On a conjecture of Cheeger

This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show that the push-forward of the measure from a Lipschitz differentiability space under a chart is absolutely continuous with respect to Lebesgue measure.


Introduction
In [Che99] Cheeger proved that in every doubling metric measure space (X, ρ, µ) satisfying a Poincaré inequality, Lipschitz functions are differentiable µ-almost everywhere. More precisely, he showed the existence of a family {(U i , ϕ i )} i∈N of Borel charts (that is, U i ⊂ X is a Borel set, X = i U i up to a µ-negligible set, and ϕ i : X → R d(i) is Lipschitz) such that for every Lipschitz map f : X → R at µ-almost every x 0 ∈ U i there exists a unique (co-)vector df (x 0 ) ∈ R d(i) with lim sup This fact was later axiomatized by Keith [Kei04], leading to the notion of Lipschitz differentiability space, see Section 2 below. Cheeger also conjectured that the push-forward of the reference measure µ under every chart ϕ i has to be absolutely continuous with respect to the Lebesgue measure, that is, see [Che99,Conjecture 4.63]. Some consequences of this fact concerning existence of bi-Lipschitz embeddings of X into some R N are detailed in [Che99,Section 14], also see [CK06,CK09] Let us assume that (X, ρ, µ) = (R d , ρ E , ν) with ρ E the Euclidean distance and ν a positive Radon measure, is a Lipschitz differentiability space when equipped with the (single) identity chart (note that it follows a-posteriori from the validity of Cheeger's conjecture that no mapping into a higher-dimensional space can be a chart in a Lipschitz differentiability structure of R d ). In this case the validity of Cheeger's conjecture reduces to the validity of the (weak) converse of Rademacher's theorem, which states that a positive Radon measure ν on R d with the property that all Lipschitz functions are differentiable νalmost everywhere must be absolutely continuous with respect to L d . Actually, it is well known to experts that this converse of Rademacher's theorem implies Cheeger's conjecture in any metric space, see for instance [Kei04, Section 2.4], [Bat15, Remark 6.11], and [Gon12].
The (strong) converse of Rademacher's theorem has been known to be true in R since the work of Zahorski [Zah46], where he characterized the sets E ⊂ R that are sets of non-differentiability points of some Lipschitz function. In particular, he proved that for every Lebesgue negligible set E ⊂ R there exists a Lipschitz function which is nowhere differentiable on E.
The same result for maps f : R d → R d has been proved by Alberti, Csörnyei & Preiss for d = 2 as a consequence of a deep structural result for negligible sets in the plane [ACP05,ACP10]. In 2011, Csörnyei & Jones [Jon11] announced the extension of the above result to every Euclidean space. For Lipschitz maps f : R d → R m with m < d the situation is fundamentally different and there exists a null set such that every Lipschitz function is differentiable at at least one point from that set, see [Pre90,PS15]. We finally remark that the weak converse of Rademacher's theorem in R 2 can also be obtained by combining the results of [Alb93] and [AM16], see [AM16, Remark 6.2 (iv)].
Recently, a result concerning the singular structure of measures satisfying a differential constraint was proved in [DR16]. When combined with the main result of [AM16], this proves the weak converse of Rademacher's theorem in any dimension, see [DR16,Theorem 1.14].
In this note we detail how the results in [AM16,DR16] in conjunction with Bate's result on the existence of a sufficient number of independent Alberti representations in a Lipschitz differentiability space [Bat15] imply Cheeger's conjecture; see Section 2 for the relevant definitions.
Note that by the same arguments of this paper Cheeger's conjecture would also follow from the results announced in [ACP05] and [Jon11].
After we finished writing this note we learned that similar results have been proved by Kell and Mondino [KM16] and by Gigli and Pasqualetto [GP16].
2. Setup 2.1. Lipschitz differentiability spaces. In the sequel, the triple (X, ρ, µ) will always denote a metric measure space, that is, (X, ρ) is a separable, complete metric space and µ ∈ M + (X) is a positive Radon measure on X.
We call a pair (U, ϕ) such that U ⊂ X is a Borel set and ϕ : X → R d is Lipschitz, a d-dimensional chart, or simply a d-chart. A function f : X → R is said to be differentiable with respect to a d-chart We call a metric measure space (X, ρ, µ) a Lipschitz differentiability space (also called a metric measure space that admits a measurable differentiable structure) if there exists a countable family of

Alberti representations.
