The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for ametric-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, d, m) (so that (supp(m), d) is a length-space and m( X) < infinity) verifying the local Curvature-Dimension condition CDloc( K, N) with parameters K is an element of R and N is an element of(1, infinity), 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 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.
The globalization theorem for the Curvature-Dimension condition / Cavalletti, Fabio; Milman, Emanuel. - In: INVENTIONES MATHEMATICAE. - ISSN 0020-9910. - 226:1(2021), pp. 1-137. [10.1007/s00222-021-01040-6]
The globalization theorem for the Curvature-Dimension condition
Cavalletti, Fabio
;
2021-01-01
Abstract
The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for ametric-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, d, m) (so that (supp(m), d) is a length-space and m( X) < infinity) verifying the local Curvature-Dimension condition CDloc( K, N) with parameters K is an element of R and N is an element of(1, infinity), 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 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.