arXiv subjects
Publications and source records attributed to Fabio Cavalletti.
We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.
We introduce a variational time discretization for the multi-dimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each timestep requires the minimization of a functional measuring the acceleration of fluid elements, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations. For one space dimension, we obtain sticky particle solutions for the pressureless case.
The scope of this note is to make a self-contained survey of the recent developments and achievements of the theory of L1-Optimal Transportation on metric measure spaces. Among the results proved in the recent papers [20, 21] where the author, together with A. Mondino, proved a series of sharp (and in some cases rigid) geometric and functional inequalities in the setting of metric measure spaces enjoying a weak form of Ricci curvature lower bound, we review the proof of the Lévy-Gromov isoperimetric inequality.
Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.
We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger formulation in terms of the perimeter.
In this note we prove that in a metric measure space $(X, d, m)$ verifying the measure contraction property with parameters $K \in \mathbb{R}$ and $1< N< \infty$, any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $m$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.
The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Levy-Gromov inequality and prove that, in two-dimensions, this criticality condition is quite rigid, as it characterizes round spheres and projective planes.
We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds and Alexandrov spaces with curvature bounded from below; the result seems new even in these celebrated classes of spaces.
For metric measure spaces verifying the reduced curvature-dimension condition $CD^*(K,N)$ we prove a series of sharp functional inequalities under the additional assumption of essentially non-branching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and more generally $RCD^*(K,N)$-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds, etc. In particular we prove Brunn-Minkowski inequality, $p$-spectral gap (or equivalently $p$-Poincaré inequality) for any $p\in [1,\infty)$, log-Sobolev inequality, Talagrand inequality and finally Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; if this extra sharpening is suppressed, all the previous inequalities for essentially non-branching $CD^*(K,N)$ spaces take the same form of the corresponding ones holding for a weighted Riemannian manifold verifying curvature-dimension condition $CD(K,N)$ in the sense of Bakry-Émery. In this sense inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the $p$-spectral gap. Finally let us mention that for essentially non-branching metric measure spaces, the local curvature-dimension condition $CD_{loc}(K,N)$ is equivalent to the reduced curvature-dimension condition $CD^*(K,N)$. Therefore we also have shown that sharp Brunn-Minkowski inequality in the \emph{global} form can be deduced from the \emph{local} curvature-dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature-dimension condition $CD(K,N)$.
The measure contraction property, $\mathsf{MCP}$ for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space. We start our investigation from the euclidean case by proving that if a positive Radon measure $\mathsf{m}$ over $\mathbb{R}^{d}$ is such that $(\mathbb{R}^{d},|\cdot |, \mathsf{m})$ verifies a weaker variant of $\mathsf{MCP}$, then its support $\text{spt}(\mathsf{m})$ must be convex and $\mathsf{m}$ has to be absolutely continuous with respect to the relevant Hausdorff measure of $\text{spt}(\mathsf{m})$. This result is then used as a starting point to investigate the rigidity of $\mathsf{MCP}$ in the metric framework. We introduce the new notion of $reference \ measure$ for a metric space and prove that if $(X,\mathsf{d},\mathsf{m})$ is essentially non-branching and verifies $\mathsf{MCP}$, and $μ$ is an essentially non-branching $\mathsf{MCP}$ reference measure for $(\text{spt}(\mathsf{m}), \mathsf{d})$, then $\mathsf{m}$ is absolutely continuous with respect to $μ$, on the set of points where an inversion plan exists. As a consequence, an essentially non-branching $\mathsf{MCP}$ reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability property for reference measures under measured Gromov-Hausdorff convergence, provided an additional uniform bound holds. In the final part we present concrete examples of metric spaces with reference measures, both in smooth and non-smooth setting.
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space contains at least $n$ distinct tangent lines, obtained as the blow-up of $n$ Lipschitz curves, where $n$ is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these $n$ distinct tangent lines span an $n$-dimensional part of the tangent space.
Sticky particle solutions to the one-dimensional pressureless gas dynamics equations can be constructed by a suitable metric projection onto the cone of monotone maps, as was shown in recent work by Natile and Savaré. Their proof uses a discrete particle approximation and stability properties for first order differential inclusions. Here we give a more direct proof that relies on a result by Haraux on the differentiability of metric projections. We apply the same method also to the one-dimensional Euler-Poisson system, obtaining a new proof for the global existence of weak solutions.
Let $(X,d,m)$ be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided $(X,d,m)$ satisfies a new weak property concerning the behavior of $m$ under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property. We also prove a stability property for Assumption 1: If $(X,d,m)$ satisfies Assumption 1 and $\tilde m = g\cdot m$, for some continuous function $g >0$, then also $(X,d,\tilde m)$ verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.
We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space $(X,d,m)$ enjoying the Riemannian curvature-dimension condition $\RCD(K,N)$, with $N < \infty$. For the first marginal measure, we assume that $μ_{0} \ll m$. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value. Moreover we prove a structure theorem for $d$-cyclically monotone sets: neglecting a set of zero $m$-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.
We will prove a decomposition for Wasserstein geodesics in the following sense: let $(X,d,m)$ be a non-branching metric measure space verifying $\mathsf{CD}_{loc}(K,N)$ or equivalently $\mathsf{CD}^{*}(K,N)$. We prove that every geodesic $μ_{t}$ in the $L^{2}$-Wasserstein space, with $μ_{t} \ll m$, is decomposable as the product of two densities, one corresponding to a geodesic with support of codimension one verifying $\mathsf{CD}^{*}(K,N-1)$, and the other associated with a precise one dimensional measure, provided the length map enjoys local Lipschitz regularity. The motivation for our decomposition is in the use of the component evolving like $\mathsf{CD}^{*}$ in the globalization problem. For a particular class of optimal transportation we prove the linearity in time of the other component, obtaining therefore the global $\mathsf{CD}(K,N)$ for $μ_{t}$. The result can be therefore interpret as a globalization theorem for $\mathsf{CD}(K,N)$ for this class of optimal transportation, or as a ``self-improving property'' for $\mathsf{CD}^{*}(K,N)$. Assuming more regularity, namely in the setting of infinitesimally strictly convex metric measure space, the one dimensional density is the product of two differentials giving more insight on the density decomposition.
We prove that every element of the polar cone to the closed convex cone of monotone transport maps can be represented as the divergence of a measure field taking values in the positive definite matrices.
Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous of a result proved for real valued Lipschitz maps defined on R2 by Alberti, Bianchini and Crippa in [1].
Let $(X,d,m)$ be a geodesic metric measure space. Consider a geodesic $μ_{t}$ in the $L^{2}$-Wasserstein space. Then as $s$ goes to $t$ the support of $μ_{s}$ and the support of $μ_{t}$ have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. We consider for each $t$ the set of times for which a geodesic belongs to the support of $μ_{t}$ and we prove that $t$ is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying $\mathsf{CD}(K,\infty)$. The non branching property is not needed.