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Nicola Gigli

Publications and source records attributed to Nicola Gigli.

At least 73 records · Page 4Linked to original sources
arXiv2016

Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces

We prove that for a suitable class of metric measure spaces, the abstract notion of tangent module as defined by the first author can be isometrically identified with the space of $L^2$-sections of the `Gromov-Hausdorff tangent bundle'. The class of spaces $({\rm X},{\sf d},{\mathfrak m})$ we consider are PI spaces that for every $\varepsilon>0$ admit a countable collection of Borel sets $(U_i)$ covering ${\mathfrak m}$-a.e.\ ${\rm X}$ and corresponding $(1+\varepsilon)$-biLipschitz maps $φ_i:U_i\to{\mathbb R}^{k_i}$ such that $(φ_i)_*{\mathfrak m}\lower3pt\hbox{$|_{U_i}$}\ll\mathcal L^{k_i}$. This class is known to contain ${\sf RCD}^*(K,N)$ spaces. Part of the work we carry out is that to give a meaning to notion of $L^2$-sections of the Gromov-Hausdorff tangent bundle, in particular explaining what it means to have a measurable map assigning to ${\mathfrak m}$-a.e.\ $x\in {\rm X}$ an element of the pointed-Gromov-Hausdorff limit of the blow-up of ${\rm X}$ at $x$.

math.DG↗
arXiv2015

Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds

The aim of the present paper is to bridge the gap between the Bakry-Émery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start from a strongly local Dirichlet form ${\mathcal{E}}$ admitting a Carré du champ $Γ$ in a Polish measure space $(X,\mathfrak{m})$ and a canonical distance ${\mathsf{d}}_{\mathcal{E}}$ that induces the original topology of $X$. We first characterize the distinguished class of Riemannian Energy measure spaces, where ${\mathcal{E}}$ coincides with the Cheeger energy induced by ${\mathsf{d}}_{\mathcal{E}}$ and where every function $f$ with $Γ(f)\le1$ admits a continuous representative. In such a class, we show that if ${\mathcal{E}}$ satisfies a suitable weak form of the Bakry-Émery curvature dimension condition $\mathrm {BE}(K,\infty)$ then the metric measure space $(X,{\mathsf{d}},\mathfrak{m})$ satisfies the Riemannian Ricci curvature bound $\mathrm {RCD}(K,\infty)$ according to [Duke Math. J. 163 (2014) 1405-1490], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Émery $\mathrm {BE}(K,N)$ condition (and thus the corresponding one for $\mathrm {RCD}(K,\infty)$ spaces without assuming nonbranching) and the stability of $\mathrm {BE}(K,N)$ with respect to Sturm-Gromov-Hausdorff convergence.

math.FA↗
arXiv2014

The continuity equation on metric measure spaces

Aim of this paper is to show that it makes sense to write the continuity equation on a metric measure space $(X,d,m)$ and that absolutely continuous curves $μ_t$ w.r.t. the distance $W_2$ can be completely characterized as solutions of the continuity equation itself, provided we impose the condition $μ_t\leq Cm$ for every $t$ and some $C>0$.

math.AP↗
arXiv2014

The abstract Lewy-Stampacchia inequality and applications

We prove an abstract and general version of the Lewy-Stampacchia inequality. We then show how to reproduce more classical versions of it and, more importantly, how it can be used in conjunction with Laplacian comparison estimates to produce large class of functions with bounded Laplacian on spaces with a lower bound on the Ricci curvature.

math.MG↗
arXiv2013

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured-Gromov-Hausdorff convergence. Then we show that the curvature conditions $CD(K,\infty)$ and $RCD(K,\infty)$ are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the $L^2$-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted $Γ$-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the $RCD(K,\infty)$ condition with $K>0$. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.

math.MG↗
arXiv2013

On the differential structure of metric measure spaces and applications

The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Borel, non negative and locally finite. ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $Δg=μ$, where $g$ is a function and $μ$ is a measure. iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.

math.MG↗
arXiv2013

Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian $CD^*(0,N)$-spaces.

math.MG↗
arXiv2013

Riemannian Ricci curvature lower bounds in metric measure spaces with $σ$-finite measure

Using techniques of optimal transportation and gradient flows in metric spaces, we extend the notion of Riemannian Curvature Dimension condition $RCD(K,\infty)$ introduced (in case the reference measure is finite) by Giuseppe Savare', the first and the second author, to the case the reference measure is $σ$-finite; in this way the theory includes natural examples as the euclidean $n$-dimensional space endowed with the Lebesgue measure, and noncompact manifolds with bounded geometry endowed with the Riemannian volume measure. Another major goal of the paper is to simplify the axiomatization of $RCD(K,\infty)$ (even in case of finite reference measure) replacing the assumption of strict $CD(K,\infty)$ with the classic notion of $CD(K,\infty)$.

math.DG↗
arXiv2013

Gromov-Hausdorff convergence of discrete transportation metrics

This paper continues the investigation of `Wasserstein-like' transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N converge, when $N\to\infty$, to the standard 2-Wasserstein distance W_2 on the continuous torus in the sense of Gromov-Hausdorff. This is the first convergence result for the recently developed discrete transportation metrics. The result shows the compatibility between these metrics and the well-established 2-Wasserstein metric.

math.MG↗
arXiv2012

A PDE approach to nonlinear potential theory in metric measure spaces

We show that the tools recently introduced by the first author in [9] allow to give a PDE description of p-harmonic functions in metric measure setting. Three applications are given: the first is about new results on the sheaf property of harmonic functions, the second is a PDE proof of the fact that the composition of a subminimizer with a convex and non-decreasing function is again a subminimizer, and the third is the fact that the Busemann function associated to a line is harmonic on infinitesimally Hilbertian CD(0,N) spaces.

math.MG↗
arXiv2012

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincaré inequality.

math.MG↗