The globalization theorem for the Curvature-Dimension condition

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.