0000000000932355
AUTHOR
Nicola Gigli
showing 13 related works from this author
On the notion of parallel transport on RCD spaces
2019
We propose a general notion of parallel transport on RCD spaces, prove an unconditioned uniqueness result and existence under suitable assumptions on the space. peerReviewed
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below
2013
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 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.
The Theory of Normed Modules
2020
This chapter is devoted to the study of the so-called normed modules over metric measure spaces. These represent a tool that has been introduced by Gigli in order to build up a differential structure on nonsmooth spaces. In a few words, an \(L^2({{\mathfrak {m}}})\)-normed \(L^\infty ({{\mathfrak {m}}})\)-module is a generalisation of the concept of ‘space of 2-integrable sections of some measurable bundle’; it is an algebraic module over the commutative ring \(L^\infty ({{\mathfrak {m}}})\) that is additionally endowed with a pointwise norm operator. This notion, its basic properties and some of its technical variants constitute the topics of Sect. 3.1.
Quasi-Continuous Vector Fields on RCD Spaces
2021
In the existing language for tensor calculus on RCD spaces, tensor fields are only defined $\mathfrak {m}$ -a.e.. In this paper we introduce the concept of tensor field defined ‘2-capacity-a.e.’ and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.
Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below
2013
We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.
First-Order Calculus on Metric Measure Spaces
2020
In this chapter we develop a first-order differential structure on general metric measure spaces. First of all, the key notion of cotangent module is obtained by combining the Sobolev calculus (discussed in Chap. 2) with the theory of normed modules (described in Chap. 3). The elements of the cotangent module L2(T∗X), which are defined and studied in Sect. 4.1, provide a convenient abstraction of the concept of ‘1-form on a Riemannian manifold’.
Differential of metric valued Sobolev maps
2020
We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is $\mathbb{R}$.
Differential structure associated to axiomatic Sobolev spaces
2020
The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure. peerReviewed
Sobolev Calculus on Metric Measure Spaces
2020
Several different approaches to the theory of weakly differentiable functions over abstract metric measure spaces made their appearance in the literature throughout the last twenty years. Amongst them, we shall mainly follow the one (based upon the concept of test plan) that has been proposed by Ambrosio, Gigli and Savare. The whole Sect. 2.1 is devoted to the definition of such notion of Sobolev space W1, 2(X) and to its most important properties.
Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure
2015
In a prior work of the first two authors with Savar´e, a new Riemannian notion of a lower bound for Ricci curvature in the class of metric measure spaces (X, d, m) was introduced, and the corresponding class of spaces was denoted by RCD(K,∞). This notion relates the CD(K, N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In this prior work the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In the above-mentioned work only finite reference measure…
Second-Order Calculus on RCD Spaces
2020
In this conclusive chapter we introduce the class of those metric measure spaces that satisfy the Riemannian curvature-dimension condition, briefly called RCD spaces, and we develop a thorough second-order differential calculus over these structures.
Heat Flow on Metric Measure Spaces
2020
In order to develop a second-order differential calculus on spaces with curvature bounds we need to make use of the regularising effects of the heat flow, to which this chapter is dedicated.
Monotonicity Formulas for Harmonic Functions in RCD(0,N) Spaces
2023
We generalize to the RCD(0,N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in RCD(K,N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric…