On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

By Matthias Erbar, Kazumasa Kuwada, Karl‐Theodor Sturm. 2013

473 citations recorded by the source corpus.

We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Émery (via energy and $$\Gamma _2$$Γ2-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $$L^2$$L2-Wasserstein distance.

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

View Paper: https://www.semanticscholar.org/paper/e000fdda5e462cbf571b693f9066a1f8cc40f53d

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, methodology): “33) has been proved at rst in [25] in the context of nite dimensional Alexandrov spaces with curvature bounded from below with a technique which, as shown in [6], generalizes to RCD(0,∞) spaces (see also [7], [20], [9]”
Obata’s Rigidity Theorem for Metric Measure Spaces (2014, background): “It is also a simple consequence of Bochner’s inequality thatwas established for general RCD*-spaces by Erbar, Kuwada and Sturm in [8].”
Local spectral convergence in RCD∗(K,N) spaces (2017, background): “ point of view more relevant in our proofs. However, the equivalence with the Lagrangian point of view, based on the theory of optimal transport ﬁrst proved in [AGS15] (in the case N = ∞) and then in [EKS15], [AMS15] (in the case N < ∞), plays indeed a key role in the proof of the results we need”
On the Bakry-\'Emery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces (2013, unknown): “The definition of RCD(K,N) adopted in [18] is actually”
The abstract Lewy-Stampacchia inequality and applications (2014, background): “In order to get this further property we need to work on infinitesimally Hilbertian CD(K,N) spaces, also called RCD (K,N) spaces ([3], [12], [2], [10]).”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “Then the finite dimensional has been studied in [16, 17] and the precise RCD (K, N) has been defined in [13] and [3] with two different approaches.”
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces (2015, result): “Apart from the stated equivalence between the Lott-Sturm-Villani and theBakryÉmery approaches, our results and techniques can hardly be compared with the recent work [29] of Erbar-Kuwada-Sturm, motivated by the same questions.”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, unknown): “The RCD(K,∞)-condition shows the relevance of the synthetic notion of EVIK-flows; its equivalence with the celebrated Bakry-Émery curvature-dimension condition has been proved in [8] and its N-dimensional version has been introduced and deeply studied by Erbar-Kuwada-Sturm in [44] (by using a refine”
Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “This circle of ideas has been closed in [25] (and later in [9]) where it has been proved that BE ( K, N ) = CD ∗ ( K, N ) + infinitesimal Hilbertianity .”
From volume cone to metric cone in the nonsmooth setting (2015, methodology): “Another remarkable result has been established by ErbarKuwada-Sturm in [11] and independently by Ambrosio-Mondino-Savaré in [5]: they proved that, in a suitable sense, RCD∗(K,N) spaces can be characterized as those spaces where the Bochner inequality with parameters K,N holds (the case N =∞ was alre”