We denote by Γ(X) the set of curves in X, that is, the set of all Lipschitz maps γ : Dom γ → X, for which the domain Dom γ ⊂ R is non-empty and compact. Note that we are not requiring Dom γ to be an interval and thus the set Γ(X) is sometimes also called the set of curve fragments on X. We equip Γ(X) with the Hausdorff metric dist H on graphs and we consider it as a subspace of the Polish space Definition 2.1. Let (X, ρ, µ) be a metric measure space. An Alberti representation of µ on a µ-measurable set A ⊂ X is a parametrized family (µ γ ) γ∈Γ(X) of positive Borel measures µ γ ∈ M + (X) with together with a Borel probability measure π ∈ P(Γ(X)) such that Here, the measurability of the integrand is part of the requirement of being an Alberti representation Remark 2.2. Note that this definition is slightly different from the one in [Bat15, Definition 2.2] since there the set Γ(X) consist of bi-Lipschitz curves. Clearly, the existence of a representation in the sense of [Bat15] implies the existence of a representation in our sense and this will suffice for our purposes. Let us, however, point out that the converse holds true as well. Indeed, the part of γ that contributes to the integral in (2.2) can be decomposed into countably many bi-Lipschitz pieces, see [Sch16, Remark 2.17].
We will further need the notion of independent Alberti-representations of a measure. Let C ⊂ R d be a closed, convex, one-sided cone, i.e. a set of the form for π-a.e. curve γ and H 1 -a.e. t ∈ Dom γ.
A number of m Alberti representations of µ are ϕ-independent if there are linearly independent cones C 1 , . . . , C m such that the i'th Alberti representation has ϕ-directions in C i . Here, linear independence of the cones C 1 , . . . , C m means that any collection of vectors v i ∈ C i \ {0} is linearly independent. In the case X = R d we will always consider ϕ = Id. One of the main results of [Bat15] asserts that a Lipschitz differentiability space necessarily admits many independent Alberti representations, also cf. [AM16, Theorem 1.1]. Recall that according to Remark 2.2 any representation in the sense of [Bat15] is also a representation in the sense of Definition 2.1.
Theorem 2.3. Let (X, ρ, µ) be a Lipschitz differentiability space with a dchart (U, ϕ). Then, there exists a countable decomposition such that every µ U k has d ϕ-independent Alberti representations.
A proof of this theorem can be found in [Bat15, Theorem 6.6].
2.3. One-dimensional currents. In order to use the results of [DR16] we need a link between Alberti representation and 1-dimensional currents. Recall that a 1-dimensional current T in R d is a continuous linear functional on the space of smooth and compactly supported differential 1-forms on R d . The boundary of T , ∂T is the distribution (0-current) defined via ∂T, f := T, df for every smooth and compactly supported function f : R d → R. The mass of T , denoted by M(T ), is the supremum of T, ω over all 1-forms ω such that |ω| ≤ 1 everywhere. In particular, finite-mass currents can be naturally identified with R d -valued Radon measures. A current T is called normal if both T and ∂T have finite mass; we denote the set of normal 1-currents by N 1 (R d ).
By the Radon-Nikodým theorem, a 1-dimensional current T with finite mass can be written in the form T = T T where T is a finite positive measure and T is a vector field in L 1 (R d , T ) with | T (x)| = 1 for T -almost every x ∈ R d . In particular, the action of T on a smooth and compactly supported 1-form ω is given by An integer-multiplicity rectifiable 1-current (in the following called simply rectifiable 1-current) T = E, τ, m is a 1-current which acts on 1-forms ω as where E is a 1-rectifiable set, τ (x) is a unit vector spanning the approximate tangent space Tan(E, x) and m is an integer-valued function such that E m dH 1 < ∞. More information on currents can be found in [Fed69].
The relation between Alberti representations and normal 1-currents is partially encoded in the following decomposition theorem, due to Smirnov [Smi93].
Theorem 2.4. Let T = T T ∈ N 1 (R d ) be a normal 1-current with | T (x)| = 1 for T -almost every x. Then, there exists a family of rectifiable 1-currents where Γ is a measure space endowed with a finite positive Borel measure π ∈ M + (Γ), such that the following assertions hold: (i) T can be decomposed as and (ii) τ γ (x) = T (x) for H 1 -almost every x ∈ E γ and for π-almost every γ ∈ Γ; (iii) T can be decomposed as where each µ γ is the restriction of H 1 to the 1-rectifiable set E γ .
