Localization and Tensorization Properties of the Curvature-Dimension Condition for Metric Measure Spaces

By Kathrin Bacher, Karl‐Theodor Sturm. 2010

240 citations recorded by the source corpus.

Abstract This is an addendum to the paper [K. Bacher, K.T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010) 28–56]. We prove the tensorization property for the curvature-dimension condition, add some detailed calculations – including explicit dependence of constants – and comment on assumptions and conjectures concerning the local-to-global statement in Bacher and Sturm (2010) [1] and Villani (2009) [6] , respectively.

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

View Paper: https://www.semanticscholar.org/paper/e81f6339e0b49d45ec8b26182ccb0b0b26c8dfd4

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Citations in the corpus

On the volume measure of non-smooth spaces with Ricci curvature bounded below (2016, methodology): “We turn to the formulation of the CD(K,N) condition, coming from [7].”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “In more recent times, two main requirements were added to that theory, namely the CD∗ condition introduced in [10] and the infinitesimal Hilbertianity introduced in [3], giving rise to the study of the so-called RCD (resp.”
Obata’s Rigidity Theorem for Metric Measure Spaces (2014, unknown): “1 (Reduced curvature-dimension condition, [6]).”
On the Bakry–Émery Condition, the Gradient Estimates and the Local-to-Global Property of $$\mathsf{RCD}^*(K,N)$$RCD∗(K,N) Metric Measure Spaces (2016, unknown): “ +∞ if κ≥ π2, sin(t √ κδ) sin( √ κδ) if 0 <κ<π2 t if κ= 0, sinh(t √ −κδ) sin( √ −κδ) if κ<0. (7.4) The so-called CD∗(K,N) condition introduced by Bacher and Sturm in [6] is based, in analogy with the case N= ∞, on a convexity inequality of UN along Wasserstein geodesics; it is a var”
New formulas for the Laplacian of distance functions and applications (2018, background): “In particular it is strictly weaker than the celebrated curvature dimension condition CD(K,N) pioneered by Lott-Sturm-Villani [42, 56, 57] and than the (weaker) reduced curvature dimension condition CD∗(K,N) [7].”
The globalization theorem for the Curvature-Dimension condition (2016, background): “However, a counterexample to the latter implication was constructed by Deng and Sturm in [34], thereby showing that:”
Cones over metric measure spaces and the maximal diameter theorem (2013, background): “CD(K,N) ⇒ CD∗(K,N), CD∗(K,N) ⇔ CDloc(K,N), CD∗(0, N) ⇔ CD(0, N) (see [10]) where the definition of CDloc(K,N) can be found for example in [47].”
An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, background): “1 Bacher and Sturm introduced in [10] a di erent notion of curvature-dimension bounds: the so called reduced-curvaturedimension, denoted as CD*(K, N).”
Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, background): “29 imply that all the curvature-dimension conditions CD(K,N), CD∗(K,N), MCP(K,N) with finite N are stable with respect to pmGconvergence (see [14, 34, 43] for the various definitions).”
Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “In any case, all our arguments are local in nature and since the local versions of CD(K,N) and CD∗(K,N) are known to be equivalent from the very first paper [11] where CD∗ has been introduced, our results are independent from [14].”
From volume cone to metric cone in the nonsmooth setting (2015, background): “Later on, mostly for technical reasons related to the local-to-global property, Bacher-Sturm introduced in [6] a variant of the CD(K,N) condition, called reduced curvature dimension condition and denoted CD∗(K,N).”
The abstract Lewy-Stampacchia inequality and applications (2014, background): “For the definition of the Curvature-Dimension condition CD(K,N) we refer to [7] (see also [32] for the ‘original’ CD(K,N) condition).”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, methodology): “The investigation about the topic began with the seminal papers of Lott-Villani [45] and Sturm [64, 65], though has been adapted considerably since the work of Bacher-Sturm [9] and Ambrosio-Gigli-Savaré [4, 5].”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, background): “For the stability of the RCD∗(K,N) condition under mGH convergence see for instance [7, 28, 35].”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “Recall that the reduced curvature dimension condition CD∗(K,N) introduced in [9] is defined as Definition C.”
On the differential structure of metric measure spaces and applications (2012, background): “mGH convergence and it is locally equivalent to the CD(K,N) condition (all this has been proved in [8]).”