The globalization theorem for the Curvature-Dimension condition

By Fabio Cavalletti, Emanuel Milman. Inventiones Mathematicae, 2016

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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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,\mathsf {d},{\mathfrak {m}})$$\end{document} (so that (supp(m),d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\text {supp}({\mathfrak {m}}),\mathsf {d})$$\end{document} is a length-space and m(X)<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {m}}(X) < \infty $$\end{document}) verifying the local Curvature-Dimension condition CDloc(K,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {CD}}_{loc}(K,N)$$\end{document} with parameters K∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \in {\mathbb {R}}$$\end{document} and N∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \in (1,\infty )$$\end{document}, also verifies the global Curvature-Dimension condition CD(K,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {CD}}(K,N)$$\end{document}. 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 L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}- and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0cec15c98e481bf1f0df7b1fb7c90d1f20d965dd

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Citations in the corpus

Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “1 Introduction In the last ten years there has been a great interest in the study of metric measure spaces with Ricci curvature bounded from below, see for instance [44], [55], [56], [29], [5], [6], [27], [7], [48], [49], [30], [24], [26], [38], [9], [47], [13], [12].”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “15) 2In [14] the case of finite measure is considered but, due to the local nature of their arguments, it is thought that the identification result extends to the general case.”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, background): “ly non-branching MCP(K,N) spaces 24 A.3. The result 25 References 27 1. Introduction The notion of RCD(K,N) metric measure spaces (m.m.s.) was proposed and analyzed in [Gig15, EKS15, AMS15] (see also [CM16a]), as a ﬁnite dimensional reﬁnement of RCD(K,∞) m.m.s. which were ﬁrst introduced and studied”