable. If such 𝜑and ℎexist, we may assume that 𝜙𝑖=𝜑 −1 ◦˜ 𝜄𝑖|𝑍𝑖is a Riemann map for both 𝑖=1,2. Then Proposition 5.10 shows that gis bi-Lipschitz. Conversely, if gis bi-Lipschitz, Theorem 1.1 provides a bi-Lipschitz homeomorphism Ψ∶ ˜ 𝑍→ 𝕊2. Then Theorem 1.6 implies the existence of a 1-quasiconformal homeomorphism 𝜋∶ 𝕊2→˜ 𝑍 such that 𝜙𝑖=𝜋 −1 ◦˜ 𝜄𝑖|𝑍𝑖is a Riemann map for 𝑖=1,2. We may also assume that Ψ◦˜ 𝜄𝑖|𝑍𝑖is orientation-preserving for 𝑖=1,2, by post-composing Ψwith a suitable reflection, if need be. Defining ℎ=Ψ◦𝜋implies that the assumptions of Proposition 5.10 hold for g. Since Theorem 1.1 and Proposition 5.10 are quantitative, so is Theorem 1.2.□
24 IKONEN 6 MAPPINGS OF FINITE DISTORTION In this section, we establish Proposition 1.4 and Theorem 1.5. Definition 6.1. Let Ω, Ω′⊂𝕊2be open. A homeomorphism 𝜓∶ Ω→Ω ′is a mapping of finite distortion if 𝜓∈𝑁 1,1(Ω, 𝕊2); second, the determinant 𝐽(𝐷𝜓) of the differential 𝐷𝜓 is non-negative and integrable; lastly, there exists a function 1⩽𝐾′ 𝜓<∞for which |𝐷𝜓|2 g⩽𝐾′ 𝜓𝐽(𝐷𝜓) 2 𝕊2-almost everywhere in Ω. (6.1) Here |𝐷𝜓|grefers to the operator norm of the differential 𝐷𝜓.Welet𝐾𝜓denote a smallest Borel function which is bounded from below by 𝜒Ωand for which Equation (6.1) holds. Definition 6.2. A smooth strictly increasing function ∶ [1, ∞) → [0, ∞) is admissible if (1) (1) = 0, (2) ∫∞ 1𝑡−2(𝑡) 𝑑1(𝑡) = ∞,and (3) 𝑡↦𝑡′(𝑡) is increasing for large values 𝑡, and converges to ∞as 𝑡→∞. We obtain the same class of admissible if we replace (2) with the condition ∫∞ 1 𝑡−1′(𝑡) 𝑑1(𝑡) = ∞. This follows from the fact that (𝑠)∕𝑠 ⩽4∫2𝑠 𝑠𝑡−2(𝑡) 𝑑1(𝑡) whenever 𝑠⩾1and the integration by parts formula. Definition 6.3. Let Ω, Ω′⊂𝕊2be open, and 𝜓∶ Ω→Ω ′a homeomorphism. We say that 𝜓is admissible if 𝜓is a mapping of finite distortion and there exists an admissible with ∫Ω 𝑒(𝐾𝜓)𝑑2 𝕊2<∞. (6.2) If (𝑡) = 𝑝𝑡 − 𝑝 for some 𝑝>0, we say that 𝜓has exponentially integrable distortion. We recall some properties of such 𝜓.First,𝜓satisfies Lusin’s condition (𝑁)[26, Theorem 1.1]. Second, 𝜓−1 ∈𝑁 1,2(Ω′,Ω)[27, Corollary 1.2]; this implies that 𝜓−1 satisfies Lusin’s condition (𝑁) [2, Theorem 3.3.7]. Third, the Jacobian 𝐽(𝐷𝜓) appearing on the right-hand side of (6.1) coincides with the Jacobian 𝐽𝜓we defined in Section 2.2 [26]. In this section, we show the following theorem. Theorem 6.4. Suppose that g∶𝕊1→𝕊1is a homeomorphism, g−1 absolutely continuous, and there exists a homeomorphism 𝜓∶ 𝑍2→ 𝑍2extending gwith 𝜓|𝑍2admissible. Then ˜ 𝑍is quasiconformally equivalent to 𝕊2. Note that Theorem 1.5 is a consequence of Theorem 6.4 so it suffices to verify Theorem 6.4.
