Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings
Full text
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-ND 4.0 https://creativecommons.org/licenses/by-nd/4.0/ Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings © 2024 the Authors Published version David, Guy C.; Eriksson-Bique, Sylvester David, G. C., & Eriksson-Bique, S. (2024). Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings. Annales de l'Institut Fourier, 74(3), 973-1016. https://doi.org/10.5802/aif.3606 2024
Université Grenoble Alpes ANNALES DE L’INSTITUT FOURIER Guy C. David & Sylvester Eriksson-Biqe Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings Tome 74, no3 (2024), p. 973-1016. https://doi.org/10.5802/aif.3606 Article mis à disposition par ses auteurs selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France http://creativecommons.org/licenses/by-nd/3.0/fr/ C E N T R E MERSENNE Les Annales de l’Institut Fourier sont membres du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org e-ISSN : 1777-5310
Ann. Inst. Fourier, Grenoble 74, 3 (2024) 973-1016 INFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS by Guy C. DAVID & Sylvester ERIKSSON-BIQUE (*) Abstract. — We study metric measure spaces that admit “thick” families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into any Euclidean space unless they admit some “infinitesimal splitting”: their tangent spaces are bi-Lipschitz equivalent to product spaces of the form Z×Rk for some k⩾1. We also provide applications to conformal dimension and give new proofs of some previously known non-embedding results. Résumé. — On étudie des espaces métriques mesurés qui possèdent des familles “épaisses” de courbes rectifiables ou de fragments de courbes, sous la forme de représentations d’Alberti ou de familles de courbes de module strictement positif. On montre que de tels espaces ne possèdent pas de plongement bi-lipschitzien dans un espace euclidien, sauf s’ils admettent une “décomposition infinitésimale”: leurs espaces tangents sont bi-lipschitz équivalents à des produits d’espaces de la forme Z×Rkpour un certain k⩾1. On donne aussi des applications à la dimension conforme et de nouvelles preuves de certains résultats de non plongement déjà connus. 1. Introduction Many natural problems in analysis on metric spaces involve studying spaces that support “thick” families of rectifiable curves (or curve fragments), in one sense or another. In this paper, we show that such spaces Keywords: bi-Lipschitz embedding, modulus, conformal dimension, Alberti representation. 2020 Mathematics Subject Classification: 30L05, 53C23, 49J52. (*) G. C. David was partially supported by the National Science Foundation under Grants No. DMS-1758709 and DMS-2054004. S. Eriksson-Bique was partially supported by the National Science Foundation under Grant No. DMS-1704215. Eriksson-Bique is also thankful for IMPAN for hosting the semester “Geometry and analysis in function and mapping theory on Euclidean and metric measure space” where part of this research was conducted. This work was also partially supported by the grant #346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
974 Guy C. DAVID & Sylvester ERIKSSON-BIQUE cannot be bi-Lipschitz embedded into any Euclidean space unless they admit some “infinitesimal splitting”: their tangent spaces are bi-Lipschitz equivalent to product spaces of the form Z×Rkfor some k⩾1. Our methods are relatively direct and admit a number of consequences. 1.1. Background In 1999, Cheeger [16] proved a deep extension of Rademacher’s theorem (Lipschitz functions are differentiable almost everywhere) to certain abstract metric measure spaces. These are the so-called PI spaces, those that are doubling and support a Poincaré inequality in the sense of [28]. As one of many consequences, he showed that if a PI space admits a bi-Lipschitz embedding into some Euclidean space, then its tangent spaces must be bi-Lipschitz equivalent to Euclidean spaces almost everywhere: Theorem 1.1 (Cheeger [16, Theorem 14.1]). — Let (X, µ)be a PI space. Suppose that Xadmits a bi-Lipschitz embedding into some Rn. Then for µ-almost-every x∈X, there is an integer k⩾1such that every tangent space (Y, y)∈Tan(X, x)is bi-Lipschitz equivalent to Rk. (Here, the notion of “tangent” is in the pointed Gromov–Hausdorff sense; see Section 2.2.) In other words, to admit a bi-Lipschitz embedding into a Euclidean space, a PI space must itself be infinitesimally Euclidean. (Extensions of this are known, see [16, Theorem 14.2] and more recent results in [17, 18, 19, 21, 23, 53].) Since we know of many abstract PI spaces that are not infinitesimally Euclidean, this consequence of Cheeger’s result can be viewed as a generalized non-bi-Lipschitz embedding theorem, i.e., a checkable criterion for a space to admit no bi-Lipschitz embedding into any Euclidean space. A Poincaré inequality is sufficient but not necessary to prove a type of Rademacher theorem in metric spaces, and hence a non-embeddability criterion like Theorem 1.1. Indeed, a number of weaker sufficient conditions implying Cheeger’s Rademacher theorem have been found since its discovery [8, 34, 53]. Most importantly for our purposes, Bate [8], building on work of Alberti [1] and Alberti–Csörnyei–Preiss [2], showed that Cheeger’s differentiable structure is equivalent to the presence of a universal family of Alberti representations. An Alberti representation of a measure is a decomposition into 1-rectifiable measures supported on fragments of rectifiable curves (see Definition 2.6). For example, Fubini’s theorem gives simple Alberti representations of Lebesgue measure on [0,1]2. ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 975 Essentially, Bate shows that if a space supports a “large enough” family of independent Alberti representations, then it supports a Rademacher theorem for Lipschitz functions, from which one can deduce an analog of the non-embeddability criterion Theorem 1.1. Given the above, it is natural to ask whether there are conditions, weaker than any of those studied above, that are not strong enough to yield a Rademacher-type theorem but that still prevent bi-Lipschitz embeddings. In this paper, we answer this question by studying spaces that support a single Alberti representation (or more generally kindependent Alberti representations), but not necessarily enough to form a “universal” family in Bate’s sense, and thus not necessarily enough to yield a differentiable structure for Lipschitz functions. Nonetheless, we show that such smaller families of Alberti representations still strongly constrain the ability of the space to bi-Lipschitz embed into any Euclidean space. The following is our main theorem. Theorem 1.2. — Let X⊆Rnbe a closed set supporting a doubling Radon measure µ0. Suppose that a non-trivial Radon measure µ≪µ0 supports kindependent Alberti representations, for some k⩾1. Then for µ-almost-every x∈X, there is a k-dimensional vector subspace V⊆Rnwith the following property: Every intrinsic tangent Y∈TanRn(X, x)of Xat xis a product Z×V, for some closed set Z⊆V⊥⊆Rn. For a vector subspace V⊂Rn, its orthogonal complement is denoted by V⊥.An “intrinsic tangent” of a subset X⊆Rnis simply a limit of rescalings of Xcentered at a fixed basepoint. For a precise definition, see Section 2.2 below. It is easy to recast Theorem 1.2 as a result that constrains bi-Lipschitz embeddings of metric spaces: Corollary 1.3. — Let Xbe a complete metric space supporting a doubling Radon measure µ0. Suppose that a non-trivial Radon measure µ≪µ0supports kindependent Alberti representations, for some k⩾1. If Xadmits a bi-Lipschitz embedding into some Euclidean space, then for µ-almost-every x∈Xand every tangent (Y, y)∈Tan(X, x),Yis bi- Lipschitz equivalent to a product Z×Rk, for some complete metric space Z. Corollary 1.3 gives a simple non-bi-Lipschitz embedding criterion that applies to a wider class of examples than Theorem 1.1. Theorem 1.2 and Corollary 1.3 are proven without recourse to any more general metric measure Rademacher theorem. Rather, their proofs rely only TOME 74 (2024), FASCICULE 3
976 Guy C. DAVID & Sylvester ERIKSSON-BIQUE on one of the preliminary results of Bate’s paper [8] (that Alberti representations induce “partial derivatives” almost everywhere, see Proposition 2.13) and an adaptation of a principle of Preiss [51] about the structure of the space of tangent objects, Proposition 2.5. Remark 1.4. — We note that there are many other conditions, independent from any of those discussed above, that prevent or constrain bi- Lipschitz embeddability. In particular, the results of [39] and [47] rest also, remarkably, on studying a single family of curves that is “thick” in some quantitative sense (although different than the senses used here). This is part of a larger program to characterize metric spaces embedding into the so-called RNP Banach spaces, and conversely to characterize RNP-Banach spaces by the spaces embedding into them; see [48]. 1.2. Corollaries of the main results We view Corollary 1.3 as a generalized non-embedding result. Since “thick” families of curves arise in many settings, it has a number of specific consequences. 1.2.1. Modulus From our perspective, the most important consequences of Corollary 1.3 involve its relationship with a well-known way of measuring the “thickness” of a family of curves Γin a metric measure space (X, µ). This is the pmodulus of the family (p⩾1), which we denote Modp(Γ, µ). This notion plays a central role in the modern theory of analysis on metric spaces [26, 29]. (We give a precise definition in Section 4.) In Proposition 4.5 and Corollary 4.9 below, we show that path families of positive modulus induce non-trivial Alberti representations. This idea is essentially already contained in work of the second-named author and his collaborators [24] and the proof below reworks that argument in a slightly different context. This is also closely related to the results of [4] and [31]. The formulation in Proposition 4.5 is slightly different and applies more directly to our setting. (To link modulus and Alberti representations, we also use ideas of Keith [33] and Bate [8, Corollary 5.8].) As a consequence of Proposition 4.5 and Corollary 1.3, we obtain the following result for spaces that contain curve families of positive modulus. ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 977 Corollary 1.5. — Let Xbe a complete metric space admitting a Radon measure µthat is absolutely continuous with respect to a doubling measure µ0. Suppose that Xcontains a family Γ⊆Curv(X)of non-constant curves with Modp(Γ, µ)>0for some p∈[1,∞). If Xadmits a bi-Lipschitz embedding into some Euclidean space, then there exists a non-trivial Radon measure µ′≪µ, such that, for µ′-almost every x∈X, every tangent Yof Xat xis bi-Lipschitz to a product Z×R, for some complete metric space Z. For many metric measure spaces, it is known, or not difficult to check, that they contain a path family of positive modulus and do not have tangents that split as Z×R. Therefore such metric spaces cannot bi-Lipschitz embed into any Euclidean space. We give some applications of this argument below. 1.2.2. Conformal dimension An important problem in metric geometry is to understand the conformal dimension of a metric space, a quasisymmetric invariant first introduced by Pansu [49], and much used since [42]. There are a number of variations of this quantity, but we focus on the Ahlfors regular conformal dimension. This is a variant first named by Bonk–Kleiner in [11], where they attribute the idea to Bourdon–Pajot [13]. We recall that a metric space Xis Ahlfors Q-regular if there is a constant C⩾1such that C−1rQ⩽HQ(B(x, r)) ⩽CrQfor all x∈Xand r⩽diam(X), where HQdenotes the Q-dimensional Hausdorff measure. The Ahlfors regular conformal dimension of Xmeasures the infimal dimension Qof all Ahlfors regular quasisymmetric deformations of X: Definition 1.6. — The Ahlfors regular conformal dimension of a metric space Xis (1.1) cdimAR(X) = inf{Q:Yis Ahlfors Q-regular and quasisymmetric to X}. We refer the reader to [26, Chapter 10] for a precise definition of quasisymmetric mappings, and to [11, 42] for more background on the Ahlfors regular conformal dimension, which we now discuss briefly. By definition, the Ahlfors regular conformal dimension and its variations are quasisymmetric invariants. They have thus played an important role in TOME 74 (2024), FASCICULE 3
