Non-Hilbertian tangents to Hilbertian spaces
We provide examples of infinitesimally Hilbertian, rectifiable, Ahlfors regular metric measure spaces having pmGH-tangents that are not infinitesimally Hilbertian.
math.MG↗
arXiv subjects
Enrico Pasqualetto
Publications and source records attributed to Enrico Pasqualetto.
Abstract and concrete tangent modules on Lipschitz differentiability spaces
We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from a recent article by Bate--Kangasniemi--Orponen, this equivalence is used to show that the ${\rm Lip}-{\rm lip}$ -type condition ${\rm lip} f\le C|Df|$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to ${\rm lip} f =|Df|$. We also provide a direct proof of a result by Gigli and the second author that, for a space with a strongly rectifiable decomposition, Gigli's tangent module admits an isometric embedding into the so-called Gromov--Hausdorff tangent module, without any a priori reflexivity assumptions.
math.MG↗
Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces
The aim of this paper is threefold. We first prove that, on $\mathrm{RCD}(K,N)$ spaces, the boundary measure of any set with finite perimeter is concentrated on the $n$-regular set $\mathcal{R}_n$, where $n\le N$ is the essential dimension of the space. After, we discuss localization properties of the unit normal providing representation formulae for the perimeter measure of intersections and unions of sets with finite perimeter. Finally, we study Gauss-Green formulae for essentially bounded divergence measure vector fields, sharpening the analysis in \cite{BuffaComiMiranda19}.\\ These tools are fundamental for the development of a regularity theory for local perimeter minimizers on $\mathrm{RCD}(K,N)$ spaces in \cite{MondinoSemola21}.
math.MG↗
Isoperimetric sets in spaces with lower bounds on the Ricci curvature
In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathcal{H}^N)$, with $K\in\mathbb R$, $N\geq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in ${\sf RCD}(K,N)$ spaces $({\rm X},{\sf d},\mathfrak m)$ and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters. The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.
math.DG↗
On master test plans for the space of BV functions
We prove that on an arbitrary metric measure space a countable collection of test plans is sufficient to recover all $\rm BV$ functions and their total variation measures. In the setting of non-branching ${\sf CD}(K,N)$ spaces (with finite reference measure), we can additionally require these test plans to be concentrated on geodesics.
math.MG↗
Representation theorems for normed modules
In this paper we study the structure theory of normed modules, which have been introduced by Gigli. The aim is twofold: to extend von Neumann's theory of liftings to the framework of normed modules, thus providing a notion of precise representative of their elements; to prove that each separable normed module can be represented as the space of sections of a measurable Banach bundle. By combining our representation result with Gigli's differential structure, we eventually show that every metric measure space (whose Sobolev space is separable) is associated with a cotangent bundle in a canonical way.
math.FA↗
Parallel transport on non-collapsed $\mathsf{RCD}(K,N)$ spaces
We provide a general theory for parallel transport on non-collapsed ${\sf RCD}$ spaces obtaining both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of sufficiently regular time dependent vector fields: the price that we pay for this generality is that we cannot study parallel transport along a single such curve, but only along almost all of these (in a sense related to the notions of Sobolev vector calculus and Regular Lagrangian Flow in the nonsmooth setting). The class of ${\sf ncRCD}$ spaces contains finite dimensional Alexandrov spaces with curvature bounded from below, thus our construction provides a way of speaking about parallel transport in this latter setting alternative to the one proposed by Petrunin (1998). The precise relation between the two approaches is yet to be understood.
math.DG↗
Decomposition of integral metric currents
In the setting of complete metric spaces, we prove that integral currents can be decomposed as a sum of indecomposable components. In the special case of one-dimensional integral currents, we also show that the indecomposable ones are exactly those associated with injective Lipschitz curves or injective Lipschitz loops, therefore extending Federer's characterisation to metric spaces. Moreover, some applications of our main results will be discussed.
math.MG↗
Ultralimits of pointed metric measure spaces
The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured Gromov-Hausdorff convergence. We also introduce a weaker variant of pointed measured Gromov-Hausdorff convergence, for which we can prove a version of Gromov's compactness theorem by using the ultralimit machinery. This compactness result shows that, a posteriori, our newly introduced notion of convergence is equivalent to the pointed measured Gromov one. Another byproduct of our ultralimit construction is the identification of direct and inverse limits in the category of pointed metric measure spaces.
math.MG↗
Gamma-convergence of Cheeger energies with respect to increasing distances
We prove a $Γ$-convergence result for Cheeger energies along sequences of metric measure spaces, where the measure space is kept fixed, while distances are monotonically converging from below to the limit one. As a consequence, we show that the infinitesimal Hilbertianity condition is stable under this kind of convergence of metric measure spaces.
math.MG↗
Direct limits of infinite-dimensional Carnot groups
We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.
math.MG↗
Characterisation of upper gradients on the weighted Euclidean space and applications
In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.
math.FA↗
Testing the Sobolev property with a single test plan
We prove that in a vast class of metric measure spaces (namely, those whose associated Sobolev space is separable) the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in the weak upper gradient inequality, it suffices to consider the negligible sets of a suitable Borel measure on curves, rather than the ones of the $p$-modulus. Moreover, on $\sf RCD$ spaces we can improve our result, showing that the test plan can be also chosen to be concentrated on an equi-Lipschitz family of curves.
math.FA↗
A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
math.FA↗
Rectifiability of RCD(K,N) spaces via $δ$-splitting maps
In this note we give new proofs of rectifiability of RCD(K,N) spaces as metric measure spaces and lower semicontinuity of the essential dimension, via $δ$-splitting maps. The arguments are inspired by the Cheeger-Colding theory for Ricci limits and rely on the second order differential calculus developed by Gigli and on the convergence and stability results by Ambrosio-Honda.
math.MG↗
Universal infinitesimal Hilbertianity of sub-Riemannian manifolds
We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.
math.MG↗
Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces
This note is devoted to the study of sets of finite perimeter over RCD$(K,N)$ metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss-Green integration by parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits.
math.MG↗
Indecomposable sets of finite perimeter in doubling metric measure spaces
We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincaré inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure.
math.MG↗