METRIC SPHERES FROM GLUING HEMISPHERES 31 There exists a unique homeomorphism g∶𝕊1→𝕊1satisfying g◦𝜃=𝜃◦ℎ−1.Weseefrom Equation (7.2) that g−1 is 𝐿-Lipschitz and˜ 𝜄1is 𝐿-bi-Lipschitz with a constant 𝐿depending only on 1(𝐸). In particular, ˜ 𝑍=(𝑍,𝑑 𝑍). We denote 𝐸′=˜ 𝜄2(𝜃(𝐸)) ⊂ ˜ 𝑍, and apply [22, Theorem 1.3] as in Section 7.1, and find a 1-quasiconformal embedding 𝜓∶ ˜ 𝑍⧵𝐸 ′→𝕊2. Consider on ℝ2the distance 𝑑𝐸obtained as follows: For each absolutely continuous 𝛾∶ [0,1]→ ℝ2, denote 𝓁𝐸(𝛾) ∶=∫𝛾𝜒ℝ2⧵𝐸 𝑑𝑠.Weset𝑑𝐸(𝑥, 𝑦) = inf 𝓁𝐸(𝛾), the infimum taken over absolutely continuous paths joining 𝑥to 𝑦. We denote 𝑋=( ℝ2,𝑑 𝐸). The change of distance map 𝐻∶ ℝ2→𝑋is a 1-Lipschitz homeomorphism that is a local isometry on ℝ2⧵𝐸.Moreover,if𝜃∶ [0,1]→ℝ2is absolutely continuous, the metric speeds satisfy 𝑣𝐻◦𝜃=(𝜒ℝ2⧵𝐸 ◦𝜃)⋅𝑣𝜃1-almost everywhere.(7.3) The composition 𝐺= ˜ 𝜄2◦𝜃◦(𝐻|[−1,2]×[0,1])−1 is a 1-quasiconformal homeomorphism. This follows from Lemma 2.2, the equalities 1 ˜ 𝑍(𝐸′)=0=1 𝑋(𝐻(𝐸)), together with Proposition 3.6 and Equation (7.3). We consider a Cantor set 𝐸obtained from [23, Example 6.1]. The key property of 𝐸is the following: there exists a path family Γon [0, 1]2, each path joining (0,0) to (1,0), such that mod 𝐻Γ ⩾(4𝜋)−1 and mod Γ = 0. Given that 𝐺is 1-quasiconformal, the points ˜ 𝜄2(𝜃(𝑥)), where 𝑥 = (0, 0), (1, 0), fail Equation (2.6). Consequently, ˜ 𝑍is not quasiconformally equivalent to 𝕊2, and the embedding 𝜓does not have a quasiconformal extension Ψ∶ ˜ 𝑍→𝕊2. Question 7.1. Are there Cantor sets 𝐸with 1(𝐸) > 0 such that a quasiconformal embedding 𝜓∶ ˜ 𝑍⧵𝐸 ′→𝕊2extends to a quasiconformal homeomorphism Ψ∶ ˜ 𝑍→𝕊2? Given a compact set 𝐹⊂𝑌with 𝑌=ℝ2or 𝑌=𝕊2, we say that 𝐹has zero absolute area if every 1-quasiconformal embedding 𝑓∶ 𝑌⧵𝐹 →𝕊2satisfies 2 𝕊2(𝕊2⧵𝑓(𝑌⧵𝐹))=0. We expect that the quasiconformal extension Ψexists if and only if the set 𝐹=𝕊2⧵𝜓(˜ 𝑍⧵𝐸 ′) has zero absolute area; the “only if”-direction follows by applying the techniques used in Section 4, by noting that the composition (𝑓 ◦𝜓)−1 has a continuous, monotone, and surjective extension ˜ 𝜋with mod Γ ⩽mod ˜ 𝜋Γ for all path families. We expect that the “if”-direction follows from [23, Theorems 1.3 and 1.4, together with Lemma 5.1]. If 𝐸in Question 7.1 has zero absolute area, [23, Theorem 1.3] implies that the change of distance map 𝐻is a 1-quasiconformal homeomorphism. Given that the 𝐺above is 1-quasiconformal, one readily verifies that˜ 𝜄2is a 1-quasiconformal homeomorphism onto its image. We ask the following. Question 7.2. Let 𝐸,g,and𝜓be as in Question 7.1.If˜ 𝜄2∶ 𝑍2→˜ 𝑍is a 1-quasiconformal parametrization of its image, does 𝜓∶ ˜ 𝑍⧵𝐸 ′→𝕊2extend to a quasiconformal homeomorphism Ψ∶ ˜ 𝑍→𝕊2? In particular, if 𝐸has zero absolute area, does 𝐹=𝕊2⧵𝜓(˜ 𝑍⧵𝐸 ′)have zero absolute area? As a related note, it is clear, for example, by [21, Theorem 1.1 and Proposition 1.2], that the inclusion map ˜ 𝜄2is a 1-quasiconformal homeomorphism if and only if there exists a quasiconformal homeomorphism ℎ∶˜ 𝜄2(𝑍2)→𝔻, where 𝔻is the closed Euclidean unit disk.