978 Guy C. DAVID & Sylvester ERIKSSON-BIQUE geometric group theory and quasiconformal geometry, and their properties are connected to many deep questions. We refer to [42] for a book-length account of many of these connections. In particular, it is a difficult problem to understand for which metric spaces the conformal dimension is actually achieved as a minimum. This problem is closely related to an approach to Cannon’s conjecture initiated by Bonk and Kleiner [10, 11]. As an example, it is known that cdimAR(S)is strictly less than the Hausdorff dimension of the standard Sierpiński carpet S[35], but not whether the infimum in (1.1) is achieved by some Ahlfors regular space Ywhen X=S. See [11, 42] for additional details. (Finding the exact value of cdimAR(S)is also a well-known open problem; see, e.g., [38] for recent progress.) Complicating the problem further, even if the conformal dimension of a subset X⊆RNis achieved by a space Yquasisymmetric to X, there is no reason that Yshould be a subset of Euclidean space. Below, we show that this non-embedding phenomenon should be expected quite generally. We need the following important result of Keith and Laakso. Theorem 1.7 (Keith–Laakso [35, Corollary 1.0.2]). — Let Q⩾1and let Xbe a complete, Ahlfors Q-regular metric space. Then cdimAR(X) = Qif and only if there is a weak tangent of Xthat contains a family of non-constant curves with positive p-modulus, for some p⩾1. (As remarked on [35, p. 1279], this version follows from the version stated there. The notion of a “weak tangent” is defined in Section 2.2.) As a consequence of Corollary 1.5 and Theorem 1.7, we obtain: Corollary 1.8. — Let Q⩾1and let Xbe a complete, Ahlfors Q- regular metric space, where Q= cdimAR(X). If Xadmits a bi-Lipschitz embedding into some Euclidean space RN, then there is a complete metric space Zand a weak tangent of Xthat is bi-Lipschitz equivalent to Z×R. Corollary 1.8 shows that we should not expect minimizers for conformal dimension to appear within Euclidean space except under quite special circumstances, i.e., in the presence of some form of splitting. We illustrate a more concrete special case here. For this we introduce the terminology of linear connectedness: A metric space Xis linearly connected (with constant C⩾1) if every pair of points x, y ∈Xcan be joined by a compact, connected subset E⊆Xwith diam(E)⩽Cd(x, y). In particular, quasiconvex spaces (in which every pair ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 979 of points can be joined by a curve with length comparable to the distance between the points) are linearly connected. Corollary 1.9. — Let Xbe a linearly connected metric space. Suppose that 1<cdimAR(X)<2and that cdimAR(X)is achieved by a space Y. (In other words, Yis quasisymmetric to Xand Ahlfors Q-regular with Q= cdimAR(X) = cdimAR(Y).) Then Yadmits no bi-Lipschitz embedding into any Euclidean space. As an example, consider the classical Sierpiński carpets Spfor odd integers p > 1(the most famous example being S=S3). These are plane fractals formed by dividing the unit square into p−1×p−1squares and removing the middle square, then iterating this construction on the remaining squares. See [12] for details on this notation. The spaces Spare all linearly connected and have cdimAR(Sp)∈(1,2) (see [12, p. 595]). It is an open question whether the Ahlfors regular conformal dimensions of these spaces are actually achieved. Corollary 1.9 shows that they cannot be achieved by subsets of any Euclidean space. We note that the linear connectedness condition cannot be removed from Corollary 1.9: The product C × [0,1] ⊆R2of the standard Cantor set with the unit interval is Ahlfors Q-regular (for Q= 1 + log(2) log(3) ∈(1,2)) and known to be minimal for conformal dimension (by a theorem of Tyson [56]). However, it sits isometrically in R2. One may even make this example connected by taking its union with [0,1] × {0}, showing that “linearly connected” cannot be replaced by “connected” in Corollary 1.9. This example is also easily seen to have a weak tangent that is bi-Lipschitz equivalent to a product Z×R. 1.2.3. The slit carpet The slit carpet Mis a metric space homeomorphic to the standard Sierpiński carpet with a number of interesting properties. It was first proposed by Bonk and Kleiner and first studied in print by Merenkov [45]. Following [45], we define the space as follows, mostly using notation from [22]. We will be rather brief here, referring the reader to [22] or [45] for more details. Let Q0= [0,1]2denote the unit square in R2. For each dyadic subsquare Q⊆Q0, let sQdenote a central vertical “slit” in Qof half the side length. More specifically, if Q= [a2−k,(a+ 1)2−k]×[b2−k,(b+ 1)2−k], TOME 74 (2024), FASCICULE 3
986 Guy C. DAVID & Sylvester ERIKSSON-BIQUE (3) In the subsequence from (2), we may also obtain that •the functions λ−1 jk(f(·)−f(a)) converge to a Lipschitz function b f:b A→Rmas above, to yield (b A, b f)∈TanRn(A, f, a), •the sequence λ−1 jk(B−f(a)) converges in the pointed Hausdorff sense to an element b Bof TanRm(B, f(a)), and •b f(b A)⊆b B. (4) If fis L-bi-Lipschitz, then so is b f. (5) If Xand Aare Ahlfors Q-regular, then so are every element of WTan(X)and TanRn(A, a). (6) If µis a doubling measure on Aand ais a point of density of a subset A′⊆A, then TanRn(A′, f, a) = Tan(A, f, a). (7) If Y∈WTan(X), then WTan(Y)⊆WTan(X). Proof. — For (1), (2), and (3), see [20, Lemmas 8.6 and 8.13]. For (4), see [20, Lemma 8.20]. For (5), see [20, Lemma 8.28]. For (6) concerning the tangent spaces, see [20, Lemma 9.6] or [40, Proposition 3.1]; the extension to the tangent mappings is simple, as remarked in [21]. For (7), see [20, Lemma 9.5]. □ We will need one more fact about tangents, a principle that appears in many different forms and goes back to Preiss [51]. Versions appear in, e.g., [6, 21, 40, 44]. Informally, this is the principle that “tangents with moved basepoints are still tangents”. Proposition 2.5. — Let A⊆Rnbe a closed set supporting a doubling measure µ. Let f:A→Rmbe a Lipschitz mapping. Then for µ-a.e. a∈A, the following holds: For all (b A, b f)∈TanRn(A, f, a)and all b∈b A, we have (b A−b, b f(·+b)−b f(b)) ∈TanRn(A, f, a). The proof of Proposition 2.5 is a minor modification of facts in the literature, and so postponed until the Appendix (Section A). 2.3. Curves and fragments The key objects in this paper are families of curves (or curve fragments) in metric spaces. We introduce some notation to discuss these objects. Our definitions and notation follow those in [53] for the most part, with some minor changes. ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 987 Fix a separable, locally compact metric space X. A fragment in Xis a bi-Lipschitz map γ:C→X, where C⊆Ris compact and the onedimensional Lebesgue measure L1(C)is positive. We write Frag(X)for the collection of fragments in X. If γ∈Frag(X), then the domain Cof γis denoted dom(γ)and the image in Xis denoted im(γ). If f:X→Rmis any function, then we define (f◦γ)′(t) = lim t′→t,t′∈C f(γ(t′)) −f(γ(t)) t′−t, when the limit exists and t∈C= dom(γ)is a density point. If X⊂ Rn, then we simply write γ′(t), when f= id is the identity map. (As a reminder, a density point of a compact set C⊂Ris a t∈Cwhere limh→0 L1(C∩(t−h,t+h)) 2h= 1.) In Section 4, we will also consider Curv(X), the collection of all nonconstant, Lipschitz maps γ:I→X, where Iis a compact interval in Rof positive length. Thus, elements of Curv(X)represent honest curves in X. We will also use the notation dom(γ)to denote the domain of an element γ∈Curv(X). Note that neither Curv(X)nor Frag(X)is a subspace of the other. We now discuss the appropriate topologies on Frag(X)and Curv(X), borrowing from [53, Section 2]. The spaces Frag(X)and Curv(X)both admit embeddings into the space Haus(R×X)of non-empty compact subsets of R×X, by γ7→ {(t, γ(t)) : t∈dom(γ)}. The space Haus(R×X)is given the Hausdorff metric and the induced topology. If Xis complete, then so is Haus(R×X). Therefore, we topologize Frag(X)and Curv(X)as subspaces of Haus(R× X). We note that these spaces are σ-compact if Xis proper. 2.4. Line integrals and metric derivatives Let Xbe a metric space and γ∈Curv(X). We denote by len(γ)the length of γ, as in [26, Chapter 7]. If g:X→Ris a Borel function, then Rγg ds is defined as Zlen(γ) 0 g(eγ(t)) dt, where eγis the arc length parametrization of γ; see [26, Chapter 7]. TOME 74 (2024), FASCICULE 3
988 Guy C. DAVID & Sylvester ERIKSSON-BIQUE Following [7, Definition 4.1.2], the metric derivative of γat a point t∈ dom(γ)is dγ(t) := lim h→0 d(γ(t+h), γ(t)) |h|, whenever the limit exists. By [7, Theorem 4.1.6], dγ(t)does exist for a.e. t∈dom(γ), and len(γ) = Zdom(γ) dγ(t) dt. It follows that the arc length parametrization eγsatisfies d˜γ(t) = 1 for a.e. t∈dom(eγ). 2.5. Alberti representations Fix a complete, locally compact, separable metric space X. Recall that M(X)denotes the space of Radon measures on X. We equip M(X)with the weak∗topology arising from viewing M(X)as the dual space of Cc(X), the space of compactly supported continuous functions on X. See [53, Assumption 2.3] for details. Inside M(X), we consider the subspace P(X) consisting of probability measures. Note that elements of P(X)are Borel. Fix a metric space Xand a measure µ∈M(X). The following definition is due to Bate [8], based on earlier work of Alberti [1]. In [53], the definition was clarified and modified slightly, and this is the definition we present below. Definition 2.6. — An Alberti representation Aof µis a pair (P, ν) where (1) Pis a Radon probability measure on Frag(X), (2) ν: Frag(X)→M(X)is a Borel map with νγ≪H1|im(γ)for each γ∈Frag(X), (3) the measure µcan be represented as µ(A) = ZFrag(X) νγ(A)dP(γ), for each A⊆XBorel, (4) and, for each Borel A⊆Xand compact interval I⊆R, the map γ7→ νγ(A∩γ(dom(γ)∩I)) is Borel. Note that, given the topologies defined above, the statement that the map ν: Frag(X)→M(X)is Borel means that γ7→ ZX g(x)dνγ(x) ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 989 is a Borel map from Frag(X)to Rfor each g∈Cc(X). Acone in Rnis a set of the form Cone(w, t) := {v∈Rn:v= 0 and v·w⩾t|v|}, for some w∈Sn−1and t∈R. Note that, for any w∈Sn−1and t⩽−1, Cone(w, t) = Rn\ {0}. Remark 2.7. — Our definition of a cone departs slightly from those in [8] and [53]. In particular, our cones may have opening angle larger than π. It is clear that any cone of the types in [8] and [53] is a subset of a cone like one above. Definition 2.8. — Fix a metric space X, a Lipschitz map ϕ:X→Rn, and a cone C⊆Rn. A fragment γ∈Frag(X)is said to be in the ϕ-direction of Cif (ϕ◦γ)′(t)∈ Cfor a.e. t∈dom(γ). An Alberti representation A= (P, ν)of a measure µ∈M(X)is said to be in the ϕ-direction of Cif P-a.e. γ∈Frag(X)is in the ϕ-direction of C. For an example, consider X=Rnand the identity map ϕ= id, as well as the Lebesgue measure λ. Probably the simplest example of an Alberti representation is given by the Fubini-representation of the Lebesgue measure λon Rnby integration on lines parallel to any direction. If these lines are parallel to a given non-zero vector w∈Rn, then this Alberti-representation is in the id-direction of Cone(w, t)for any t < 1. Slightly more complicated examples are obtained by taking superpositions with different directions, and by splitting the lines to segments. Definition 2.9. — Cones C1, . . . , Ckin Rnare called independent if each collection {v1, . . . , vk:vi∈Ci} is linearly independent. A collection A1,...,Akof Alberti representations of a measure µ∈ M(X)is called ϕ-independent, for a Lipschitz ϕ:X→Rm, if there are independent cones C1, . . . , Ckin Rmsuch that each Aiis in the ϕ-direction of Ci. We call a collection A1,...,Akof Alberti representations independent if they are ϕ-independent for some Lipschitz map ϕas above. A few remarks concerning this definition are in order. Remark 2.10. — If X⊆Rnand µ∈M(X)supports k ϕ-independent Alberti representations, for some ϕ:X→Rm, then the map ϕmay be TOME 74 (2024), FASCICULE 3
990 Guy C. DAVID & Sylvester ERIKSSON-BIQUE extended to a Lipschitz map ϕ:Rn→Rmwithout altering the notion of ϕ-independence. Remark 2.11. — Traditionally (i.e., in [8]), it is assumed that m=kin Definition 2.9, but we see no need to assume this, and in fact it will be occasionally convenient not to. Remark 2.12. — In the case k= 1 of Definition 2.9, one may take all of Rm\{0}as a single independent cone. Thus, a single Alberti representation (P, ν)is independent if and only if there is a Lipschitz map ϕ:X→Rm such that (ϕ◦γ)′(t)= 0 for P-a.e. γ∈Frag(X)and a.e. t∈dom(γ). In particular, if X⊆Rn, then a single non-trivial Alberti representation is automatically independent. Indeed, take ϕto be the identity map. Since every γ∈Frag(X)is bi-Lipschitz, we have that (ϕ◦γ)′(t) = γ′(t)= 0 for all γ∈Frag(X)and a.e. t∈dom(γ). A last key fact for us will be the following result from [8]. Essentially, one would like to know that a phenomenon which happens at almost every point along each curve in an Alberti representation actually happens almost everywhere in X. This is what the following result provides. (See also the more general [8, Proposition 2.9].) Proposition 2.13 ([8, Corollary 2.13]). — Let Xbe a complete metric space with a Radon measure µ. Let ϕ:X→Rmbe Lipschitz such that µ has k ϕ-independent Alberti representations. Let f:X→Rnbe Lipschitz. Then for µ-a.e. x∈X, the following hold: (1) There are γ1, . . . , γk∈Frag(X)such that γi(0) = xand γ−1(x)is a density point of dom(γ). (2) The derivatives (ϕ◦γi)′(0) exist and form a linearly independent set in Rm. (3) The derivatives (f◦γi)′(0) exist. We briefly note that Bate assumes that k=mand n= 1 in the cited result, but the proof using [8, Lemma 2.8 and Proposition 2.9] works in this generality. 2.6. Connecting to other measures defined on curve families In Section 4, we will need to connect Alberti representations to a related type of measure defined on Curv(X). ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 991 Proposition 2.14. — Let Xbe a proper metric space, ϕ:X→Rn bi-Lipschitz, and C⊂Rna cone. Suppose that Pis a Radon measure(1) on Curv(X), so that for P-almost every γ,(ϕ◦γ)′(t)∈Cor dγ(t)=0for almost every t∈dom(γ). If the Borel measure defined by µ(A) = ZCurv(X)Zγ 1AdsdP is locally finite (hence Radon), then it admits an Alberti representation in the ϕ-direction of C. Remark 2.15. — The literature is rife with different versions of Alberti representations, see [3, 8, 19, 53] for some of them. Modifications of this argument can be used to show that, roughly speaking, if one has a representation in one of these senses, then one has also a representation in any other sense. We briefly remark that the cones considered in [8] are slightly different from ours, but the proof applies for both notions of cone. Proof. — By [8, Corollary 5.8] (see the “in particular. . . ” statement), we can decompose X=A∪N, where µ|Aadmits an Alberti representation in the ϕ-direction of C, and H1(im(γ)∩N)=0 for every γ∈Frag(X)in the ϕ-direction of C. If we can show that µ(N) = 0, then µrestricted to the full-measure set Asupports an Alberti representation in the ϕ-direction of C, and this completes the proof. To establish this, we will show that for P-almost every curve γwe have Zγ 1Nds= 0. First, for P-almost every curve γ:I→X, and almost every t∈Iwe have (ϕ◦γ)′(t)∈Cor dγ(t)=0. Let γbe any curve with such properties, and let eγ:e I→Xbe its arc length reparametrization. Then (ϕ◦eγ)′(t)∈C for almost every t∈I. By [37, Lemma 4], we can find compact sets Kjsuch that e I=SjKj∪S, |S|= 0, and eγj:= eγ|Kjis bi-Lipschitz. It follows that eγj∈Frag(X)and in the ϕ-direction of C, and hence H1(im(eγj)∩N)=0for each j. (1) While the notation may suggest so, this measure need not be a probability measure. TOME 74 (2024), FASCICULE 3
992 Guy C. DAVID & Sylvester ERIKSSON-BIQUE Thus, Zγ 1Nds=Z˜ I 1N(eγ(t)) dt=X jZKj 1N(eγ(t)) dt= 0, which completes the proof. □ 3. Proof of Theorem 1.2 In this section, we prove Theorem 1.2. The proof requires a few lemmas. Lemma 3.1. — Let a closed set X⊆Rnsupport a doubling measure µ0. Let µ≪µ0support k ψ-independent Alberti representations, for some Lipschitz ψ:Rn→Rm. Then for µ-a.e. x∈X, there are linearly independent vectors v1, . . . , vk such that the following holds: For every Y∈TanRn(X, x), every y∈Y, and every i∈ {1, . . . , k}, there is a line through yin direction vithat is contained in Y. Proof. — We apply Proposition 2.13, in the case ϕ=ψand fis the inclusion X→Rn. This tells us that, at µ-a.e. x∈X, there are γ1, . . . , γk∈Frag(X)such that the following hold: (1) For each 1⩽i⩽k, we have γi(0) = xwith 0a density point of dom(γ). (2) The vectors (ψ◦γ1)′(0),...,(ψ◦γk)′(0) are linearly independent in Rm. (3) For each 1⩽i⩽k,γ′ i(0) exists. Fix an x∈Xwhere the above hold and where the conclusion of Proposition 2.5 holds. Let wi= (ψ◦γi)′(0) ∈Rk. Let vi=γ′ i(0) ∈Rn. Note that vi= 0 as γiis bi-Lipschitz, and wi= 0 by (ii). Consider an arbitrary tangent (Y, b ψ)∈TanRn(X, ψ, x), subject to the sequence of scales λk→0. By Lemma 2.4(2), (3) and (6), we may pass to a subsequence of {λj} subject to which tangent mappings Liof each γiat 0exist. More precisely, ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 993 we have the following for each 1⩽i⩽k: (R, Li)∈TanR(dom(γi), γi,0) with Li(R)⊆Y, and (R,b ψ◦Li)∈TanR(dom(γi), ψ ◦γi,0). Moreover, since γiand ψ◦γiare differentiable at 0, their tangent maps Li and b ψ◦Liare linear. In particular, recalling γ′ i(0) = viand (ψ◦γi)′(0) = wi, we have the following properties of Li: (3.1) Li(t) = tvi∈Yand ψ(Li(t)) = twifor all t∈R. By definition, we also have 0∈Yand b ψ(0) = 0. To summarize, the above argument shows that for every element (Y, b ψ)∈ Tan(X, ψ, x), there is a line Lithrough 0with the properties in (3.1). Consider again an arbitrary (Y, b ψ)∈Tan(X, ψ, x). Proposition 2.5 therefore says that for every y∈Y, the pair (Y−y, b ψ(·+y)−b ψ(y)) is also an element of TanRn(X, ψ, x). This implies that for every y∈Yand i∈ {1, . . . , k}, there is a function Ly i:R→Y such that Ly i(t) = y+tviand ψ(Li(t)) = b ψ(y) + twifor all t∈R. In other words, Ly iis the parametrization of a line through yin direction vi (contained in Y), whose composition with b ψparametrizes a line through b ψ(y)in direction wi. It remains to show that the vectors viare linearly independent. Suppose to the contrary that there was a non-trivial linear combination k X i=1 aivi= 0. Let y0= 0 ∈Yand p0= 0. For i= 1, . . . , k + 1, inductively set yi=Lyi−1 i(ai) = yi−1+aivi and pi=b ψ(yi). Note that pi=pi−1+aiwifor each i= 1,..., k by the properties of Ly iabove. Then yk= 0. This implies that pk=b ψ(yk)=0. On the other hand pk= k X i=1 aiwi. This contradicts the linear independence of the vectors wi.□ TOME 74 (2024), FASCICULE 3
994 Guy C. DAVID & Sylvester ERIKSSON-BIQUE Lemma 3.2. — Let Y⊆Rnbe a closed set. Let v1, . . . , vkbe linearly independent in Rn. Suppose that, for each y∈Y, there are klines Li={y+tvi:t∈R} that pass through yand are contained in Y. Then Y=Z×V, where V= span({v1, . . . , vk})and Z⊆V⊥is a closed set. Proof. — Let V= span({v1, . . . , vk}), and let Z= projV⊥(Y), the projection of Yto the orthogonal complement of V. We now claim that Y=Z×V. Certainly Y⊆Z×V, by definition of orthogonal projection. For the other direction, suppose p∈Z×V. Then projV⊥(p)∈Z, so projV⊥(p) = projV⊥(y)for some y∈Y. It follows that p=y+a1v1+· · · +akvk, where ai∈R. Set y0=y∈Y. For i= 1, . . . , k, we inductively set yi=yi−1+aivi. By assumption, each point yiis in Y. Note that the last point ykis equal to p. Hence p∈Y, which proves that Z×V⊆Y. Lastly, we argue that Zis closed. Indeed, if znis a sequence in Zconverging to z∈Rn, then the points (zn,0) ∈Y⊆V⊥×V=Rn converge to (z, 0) ∈Y, since Yis closed. It follows that z∈Z.