The Globalization Theorem for the Curvature Dimension Condition

2016/12/22 by Fabio Cavalletti, Emanuel Milman, Cavalletti, Fabio +1 · 67 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Bounded function #Curvature #Dimension (graph theory) #Geodesic #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematical analysis #Mathematics #Measure (data warehouse) #Order (exchange) #Pure mathematics #Ricci curvature #Space (punctuation) #math.FA #math.MG

paper · pdf · open access · doi:10.1007/s00222-021-01040-6

published in Inventiones mathematicae 226(1), 1-137 (Springer Science+Business Media) · 92 pages; polished the introduction, added some heuristic arguments to provide insight, and corrected typos. To appear in Inventiones Mathematicae

arxiv created 2021/02/24 · arxiv updated 2021/02/26 · openalex publication_date 2021/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space (X,\mathsf d,\mathfrak m) (so that (supp \mathfrak m,\mathsf d) is a length-space and \mathfrak m(X) < ∞) verifying the local Curvature-Dimension condition CD_loc(K,N) with parameters K ∈ ℝ and N ∈ (1,∞), also verifies the global Curvature-Dimension condition CD(K,N). In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between L¹ and L² optimal-transport-based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Polar factorization and monotone rearrangement of vector‐valued functions
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• An optimal Poincaré inequality for convex domains

Cited by

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• Isoperimetric problems and lower bounds on curvature
• Mosco-convergence of Cheeger energies on varying spaces satisfying curvature dimension conditions
• Stability of local Riemannian Ricci curvature lower bounds
• Non-collapsed eGH convergence and dimension
• Comparison estimates on nonsmooth spaces with integrable Ricci lower bounds via localization
• Cheng's eigenvalue comparison on metric measure spaces and applications
• Displacement convexity of Entropy and the distance cost Optimal Transportation
• Sharp Poincaré inequalities under Measure Contraction Property
• An almost rigidity Theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth
• On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces
• Rectifiability of the reduced boundary for sets of finite perimeter over RCD(K,N) spaces
• Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces
• Second order differentiation formula on compact RCD^*(K,N) spaces
• Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian
• On perimeter minimizing sets in manifolds with quadratic volume growth
• On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth. [europepmc]
• Almost-Riemannian manifolds do not satisfy the curvature-dimension condition. [europepmc]
• Moduli spaces of compact RCD(0,N)-structures. [europepmc]
• Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics. [europepmc]
• Hawking's Singularity Theorem for Lipschitz Lorentzian Metrics. [europepmc]

Related

• Local curvature-dimension condition implies measure-contraction property
• The Globalization Theorem for CD(K,N) on locally finite Spaces
• Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
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• Almost-Riemannian manifolds do not satisfy the curvature-dimension condition