Rectifiability of spaces via splitting maps
Abstract.
In this note we give new proofs of rectifiability of spaces as metric measure spaces and lower semicontinuity of the essential dimension, via splitting maps. The arguments are inspired by the CheegerColding theory for Ricci limits and rely on the second order differential calculus developed by Gigli and on the convergence and stability results by AmbrosioHonda.
Key words and phrases:
Rectifiability, space, tangent cone2010 Mathematics Subject Classification:
26B30, 26B20, 53C23Introduction
In the last years the study of metric measure spaces has undergone a fast development. After the introduction of the curvaturedimension condition in the independent works [37, 38] and [34], the notion of space was proposed in [27] as a finitedimensional counterpart of , introduced in [3] (see also [2] for the case of finite reference measure and [10] for the introduction of the reduced curvaturedimension condition ). In the infinitedimensional case the equivalence of the original Lagrangian approach with an Eulerian one, based on the Bochner inequality, was studied in [4]. Then [25] established equivalence with the dimensional Bochner inequality for the socalled class (see also [9]). Equivalence between and has been eventually achieved in [13] in the case of finite reference measure, closing the circle. Apart from smooth weighted Riemannian manifolds (with generalized Ricci tensor bounded from below), the class includes Ricci limit spaces, whose study was initiated by CheegerColding in the nineties [17, 18, 19] (see also the survey [15]), and Alexandrov spaces [36]. We refer the reader to the survey [1] for an account about this quickly developing research area.
Many efforts have been recently aimed at understanding the structure theory of spaces. After [35] by MondinoNaber, we know that they are rectifiable as metric spaces and later, in the three independent works by De PhilippisMarcheseRindler, KellMondino and Gigli together with the second named author [23, 32, 31], the analysis was sharpened taking into account the behaviour of the reference measure and getting rectifiability as metric measure spaces. Moreover, in the recent [12], the first and the third named authors proved that spaces have constant dimension, in the almost everywhere sense.
The development of this theory was inspired in turn by the results obtained for Ricci limit spaces in the seminal papers by CheegerColding (see also [22] by ColdingNaber for constancy of dimension).
In the proofs given in [17, 19] a crucial role was played by splitting maps:
Definition 0.1.
Let be an space. Let and be given. Then a map is said to be a splitting map provided:

is harmonic and Lipschitz for every ,

for every ,

for every .
These maps provide approximations, in the integral sense and up to the second order, of independent coordinate functions in the Euclidean space. They were introduced in [16], in the study of Riemannian manifolds with lower Ricci curvature bounds.
Item ii) in the definition of splitting maps is about smallness of the norm of the Hessian, in scale invariant sense. In [17, 19] and in more recent works dealing with Ricci limits as [20], splitting maps are built only at the level of the smooth approximating sequence, where there is a clear notion of Hessian available, the metric information they encode (GH closeness to Euclidean spaces) is then passed to the limit.
Prior than [26], there was no notion of Hessian available in the framework. This, together with the absence of smooth approximating sequences, motivated the necessity to find an alternative approach to rectifiability in [35, 23, 32, 31] with respect to the CheegerColding theory. A new almost splitting via excess theorem was the main ingredient playing the role of the theory of splitting maps in [35] while, studying the behaviour of the reference measure with respect to charts, a crucial role was played in both [23, 32, 31], by a recent and powerful result obtained by De PhilippisRindler [24].
Nowadays we have at our disposal both a second order differential calculus on spaces [26] and general convergence and stability results for Sobolev functions on converging sequences of spaces [5, 6]. In our previous paper [11] we exploited all these tools to prove rectifiability for reduced boundaries of sets of finite perimeter in this context. The study of [11] was devoted to the theory in codimension one, which required some additional ideas and technical efforts, but it was evident that similar arguments could provide new and more direct proofs of rectifiability for spaces in the spirit of those in [17, 19].
Taking as a starting point existence of Euclidean tangents almost everywhere with respect to the reference measure, obtained by GigliMondinoRajala in [29], in this short note we provide the arguments to get uniqueness (almost everywhere) of tangents and rectifiability of spaces as metric measure spaces via splitting maps. Moreover, we recover via a different strategy the result about lower semicontinuity of the so called essential dimension proved firstly in [33]. After Section 1, dedicated to review some preliminaries and establish the basic tool about propagation of the splitting property, the remaining Subsection 2.1, Subsection 2.2 and Subsection 2.3 are devoted to uniqueness of tangents and lower semicontinuity of the essential dimension, metric rectifiability and the behaviour of the reference measure under charts, respectively.
Acknowledgements
The authors wish to thank Luigi Ambrosio, Nicola Gigli, Andrea Mondino and Tapio Rajala for useful comments on an earlier version of this note.
The second named author was partially supported by the Academy of Finland, projects 307333 and 314789.
Part of this work was developed while the first and third named authors were visiting the Department of Mathematics and Statistics of the University of Jyvaskyla: they wish to thank the institute for the excellent working conditions and the stimulating atmosphere.
1. Preliminaries and notation
1.1. Differential calculus on metric measure spaces
For our purposes, a metric measure space is a triple , where is a proper metric space, while is a Radon measure on . Given a Lipschitz function , we will denote by its slope, which is the function defined as
and elsewhere. Given any open set , we denote by the family of all Lipschitz functions whose support is bounded and satisfies .
1.1.1. Sobolev space
Following [14], we define the Sobolev space as
where the Cheeger energy is the convex, lower semicontinuous functional
It holds that is a Banach space if endowed with the norm , given by
Given any , one can select a canonical object
– called the minimal relaxed slope of
– for which admits the integral representation
.
We have chosen to stress the dependence on the measure for the gradient and the other differential objects, here and in the sequel, to avoid confusion.
Given an open set , we define as the space of all those such that holds for every . Thanks to the locality property of the minimal relaxed slope, it makes sense to define as
Finally, we define as the space of all such that .
1.1.2. Tangent module
Whenever is a Hilbert space, we will say that is infinitesimally Hilbertian. In this case, we recall from [26] that the tangent module and the corresponding gradient map can be characterised as follows: is an normed module (in the sense of [28, Definition 1.3]) that is generated by , while is a linear map satisfying a.e. on for all . The pointwise scalar product ,
is a symmetric bilinear form, as a consequence of the infinitesimal Hilbertianity assumption.
The dual module of is denoted by and called the cotangent module of .
In the framework of weighted Euclidean spaces, we have another notion of tangent module at our disposal. Given a Radon measure on , we denote by the space of all maps from to itself. It turns out that is an normed module generated by , where stands for the ‘classical’ gradient of .
1.1.3. Divergence and Laplacian
In the setting of infinitesimally Hilbertian spaces , one can consider the following notions of divergence and Laplacian:

Divergence. We declare that belongs to provided there exists a (uniquely determined) function such that

Laplacian. Given any open set , we declare that belongs to provided there exists a (uniquely determined) function such that
where stands for the closure of in . For brevity, we shall write in place of .
The domains and are vector subspaces of and , respectively. Moreover, the operators and are linear.
It can be readily checked that a given function belongs to if and only if its gradient belongs to . In this case, it also holds that .
1.2. spaces
We assume the reader to be familiar with the language of spaces and the notion of pointed measured Gromov–Hausdorff convergence (often abbreviated to pmGH).
We recall the following scaling property: if is an space, then is an space for any choice of . Furthermore, there exists a distance on the set (of isomorphism classes) of spaces that metrises the pmGHtopology [30].
Remark 1.1.
Any sequence , of pointed spaces converges, up to the extraction of a subsequence, to some pointed space with respect to the pmGHtopology. This follows from a compactness argument due to Gromov and the stability of the condition.
1.2.1. Test functions
Let be an space. A fundamental class of Sobolev functions on is given by the algebra of test functions [26]:
Since spaces enjoy the SobolevtoLipschitz property, we know that any element of admits a Lipschitz representative. Moreover, it holds that is dense in and that for every .
Lemma 1.2 (Good cutoff functions [8, 35]).
Let be an space. Let and . Then there exists such that on , the support of is compactly contained in , and on .
We recall the notion of Hessian of a test function [26]: given , we denote by the unique element of the tensor product (cf. [26, Section 1.5]) such that
holds for every . The pointwise norm of belongs to .
Given an open set and a function , we say that is harmonic if . If in addition is Lipschitz, then one can define (the modulus of) its Hessian as follows:
(1.1) 
This way we obtain a welldefined function , thanks to the locality property of the Hessian and the fact that for every as in (1.1).
1.3. Splitting maps on spaces
In this subsection we collect the main properties of splitting maps that we will use in the sequel. Let us recall that their introduction in the study of spaces with lower Ricci curvature bounds dates back to [16] and that they have been extensively used in [17, 18, 19] and in more recent works on Ricci limits [21, 20]. Before the development of the second order calculus in [26], there was need for alternative arguments avoiding the use of the Hessian in order to develop the structure theory of spaces in [35]. In recent times (see [7, 11]) splitting maps have come into play also in the theory thanks to [26] and the stability results of [5, 6].
The results connecting splitting maps with isometries stated below are borrowed from [11]. Although being less local than those provided by the CheegerColding theory, they are sufficient for our purposes and allow for more direct proofs via compactness.
Definition 1.3 (Splitting map [11]).
Let be an space. Let and be given. Then a map is said to be a splitting map provided:

is harmonic and Lipschitz for every ,

for every ,

for every .
Proposition 1.4 (From GHisometry to splitting [11]).
Let be given. Then for any there exists such that the following property holds. If is an space, , with , and there is an space such that
then there exists a splitting map .
Proposition 1.5 (From splitting to GHisometry [11]).
Let be given. Then for any there exists such that the following property holds. If is an space, , and there exists a map such that is a splitting map for all , then for any it holds that
for some pointed space .
Below we state and prove a result about propagation of the splitting property at many locations with respect to the reference measure and at all scales. The proof is based on a standard maximal function argument.
Proposition 1.6 (Propagation of the splitting property).
Let be given. Then there exists a constant such that the following property holds. If is an space and is a splitting map for some , with , and , then there exists a Borel set such that and
Proof.
Thanks to a scaling argument, it is sufficient to prove the claim for and . Let us define as , where we set
It holds that is a splitting map for all and . To prove the claim, it remains to show that for all .
Given any , we can choose such that . By Vitali covering lemma, we can find a sequence such that are pairwise disjoint and . Therefore, it holds that
where we used the doubling property of , the defining property of and the fact that is a splitting map on . An analogous argument shows that for all , thus the statement is achieved. ∎
2. Structure theory for spaces
Given a pointed space and a radius , we define the normalised measure on as
The tangent cone is defined as the family of all those spaces such that
for some sequence of radii with . It follows from the scaling property of the condition and Remark 1.1 that any element of is a pointed space.
Let us briefly recall the properties that we take as a starting point for our analysis of the structure theory of spaces. The first one is a version of the iterated tangent property suited for this setting. Building upon this, in [29] it was proved that at almost every point there exists at least one Euclidean space in the tangent cone, on spaces (see Theorem 2.2 below).
Theorem 2.1 (Iterated tangent property [29]).
Let be an space. Then for a.e. point it holds that
Theorem 2.2 (Euclidean tangents to spaces [29]).
Let be an space. Then for a.e. point there exists with such that
where we set for every .
2.1. Uniqueness of tangent cones
Let be an space. Then we define
With a terminology borrowed from [17] and inspired by geometric measure theory, points in are called regular points of . Moreover, given any point and any , we say that an element splits off a factor provided
for some pointed space .
In [35] uniqueness of tangents (almost everywhere w.r.t. the reference measure ) was proved together with rectifiability relying on a new splitting via excess theorem (cf. [35, Theorem 6.7] and [35, Theorem 5.1]). Below we provide a new proof of uniqueness of tangents based on the same principle about propagation of regularity but more similar to the one given in [17] for Ricci limits.
Theorem 2.3 (Uniqueness of tangents).
Let be an space. Then it holds
Proof.
Step 1. Fix any with . We define the auxiliary sets as follows:

is the set of all points such that , but no other element of splits off a factor .

is the set of all points which satisfy and for every with .
Observe that . The measurability
of the sets can be proven adapting the proof
of [35, Lemma 6.1]. It also follows from Theorem 2.2
that .
Step 2. We aim to prove that .
We argue by contradiction: suppose .
Then we can find where the iterated tangent property
of Theorem 2.1 holds. Since , there exists a pointed
space with such that
Theorem 2.2 yields the existence of a point such that , for some with . This implies that
Therefore, Theorem 2.1 guarantees that
belongs to , which contradicts the fact that .
Consequently, we have proven that , as desired.
Step 3. In order to complete the proof of the statement,
it suffices to show that
(2.1) 
Let and be fixed. Choose any associated with as in Proposition 1.5. Moreover, choose any associated with as in Proposition 1.4. Given a point , we can find such that and
By applying Vitali covering lemma to the family , we obtain a sequence such that are pairwise disjoint and . For any , we know from Proposition 1.4 that there exists a splitting map . Furthermore, by Proposition 1.6 there exists a Borel set such that and is a splitting map for every and . Hence, by Proposition 1.5, for any the following property holds:
(2.2) 
Then let us define . Clearly, each element of satisfies (2.2). Moreover, it holds
(2.3) 
Now consider the Borel set . It follows from (2.3) that . Moreover, let and be fixed. Then by using (2.2) we can find a sequence of pointed spaces such that
(2.4) 
Up to a not relabelled subsequence, we can suppose that in the pmGHtopology, for some pointed space . Consequently, (2.4) ensures that is isomorphic to . Given that , we deduce that must be a singleton. In other words, we have proven that any element of is isomorphic to , so that . This shows that , whence the claim (2.1) follows. ∎
By combining Theorem 2.3 with the properties of splitting maps discussed in Section 1.3, we can give a direct proof of the following result, that was proved for the first time in [33]:
Theorem 2.4.
Let be an space. Let , be the maximal number such that . Then for any and we have that no element of splits off a factor . In particular, it holds that for every .
Proof.
First of all, we claim that for any given there exists such that
(2.5) 
for every and for every pointed space . This can be easily checked arguing by contradiction.
We prove the main statement by contradiction: suppose there exist and such that
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