□ Proof of Theorem 1.2. — Let X⊆Rnbe a closed set admitting a doubling Radon measure µ0. Let µbe a measure absolutely continuous to µ0that admits k ϕ-independent Alberti representations. Let xbe a point at which the conclusion of Lemma 3.1 holds (a set of points that has full µ-measure). Let v1, . . . , vkbe the associated linearly independent vectors in Rn. Let Y∈TanRn(X, x). Then each point y∈Yadmits klines L1, . . . , Lk through y, in directions vi, that are contained in Y. By Lemma 3.2, this implies that Y=Z×Vfor a k-dimensional subspace V= span({v1, . . . , vk})and some closed set Z⊆V⊥⊆Rn. This completes the proof. □ ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 995 4. Modulus and Alberti representations In this section, we relate Alberti representations to the more classical notion of the modulus of a family of curves. The main result in this section is Proposition 4.5, which may be of independent interest. (See Remark 4.6 for more on the provenance of this result.) We first recall the definition of the modulus of a family of curves. It is worth noting, that for us Curv(X)consists only of curves with Lipschitz parametrizations, while traditionally Modulus is defined for an a priori larger class of collections of γ:I→X, which are merely continuous. However generality is not lost, as the modulus of non-rectifiable curves vanishes by convention, and rectifiable curves can be reparametrized as Lipschitz curves without affecting the modulus. Definition 4.1. — Let Xbe a metric space with a Radon measure µ, let Γ⊆Curv(X), and let p⩾1. A Borel measurable function ρ:X→[0,∞]is called admissible for Γ if Rγρds⩾1for each γ∈Γ. We set A(Γ) to be the collection of all admissible functions for Γ. The p-modulus of Γ, with respect to the measure µ, is denoted (4.1) Modp(Γ, µ) = inf ρ∈A(Γ) ZX ρpdµ. For our duality argument, we will need to work with continuous functions ρ. Thus, we define Modc p(Γ, µ)by replacing the infimum in (4.1) with the infimum over all admissible ρ:X→[0,∞)that are in addition continuous with compact support. In general, Modc p(Γ, µ)may be larger that Modp(Γ, µ), and our first goal is to identify an assumption under which they are equal. We will need the following basic continuity fact both for the equality of Modpand Modc pand for the duality argument below. It is a version of [33, Proposition 4] in our topology. Lemma 4.2. — Let ρn:X→Rbe an increasing sequence of lower semicontinuous functions converging pointwise to ρ:X→R. If γn: [a, b]→X converge uniformly to γ: [a, b]→X, or if γn∈Curv(X)converge to γ∈Curv(X), then lim inf n→∞ Zγn ρnds⩾Zγ ρds. Further, the map γ7→ Rγρdsis lower semi-continuous on Curv(X). TOME 74 (2024), FASCICULE 3
1002 Guy C. DAVID & Sylvester ERIKSSON-BIQUE 5. Proofs of the corollaries In this section, we prove all the corollaries of our main result stated in the introduction. Before beginning the proofs, the following basic lemma allows us to reduce problems of bi-Lipschitz embedding to Theorem 1.2. Lemma 5.1. — Let Xbe a metric space with a doubling measure µ0. Suppose that µ≪µ0supports k ϕ-independent Alberti representations, for some ϕ:X→Rm. Let f:X→Ybe a bi-Lipschitz homeomorphism. Then (1) bµ0:= f∗(µ0)is a doubling measure supported on Y. (2) bµ:= f∗(µ)≪bµ0. (3) bµsupports k ϕ ◦f−1-independent Alberti representations. Proof. — The first two statements are immediate from the definitions. For the third statement, let Ai= (Pi, νi)be independent Alberti representations for µ, for i= 1, . . . , k. Note that f:X→Yand f−1:Y→Xinduce continuous maps F: Frag(X)→Frag(Y)and F−1: Frag(Y)→Frag(X), by post-composition. For each Alberti representation Ai= (Pi, νi), we may define a map bνi: Frag(Y)→M(Y)by bνi γ=f∗(νi F−1(γ))for each γ∈Frag(Y). It is immediate that bνi γ≪ H1|im(γ)for each γ∈Frag(Y), since bi-Lipschitz maps preserve sets of zero H1-measure. We therefore define the Alberti representations c Ai= (F∗(Pi),bνi) for i= 1, . . . , k. It is easy to check that these satisfy conditions (1), (2), and (4) of Definition 2.6. For condition (3), observe that if A⊆Yis Borel and i∈ {1, . . . , k}, ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 1003 then bµ(A) = µ(f−1(A)) =ZFrag(X) νi γ(f−1(A)) dPi(γ) =ZFrag(X) νi F(γ)(A) dPi(γ) =ZFrag(Y)bνi α(A) dF∗(Pi)(α), as desired. Lastly, we check the independence of the new Alberti representations on Y. Let A= (P, ν)denote any one of the koriginal Alberti representations Aiabove. Then there is a cone C⊆Rksuch that (ϕ◦γ)′(t)∈Cfor P-a.e. γ∈Frag(X)and a.e. t∈dom(γ). Let G⊆Frag(X)be the full P-measure set on which this holds. Let b G=F(G)⊆Frag(Y), a set of full b P-measure in Frag(Y). Consider any α∈b G⊆Frag(Y). Then the fragment γdefined by t7→ f−1(α(t)) is in G. Therefore, for a.e. t∈dom(α) = dom(γ), (ϕ◦f−1◦α)′(t)=(ϕ◦γ)′(t) is in C. Thus, each new Albert representation c Aiis in the (ϕ◦f−1)-direction of the same cone of which Aiwas in the ϕ-direction. Thus, the representations c Aiare (ϕ◦f−1)-independent. □ As a consequence, we can now prove Corollary 1.3. Proof of Corollary 1.3. — Let Xbe a complete metric space admitting a doubling Radon measure µ0. Suppose that a measure µ≪µ0supports k independent Alberti representations, for some k⩾1. Suppose that f:X→Rnis a bi-Lipschitz embedding. Let X′=f(X), µ′ 0=f∗(µ0), and µ′=f∗(µ). It follows from Lemma 5.1 that µ′admits kindependent Alberti representations. Therefore, by Theorem 1.2, at µ′-a.e. point x′∈X′every tangent Y′∈TanRn(X′, x′)is isometric to Z×Rk, for some closed set Z⊆Rn−k. The set of all preimages under fof such points x′∈X′forms a set of full µ-measure in X. At such a point x=f−1(x′)∈X, each tangent (Y, y)∈Tan(X, x)is bi-Lipschitz equivalent to an element of TanRn(X′, x′), TOME 74 (2024), FASCICULE 3
1004 Guy C. DAVID & Sylvester ERIKSSON-BIQUE with a bi-Lipschitz map given by a tangent map of f. Thus, any such tangent Yis bi-Lipschitz equivalent to a product Z×Rk, for some complete metric space Z. This proves the corollary. □ 5.1. Modulus and conformal dimension Here we prove Corollaries 1.5, 1.8, and 1.9. Proof of Corollary 1.5. — Let Xbe a complete metric space admitting a Radon measure µthat is absolutely continuous with respect to a doubling measure µ0. Suppose that Xcontains a family of (non-constant) curves Γso that Modp(Γ, µ)>0for some p∈[1,∞), and that Xadmits a bi-Lipschitz embedding ϕinto some Rn. By Corollary 4.9, there is a non-trivial measure on Xthat is absolutely continuous to µ, hence to µ0, and supports a ϕ-independent Alberti representation. The corollary then follows from Corollary 1.3. □ Proof of Corollary 1.8. — Let Xsatisfy the assumptions of the corollary. Thus, Xis Ahlfors Q-regular with Q= cdimAR(X)and Xadmits a bi- Lipschitz embedding into some Rn. By the Keith–Laakso Theorem 1.7, there is a weak tangent Wof X that contains a family of non-constant curves with positive modulus. By Lemma 2.4, the space Walso admits a bi-Lipschitz embedding into Rn. By Corollary 1.5, there is a tangent Yof Wthat is bi-Lipschitz equivalent to Z×Rfor some complete metric space Z. As Yis also a weak tangent of the original space X(see Lemma 2.4(7)), this completes the proof. □ Proof of Corollary 1.9. — Let Xbe linearly connected and let Ybe quasisymmetric to Xand Ahlfors Q-regular, where Q= cdimAR(X) = cdimAR(Y)∈(1,2). Suppose that Ydid admit a bi-Lipschitz embedding into some Euclidean space, RN. By Corollary 1.8, Ywould then admit a weak tangent Wthat is bi- Lipschitz equivalent to a product Z×R, for some complete metric space Z. Let ϕ:Z×R→Wbe bi-Lipschitz. Let π:Z×R→Zbe the projection to the Zfactor. The linear connectedness condition is preserved under both quasisymmetry and passage to weak tangents. (The former assertion is immediate ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 1005 from the definitions, and the latter is contained in the proof of [36, Proposition 5.4].) Thus, Wis linearly connected. By Lemma 2.4(5), Wis also Ahlfors Q-regular. Next, we observe that Zmust contain at least two points: if not, then Wwould be bi-Lipschitz equivalent to Z×R∼ =R, which would contradict the fact that Wis Ahlfors Q-regular for Q > 1. We now observe that Zmust contain a compact, connected set Kwith at least two points. To see this, fix distinct points z, z′∈Z. Since Wis linearly connected, there is a compact, connected set Jin Wthat contains ϕ(z, 0) and ϕ(z′,0). The set K=π(ϕ−1(J)) is then a continuum containing zand z′. Thus, Zcontains a non-trivial continuum Kand so Wcontains a bi- Lipschitz image of the space K×[0,1]. We now argue that (5.1) dimH(K×[0,1]) ⩾2. That the Hausdorff dimension of a product is at least the sum of the dimensions of the factors is standard for compact subsets of Euclidean space (see, e.g., [9, Theorem 3.2.1]); we give a brief argument in our setting here: Since Kis compact and connected, H1(K)>0. By Frostman’s Lemma [43, Theorem 8.17], Ksupports a Radon measure µsatisfying µ(B(x, r)) ⩽rfor all x∈Kand 0< r ⩽diam(K). If L1denotes Lebesgue measure on [0,1], then the measure µ× L1on K×[0,1] satisfies (µ× L1)(B(p, r)) ⩽r2for all p∈K×[0,1] and 0< r ⩽diam(K×[0,1]). By the “mass distribution principle” (see [26, p. 61]), we obtain (5.1). We therefore arrive at 2> Q = dimH(W)⩾dimH(K×[0,1]) ⩾2, a contradiction. □ 5.2. Slit carpet Here, we prove Corollary 1.10. This will follow from Corollary 1.5 and some facts about the slit carpet. We first summarize some results of Merenkov [45]. TOME 74 (2024), FASCICULE 3
1006 Guy C. DAVID & Sylvester ERIKSSON-BIQUE Proposition 5.2 ([45, Lemma 2.1, Proposition 2.4, and Lemma 4.2]). The slit carpet Mhas the following properties: (1) It is homeomorphic to the standard Sierpiński carpet. In particular, it is compact and has topological dimension 1.(3) (2) It is geodesic and Ahlfors 2-regular. (3) It admits a family of non-constant curves with positive 2-modulus (with respect to the measure H2). Proof of Corollary 1.10. — By Proposition 5.2, we may fix a point p∈M where the conclusion of Corollary 1.5 holds. The self-similarity of Measily implies the following: There is a constant c > 0such that, for each r > 0, there is a point qrand a compact set Kr⊆B(p, r)such that B(qr, cr)⊆Kr⊆B(p, r), and Kris isometric to tMfor some t∈(2cr, 2r). Let (Y, y)∈Tan(M, p)be obtained by rescaling along a sequence λi→0. Let qi=qλiand Ki=Kλi. Passing to a (subsequential) limit, the above properties imply (see, e.g., [20, Lemma 8.31]) that there is a point q∈Y and a compact set K⊆Ysuch that B(q, c)⊆K⊆B(y, 1) and Kis bi-Lipschitz equivalent to M. By Corollary 1.5, Yis bi-Lipschitz equivalent to Z×Rfor some complete (and necessarily doubling) metric space Z. As a tangent of M,Yis quasiconvex and therefore so is Z. It follows that Zcontains a non-trivial topological arc through each point. Hence, there is a homeomorphic image of [0,1]2contained in B(q, c). This implies that B(q, c)⊆Kmust have topological dimension 2. However, this contradicts the fact that Kis bi-Lipschitz equivalent to M, which has topological dimension 1.□ (3) Here, “topological dimension” can refer to Lebesgue covering dimension or (small) inductive dimension, which are equivalent for compact metric spaces [46]. All we will need to know is that Mdoes not contain a topologically embedded copy of any open subset of R2. ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 1007 5.3. Heisenberg group Here we give a brief introduction to the Heisenberg group and prove Corollary 1.11. 5.3.1. Preliminaries on the Heisenberg group We now fix some notation and definitions. We will be very brief, referring the reader to [15, 41] for details. The group His R3endowed with the non-abelian group law (x1, y1, z1)·(x2, y2, z2)=(x1+x2, y1+y2, z1+z2+1 2(x1y2−y1x2)). There are many standard, bi-Lipschitz equivalent ways to equip Hwith a metric. For simplicity, we fix the so-called Korányi distance, though it will make little difference below. Definition 5.3. — The Korányi norm of (x, y, z)∈His ∥(x, y, z)∥= ((x2+y2)2+ 16z2)1/4. The Korányi distance between p, q ∈His d(p, q) = ∥p−1q∥. The Korányi distance on Hinduces the usual topology from R3and has the following features (see, e.g. [41, Example 1.3]): (1) Left-invariance: d(p, q) = d(p′·p, p′·q)for all p, q, p′∈H. (2) Dilations: For each t > 0, the map δt(x, y, z)=(tx, tx, t2z) is a group homomorphism with the property that d(δt(p), δt(q)) = td(p, q)for all p, q ∈H (3) Doubling: The Lebesgue measure Lon R3is doubling on (H, d). In fact, it satisfies the Ahlfors 4-regularity property L(B(p, r)) ≈r4 for all p∈H,r > 0and a fixed positive implied constant. In particular, item (3) implies that every open set in Hhas Hausdorff dimension 4. The Korańyi distance is also bi-Lipschitz equivalent to the more wellknown Carnot–Carathéodory distance, which we do not define here, as both satisfy properties (1)–(3) above [41]. From these properties, we first derive the following: TOME 74 (2024), FASCICULE 3
