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 (text {supp}({mathfrak {m}}),mathsf {d}) is a length-space and {mathfrak {m}}(X) < infty ) verifying the local Curvature-Dimension condition {mathsf {CD}}_{loc}(K,N) with parameters K in {mathbb {R}} and N in (1,infty ), also verifies the global Curvature-Dimension condition {mathsf {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 L^1- and L^2-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.
Highlights
The Curvature-Dimension condition CD(K, N ) was first introduced in the 1980’s by Bakry and Émery [15,16] in the context of diffusion generators, having in mind primarily the setting of weighted Riemannian manifolds, namely smooth Riemannian manifolds endowed with a smooth density with respect to the Riemannian volume
Following the works of Cordero-Erausquin– McCann–Schmuckenschläger [33], Otto–Villani [62] and von Renesse–Sturm [70], it was realized that the CD(K, ∞) condition in the smooth setting may be equivalently formulated synthetically as a certain convexity property of an entropy functional along W2 Wasserstein geodesics
We conclude this work with several brief remarks and suggestions for further investigation
Summary
Milman 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. Numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, all be equivalent in the above setting, thereby unifying the theory. 12 Combining change-of-variables formula with Kantorovich 3rd order information . 12 Combining change-of-variables formula with Kantorovich 3rd order information . . . 107
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.