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 $$(X,\\mathsf {d},{\\mathfrak {m}})$$\n \n (\n X\n ,\n d\n ,\n m\n )\n \n (so that $$(\\text {supp}({\\mathfrak {m}}),\\mathsf {d})$$\n \n (\n supp\n (\n m\n )\n ,\n d\n )\n \n is a length-space and $${\\mathfrak {m}}(X) < \\infty $$\n \n m\n (\n X\n )\n <\n ∞\n \n ) verifying the local Curvature-Dimension condition $${\\mathsf {CD}}_{loc}(K,N)$$\n \n \n CD\n \n loc\n \n \n \n (\n K\n ,\n N\n )\n \n \n with parameters $$K \\in {\\mathbb {R}}$$\n \n K\n ∈\n R\n \n and $$N \\in (1,\\infty )$$\n \n N\n ∈\n (\n 1\n ,\n ∞\n )\n \n , also verifies the global Curvature-Dimension condition $${\\mathsf {CD}}(K,N)$$\n \n CD\n (\n K\n ,\n N\n )\n \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 $$L^1$$\n \n L\n 1\n \n - and $$L^2$$\n \n L\n 2\n \n -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.\n