32 IKONEN 7.3 Welding homeomorphisms We consider a welding homeomorphism g∶𝕊1→𝕊1with welding curve ⊂𝕊2. Consider the monotone mapping ˜ 𝜋∶ 𝕊2→˜ 𝑍obtained from Equation (4.1). Question 7.3. If ˜ 𝜋is a homeomorphism, is it a 1-quasiconformal homeomorphism? We showed in Proposition 4.1 that if ˜ 𝜋is not a homeomorphism, then ˜ 𝑍is not quasiconformally equivalent to 𝕊2; the collapsing creates points of positive capacity—by which we mean that Equation (2.6) fails—in ˜ 𝑍. Question 7.3 asks if the collapsing is the only obstruction for quasiconformal uniformization. Lemma 4.8 reduces the question to understanding when ˜ 𝜋−1 ∈𝑁 1,2(˜ 𝑍,𝕊2). 7.4 Quasisymmetries Observe that the assumptions of Proposition 1.4 are satisfied by every quasisymmetry g∶𝕊1→𝕊1 that is strongly quasisymmetric [3, 5 8, 33]: for every 𝜖>0there exists 𝛿>0such that for every subarc 𝐼⊂𝕊1and Borel set 𝐸⊂𝐼, 1 𝕊1(𝐸) ⩽𝛿1 𝕊1(𝐼) implies 1 𝕊1(g(𝐸)) ⩽𝜖1 𝕊1(g(𝐼)). The welding curves corresponding to strongly quasisymmetric homeomorphisms are special cases of the asymptotically conformal quasicircles; see [30]. One might ask whether or not ˜ 𝑍is quasiconformally equivalent to 𝕊2whenever gis a welding homeomorphism corresponding to such a curve. Corollary 4 of [30] provides us with an example of asymptotically conformal quasicircle which has an uncountable number of tangent points, with the tangent points dense in , but they also have zero 1D Hausdorff measure. Lemma 7.4. There exists a quasisymmetric g∶𝕊1→𝕊1with asymptotically conformal welding curve such that ˜ 𝑍is not homeomorphic to 𝕊2. Lemma 7.4 follows from Proposition 4.1, Lemma 4.5, and the cited example. Question 7.5. Is the answer to Question 7.3 yes if we also assume that g∶𝕊1→𝕊1is a quasisymmetry? To answer Question 7.5 negatively, one needs to construct a quasisymmetry 𝜓∶ 𝑍2→ 𝑍2,with g=𝜓|𝕊1, for which the measures g∗1 𝕊1and 1 𝕊1are not mutually singular in any subarc 𝐼⊂𝕊1, yet the corresponding ˜ 𝑍is not quasiconformally equivalent to ˜ 𝑍. Equivalently, one only needs to show that the homeomorphism 𝐻∶ 𝕊2→˜ 𝑍, coinciding with˜ 𝜄1in 𝑍1and with˜ 𝜄2◦𝜓in 𝑍2,isnot quasiconformal. By arguing as in the proof of Lemma 4.8, one sees that 𝐻is quasiconformal if and only if 𝐻−1 ∈𝑁 1,2(˜ 𝑍,𝕊2). ACKNOWLEDGEMENTS The author was supported by the Academy of Finland, project number 308659 and by the Vilho, Yrjö and Kalle Väisälä Foundation.
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