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

Abstract

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 $\sigma$-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 $CD(K,\infty )$.

42 pages; final version (minor changes to the old one, in particular we added some more preliminaries and explanations) to be published in Transactions of the AMS

References in corpus (3)

• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature
• Local Poincaré inequalities from stable curvature conditions on metric spaces

Cited by in corpus (90)

• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• The splitting theorem in non-smooth context
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
• Li-Yau and Harnack type inequalities in $RC{D}^{\ast }(K,N)$ metric measure spaces
• An optimal transport formulation of the Einstein equations of general relativity
• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• Independence on $p$ of weak upper gradients on RCD spaces
• New formulas for the Laplacian of distance functions and applications
• On quotients of spaces with Ricci curvature bounded below
• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth
• Nonlinear diffusion equations and curvature conditions in metric measure spaces
• Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below
• On the universal cover and the fundamental group of an $RC{D}^{\ast }(K,N)$-space
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• Harmonic functions on metric measure spaces
• Quantitative isoperimetry à la Levy-Gromov
• On Interpolation and Curvature via Wasserstein Geodesics
• Quantitative Obata's Theorem
• Sharp Cheeger-Buser type inequalities in $\mathsf{RCD}(K,\infty )$ spaces
• Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds
• Infinitesimal Hilbertianity of weighted Riemannian manifolds
• Measure rigidity of Ricci curvature lower bounds
• Gravity from thermodynamics: optimal transport and negative effective dimensions
• The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds
• Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries
• On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces
• Hamilton's Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces
• Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces
• Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces
• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• A Talenti-type comparison theorem for $\mathrm{RCD}(K,N)$ spaces and applications
• Functional inequalities for the heat flow on time-dependent metric measure spaces
• Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows
• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Rigidity for the spectral gap on $RCD(K,\infty )$-spaces
• On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry
• A new notion of angle between three points in a metric space
• An overview on the proof of the splitting theorem in non-smooth context
• An upper bound on the revised first Betti number and a torus stability result for RCD spaces
• Characterizations of monotonicity of vector fields on metric measure spaces
• Recognizing the flat torus among $RC{D}^{\ast }(0,N)$ spaces via the study of the first cohomology group
• Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
• Heat Kernel Bounds on Metric Measure Spaces and Some Applications
• Entropy-Transport distances between unbalanced metric measure spaces
• Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry
• Sharp Heat Kernel Bounds and Entropy in Metric Measure Spaces
• From Harnack inequality to heat kernel estimates on metric measure spaces and applications
• Non-branching geodesics and optimal maps in strong CD(K,{\infty})-spaces
• On fundamental groups of RCD spaces
• Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary
• Cheeger-harmonic functions in metric measure spaces revisited
• Angles between curves in metric measure spaces
• The metric measure boundary of spaces with Ricci curvature bounded below
• On the notion of Laplacian bounds on $\mathrm{RCD}$ spaces and applications
• Coarse Ricci curvature of hypergraphs and its generalization
• Convergence of Non-Symmetric Diffusion Processes on RCD spaces
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces
• Moduli spaces of compact RCD(0,N)-structures
• Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces
• Weighted Littlewood--Paley inequalities for heat flows in $\RCD$ spaces
• Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces
• A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces
• Cheeger bounds on spin-two fields
• Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching
• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Heat and entropy flows in Carnot groups
• Ricci tensor on smooth metric measure space with boundary
• A Bishop type inequality on metric measure spaces with Ricci curvature bounded below
• Dimension free Harnack inequalities on $\RCD(K, \infty)$ spaces
• Space-time Wasserstein controls and Bakry-Ledoux type gradient estimates
• Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability
• Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature
• Cheng maximal diameter theorem for hypergraphs
• Ultralimits of pointed metric measure spaces
• Littlewood--Paley--Stein inequalities on $\text{RCD}(K,\infty )$ spaces
• Characterization of equality in Zhong-Yang type (sharp) spectral gap estimates for metric measure spaces
• Characterization of Low Dimensional $RC{D}^{\ast }(K,N)$ spaces
• Existence of optimal transport maps in very strict $CD(K,\infty )$ -spaces
• An entropy generation formula on $RCD(K,\infty )$ spaces
• Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions
• Failure of topological rigidity results for the measure contraction property
• Ricci curvature and Orientability
• Rigidity of some functional inequalities on RCD spaces
• Isometric immersions of RCD spaces
• The Measure Preserving Isometry Groups of Metric Measure Spaces
• Evolution variational inequality and Wasserstein control in variable curvature context
• Margulis Lemma on $\text{RCD}(K,N)$ spaces