The Globalization Theorem for the Curvature-Dimension Condition

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 $(X,\mathrm{d},\mathrm{m})$ (so that $(\mathrm{supp}(\mathrm{m}),\mathrm{d})$ is a length-space and $\mathcal(X) < \infty$) verifying the local Curvature-Dimension condition $\mathrm{CD}_{loc}(K,N)$ with parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$, also verifies the global Curvature-Dimension condition $\mathrm{CD}(K,N)$. In other words, the Curvature-Dimension condition enjoys the local-to-global property, rendering all other synthetic notions of Curvature-Dimension such as $\mathrm{CD}^*$ and $\mathrm{CD}^e$ equivalent to $\mathrm{CD}$ in the above setting.

In the talk, we will try to sketch (at least some of) the main new ingredients of the proof: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated interpolating Kantorovich potentials; a surprising third-order bound on 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. The change-of-variables formula is obtained via a new synthetic notion of Curvature-Dimension we dub $\mathrm{CD}^{1}(K,N)$.

This is joint work with Fabio Cavalletti.

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