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, d, m) (so that (supp(m), d) is a length-space and 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 ), meaning that the Curvature-Dimension condition enjoys the globalization (or local-to-global) property. The main new ingredients of our proof are an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order derivative of an associated Kantorovich potential; a surprising third-order bound on the latter Kantorovich potential, 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. The change-of-variables formula is obtained via a new synthetic notion of Curvature-Dimension we dub CD 1 (K, N ). Contents

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds

“…The class RCD(K, N ) was proposed in [G15], motivated by the validity of the sharp Laplacian comparison and of the Cheeger-Gromoll splitting theorem, proved in [G13]. The (a priori more general) RCD * (K, N ) condition was thoroughly analysed in [EKS15] and (subsequently and independently) in [AMS15] (see also [CM16] for the equivalence betweeen RCD * and RCD in the case of finite reference measure).…”

The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

“…After the introduction, in the independent works [63,64] and [46], of the curvature dimension condition CD(K, N) encoding in a synthetic way the notion of Ricci curvature bounded from below by K and dimension bounded above by N, the definition of RCD(K, N) m.m.s. was first proposed in [40] and then studied in [41,38,12], see also [28] for the equivalence between the RCD * (K, N) and the RCD(K, N) condition. The infinite dimensional counterpart of this notion had been previously investigated in [9], see also [8] for the case of σ-finite reference measures.…”

Isoperimetric sets in spaces with lower bounds on the Ricci curvature

“…After the introduction, in the independent works [61,62] and [45], of the curvature dimension condition CD(K, N) encoding in a synthetic way the notion of Ricci curvature bounded from below by K and dimension bounded above by N, the definition of RCD(K, N) m.m.s. was first proposed in [30] and then studied in [31,26,13], see also [20] for the equivalence between the RCD * (K, N) and the RCD(K, N) condition in the case the reference measure is finite. The infinite dimensional counterpart of this notion had been previously investigated in [11], see also [8] for the case of σ-finite reference measures.…”