An Alberti representation of a Euclidean measure splits it into measures concentrated on "fragments" of curves. In general, these fragments cannot be glued together to obtain a 1-dimensional normal current since the boundary may have infinite mass. Nevertheless, the "holes" of every curve appearing in an Alberti representation of a measure ν ∈ M + (R d ) can be "filled" in such a way as to produce a normal 1-current T with ν ≪ T . Moreover, if the representation has directions in a cone C, then the constructed normal current T has orienting vector T in C \ {0} almost everywhere (with respect to T ). Indeed, we have the following lemma, which is essentially [AM16, Corollary 6.5]; it can be interpreted as a partial converse to Theorem 2.4: Lemma 2.5. Let ν ∈ M + (R d ) be a finite Radon measure. If there is an Alberti representation ν = ν γ dπ(γ) with directions in a cone C, then there exists a normal 1-current T ∈ N 1 (R d ) such that T (x) ∈ C \{0} for T -almost every x ∈ R d and ν ≪ T .
Proof. For the purpose of illustration we sketch the proof.
Step 1. Given ν as in the statement, we claim that there exists a normal 1-current T = T T with M(T ) ≤ 1 and M(∂T ) ≤ 2 such that T (x) ∈ C, for T -almost every x and that ν is not singular with respect to T .
The claim follows from the proof of [AM16, Lemma 6.12]. For the sake of completeness let us present the main line of reasoning. By arguing as in Step 1 of the proof of [AM16, Lemma 6.12], to every γ ∈ Γ(R d ) with γ ′ (t) ∈ C and a Borel measure ν γ ≪ H 1 Im γ, we can associate a 1-Lipschitz map ψ νγ : [0, 1] → R d satisfying . This map can moreover be chosen such that γ → ψ νγ coincides with a Borel measurable map π-almost everywhere once we endow the set of curves with the topology of uniform convergence, see Step 3 in the proof of [AM16, Lemma 6.12].
as measures (here we are identifying T with an R d -valued Radon measure and use the pointwise scalar product). Moreover, as a consequence of the Radon-Nikodým theorem, for every T ∈ T ν we may write Let us set M := sup T ∈Tν g T d T > 0 and let T k ∈ T ν be a sequence with In particular, T ∈ T ν and h k ≤ g T . Set m k = max 1≤j≤k h j . By the monotone convergence theorem, m k → m ∞ ≤ g T in L 1 (R d , T ) and Hence, M is actually a maximum and it is attained by T . We now claim that ν ≪ T . Indeed, assume by contradiction that ν = g T d T + ν s T with ν s T = 0. Since the Alberti representation of ν induces an Alberti representation of ν s T , we can apply Step 1 to find a normal 1current S ∈ T ν s T ⊂ T ν such that ν s T and S are not mutually singular. In particular, if ν = g S d S + ν s S , then there exists a Borel set F ⊂ R d such that Let us define W := (T + S)/2 and note that by (2.3) it holds that T , S ≪ W so that W ∈ T ν . Moreover, there are functions h T , h S ≤ g W such that for all Borel sets E. However, for F as in (2.4) we obtain a contradiction.

Proof of Cheeger's conjecture
The key tool to prove Cheeger's conjecture is the following result from [DR16, Corollary 1.12]: Then, ν ≪ L d .
Combining the above result with Lemma 2.5 we immediately get the following: Proof. Denote by C 1 , . . . , C d independent cones such that there are d Alberti representations having directions in these cones. By Lemma 2.5 there are d normal 1-dimensional currents and T i (x) ∈ C i for ν-almost every x ∈ R d . By the independence of the cones, This implies ν ≪ L d via Theorem 3.1.
In order to use the above result to prove Theorem 1.1 one further needs the following "push-forward lemma". Lemma 3.3. Let (X, ρ, µ) be a Lipschitz differentiability space with a d-chart (U, ϕ). If µ U has d ϕ-independent Alberti representations, then also the push-forward ϕ # (µ U ) ∈ M + (R d ) has d independent Alberti representations.
Proof of Theorem 1.1. Let (U, ϕ) be a d-chart. By Theorem 2.3 there are d ϕ-independent Alberti representations of µ U k , where U = k∈N U k is the decomposition from Bate's theorem. Then, via Lemma 3.3, the push-forward ϕ # (µ U k ) also has d independent Alberti representations. Finally, Lemma 3.2 yields ϕ # (µ U k ) ≪ L d and this concludes the proof.