Calculus of Variations and Geometric Measure Theory
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F. Cavalletti - E. Milman

The Globalization Theorem for the Curvature Dimension Condition

created by cavallett on 23 Dec 2016

[BibTeX]

Preprint

Inserted: 23 dec 2016
Last Updated: 23 dec 2016

Year: 2016

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 $m(X) < \infty$) verifying the local Curvature-Dimension condition $CD_{loc}(K,N)$ with parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$, 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 $CD^{1}(K,N)$.


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