1008 Guy C. DAVID & Sylvester ERIKSSON-BIQUE Lemma 5.4. — There is an open set Uin the Heisenberg group such that L|Usupports two independent Alberti representations. This fact is well-known, and of course much more is true. We include a brief proof only to show that no sophisticated tools are needed. Proof. — Let ϕ:H→R2be the canonical Lipschitz chart, given by (x, y, z)7→ (x, y). Fix any disjoint, independent cones, Cx, Cyin R2con- taining the xand y-axis respectively. For each p= (a, b)∈R2, the curves αp(t) = t, a, b −1 2at and βp(t) = a, t, b +1 2at are bi-infinite geodesics in H. Indeed, α(a,b)(t) = (0, a, b)·(t, 0,0) and β(a,b)(t)=(a, 0, b)·(0, t, 0), and the curves (t, 0,0) and (0, t, 0) are clearly geodesics. Define the maps Φαand Φβfrom −1 2,1 23to R3by (p, t)7→ αp(t) and (p, t)7→ βp(t), respectively. One directly verifies that these maps are injective and are open mappings on the interiors of their domains. Since Φα(0) = Φβ(0) = (0,0,0), there is a non-empty open set Ucontained in Φα −1 2,1 23!∩Φβ −1 2,1 23!. We will obtain two independent Alberti representations of the measure µ=L|Uon H. Computing the Jacobians gives immediately that Φαand Φβare volume preserving. Writing λfor Lebesgue measure on −1 2,1 22, we therefore obtain by change of variables that (5.2) µ(A) = ZR2ZR 1A(αp(t)) dtdλ(p) and (5.3) µ(A) = ZR2ZR 1A(βp(t)) dtdλ(p) for any Borel set A⊆U. Thus, define a probabilty measure Pon Frag(H)by the pushforward of λ|[−1 2,1 2]2under the map from −1 2,1 22given by p7→ αp|[−1 2,1 2]∈Frag(H). ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 1009 Note that this map is continuous on −1 2,1 22. For γ∈Frag(X), define νγ= 0 if γis not in the support of P. Otherwise, γ=αp|[−1 2,1 2]for some p, and we set νγ= 1U· H1|im(γ). The pair (P, ν)defines an Alberti representation of µby (5.2), supported on curves of the form αp|[−1 2,1 2]. The exact same procedure applied to the curves βpyields an Alberti representation of µsupported on curves of the form βp|[−1 2,1 2]. Since (ϕ◦αp)′(t) = (1,0) ∈Cxand (ϕ◦βp)′(t) = (0,1) ∈Cyfor each p∈R2and t∈R, the two Alberti representations are independent. □ 5.3.2. Proof of Corollary 1.11 There are a number of proofs of Corollary 1.11 in the literature. We give a proof below that avoids Pansu’s differentiation theorem from [50]. It relies only on Theorem 1.2, invariance of domain, and the basic properties of the Heisenberg group stated in the previous subsection. Of course, we still use a blowup argument, so the ideas are similar in spirit. Proof of Corollary 1.11. — Suppose that the Heisenberg group (with the Korányi metric) admitted a bi-Lipschitz embedding into some Euclidean space. By Lemma 5.4 and Corollary 1.3, some tangent of the Heisenberg group would be bi-Lipschitz equivalent to Z×R2, for some complete metric space Z. By the homogeneity and dilation structure of the metric, every tangent of the Heisenberg group is isometric to the Heisenberg group itself. Thus, in this case, His bi-Lipschitz equivalent to Z×R2. Since His proper and quasiconvex, so is Z. It follows that Zis bi-Lipschitz equivalent to a geodesic metric space, simply by replacing the metric on Zby the associated length metric. Therefore, in particular, Zcontains a Lipschitz embedding γ: [0,1] →Z. Let ϕ:Z×R2→Hbe bi-Lipschitz. The map from (0,1) ×(0,1)2into H given by (t, p)7→ ϕ(γ(t), p) is therefore a Lipschitz homeomorphism from an open set of R3into H. By invariance of domain, the image of this map must be open in H. On the other hand, this set has Hausdorff dimension at most 3, as the Lipschitz image of a subset of R3. This violates the Ahlfors 4-regularity of H.□ TOME 74 (2024), FASCICULE 3
1010 Guy C. DAVID & Sylvester ERIKSSON-BIQUE Appendix A. Proof of Proposition 2.5 Here we give a proof of Proposition 2.5. The idea is extremely similar to that of [21, Proposition 3.1] (which in turn is based on [40, 51]), which is the same statement in the setting of Gromov–Hausdorff tangents rather than intrinsic tangents. We will therefore omit many steps if they are easy to adapt from there. Recall the notion of the outer measure µ∗(A) = inf{µ(B) : BBorel , B ⊇A}, and the associated notion of a point of outer density xof a set A, where lim r→0 µ∗(B(x, r)∩A) µ(B(x, r)) →1. As explained briefly in [21], every set of positive outer measure has a point of outer density. The following lemma is a simple extension of (6) of Lemma 2.4. See also [20, Lemma 9.6] or [40, Proposition 3.1] for closely related statements whose proofs can easily be modified to yield this one. Lemma A.1. — Let A⊆Rnsupport a doubling measure µand a Lipschitz f:A→Rm. Let E⊆Ahave a point of outer µ-density at a∈A. Then TanRn(E, f, a) = TanRn(A, f, a). We now define a notion of distance that yields the correct topology. As in [21], the distance we define will not precisely be a metric, but it will suffice for our purposes. Definition A.2. — Let A, B ⊆RNbe sets and f, g:RN→RM, Lipschitz functions. Define e D((A, f),(B, g)) = inf (ϵ > 0 : d1/ϵ(A, B)< ϵ and |f−g|< ϵ on (A∪B)∩B(0,1/ϵ)). Then define D= min( e D, 1/2). To simplify notation, given A⊆RN,f:RN→RMLipschitz, λ > 0, and p∈RN, we set Ap,λ =λ−1(A−p) and fp,λ(x) = λ−1(f(λx +p)−f(p)). Note that fp,λ is Lipschitz with the same constant as f. ANNALES DE L’INSTITUT FOURIER
INFINITESIMAL SPLITTING 1011 The following is analogous to [21, Lemma 2.3]. Lemma A.3. — The function Dhas the following properties: (1) It is non-negative and symmetric. (2) If D((A, f),(B, g)) = 0 then A=Band f=gon A=B. (3) For all pairs (A, f),(B, g),(C, h), we have the quasi-triangle inequality D((A, f),(C, h)) ⩽2(D((A, f),(B, g)) + D((B, g),(C, h))) (4) (b A, b f)∈TanRN(A, f, a)if and only if some Lipschitz extensions of fand b fto all of RNand some sequence λi→0satisfy (A.1) D((Aa,λi, fa,λi),(b A, b f)) →0 Proof. — The first two items are simple, the third follows exactly as in [21, Lemma 2.3], and the fourth follows from [20, Lemma 8.7]. □ The next lemma is an analog of [21, Lemma 2.6]. Lemma A.4. — For each N, M ∈Nand L, η > 0, the collection S={(B, g) : B⊆RN, g :RN→RML-Lipschitz} is contained in a countable collection of sets Bℓwith D-diameter at most η. The “D-diameter” of a collection of pairs {(B, g)}is the supremum of the D-distance between pairs of elements in the collection. Proof. — Consider all pairs (K, h)such that K⊆QN⊆RNis finite, and h:K→QM. By Lemma A.3(3), it suffices to show, given η∈(0,1) and (B, g)∈ S, that (B, g)is within D-distance 10ηof some (K, bh), where (K, h)is as above and bhis a Lipschitz extension of hto all RN. We may also assume L > 1. Fix K⊆B(0,2η−1)∩QNto be η/L-separated and finite such that d2η−1(K, B)⩽η/L. Then, for x∈K, set h(x)to be an element of QMwithin distance η/L of g(x). Note that his 3L-Lipschitz. Extend hto a 3L-Lipschitz map bhon RNby Kirszbraun’s theorem. For x∈K∩B(0, η−1), |g(x)−bh(x)|⩽η/L < η and if x∈B∩B(0, η−1), then |g(x)−bh(x)|⩽6η+|g(y)−h(y)|⩽10η, where yis a closest element in Kto x. This completes the proof. □ TOME 74 (2024), FASCICULE 3