On quotients of spaces with Ricci curvature bounded below by Galaz-García, F, Kell, M, Mondino, A, Sosa, G

“…It is well known that a lower bound of the sectional curvature of (M,g) is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. …”

Measure rigidity of Ricci curvature lower bounds by Cavalletti, F, Mondino, A

“…The measure contraction property, for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. …”

Nonlinear diffusion equations and curvature conditions in metric measure spaces by Ambrosio, L, Mondino, A, Savaré, G

“…The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). …”

The metric measure boundary of spaces with Ricci curvature bounded below by Brué, E, Mondino, A, Semola, D

A review of Lorentzian synthetic theory of timelike Ricci curvature bounds by Cavalletti, F, Mondino, A

“…The goal of this survey is to give a self-contained introduction to synthetic timelike Ricci curvature bounds for (possibly non-smooth) Lorentzian spaces via optimal transport and entropy tools, including a synthetic version of Hawking’s singularity theorem and a synthetic characterisation of Einstein’s vacuum equations. …”

Sectional and intermediate Ricci curvature lower bounds via optimal transport by Ketterer, C, Mondino, A

“…More generally we characterize, via optimal transport, lower bounds on the so called p-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on p-dimensional planes, 1≤p≤n. …”

Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds by Mondino, A, Naber, A

“…We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. …”

On the volume measure of non-smooth spaces with Ricci curvature bounded below by Kell, M, Mondino, A

Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures by Barilari, D, Mondino, A, Rizzi, L

“…Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a “great unification” of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds, as put forth in [87, Sec. 9]. …”

Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds by Cavalletti, F, Mondino, A

“…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 [15, 16] 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.…”

Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures by Barilari, D, Mondino, A, Rizzi, L

“…Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds. With the aim of achieving such a unification program, in this paper we initiate the study of gauge metric measure spaces.…”

Almost euclidean isoperimetric inequalities in spaces satisfying local Ricci curvature lower bounds by Cavalletti, F, Mondino, A

“…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.…”

Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds by Cavalletti, F, Mondino, A

“…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. …”

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds by Cavalletti, F, Mondino, A

“…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, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.…”

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds by Mondino, A, Semola, D

“…The class of RCD(K,N) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. …”

Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications by Cavalletti, F, Mondino, A

“…The second one is to give a synthetic notion of “timelike Ricci curvature bounded below and dimension bounded above” for a measured Lorentzian pre-length space using optimal transport. …”

Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions by Mondino, A, Nardulli, S

“…We prove existence of isoperimetric regions for every volume in non-compact Riemannian n-manifolds (M, g), n ≥ 2, having Ricci curvature Ricg ≥ (n -1) κ0g and being C0-locally asymptotic to the simply connected space form of constant sectional curvature κ0 ≤ 0; moreover in case κ0=0 we show that the isoperimetric regions are indecomposable. …”

Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature by Ikoma, N, Malchiodi, A, Mondino, A

“…The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. …”

Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below by Mondino, A, Semola, D

“…We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by and dimension bounded above by in a synthetic sense, the so called spaces. …”

Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces by De Ponti, N, Mondino, A

“…Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called RCD(K,∞) spaces.…”