Alexandrov meets Lott-Villani-Sturm

By Anton Petrunin. 2010

168 citations recorded by the source corpus.

Here I show the compatibility of two definitions of generalized curvature bounds: the lower bound for sectional curvature in the sense of Alexandrov and the lower bound for Ricci curvature in the sense of Lott-Villani-Sturm.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9e2992c03bdeb7d175518519e973acfac902cc21

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “2 ELIA BRUÈ, ENRICO PASQUALETTO, AND DANIELE SEMOLA We do know nowadays that, apart from smooth weighted Riemannian manifolds (with generalized Ricci tensor bounded from below), this class includes Ricci limits (see [16, 17, 18]) and Alexandrov spaces [43].”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, methodology): “The curvature-dimension condition CD(K,N) has been verified for Riemannian manifolds [39], Finsler spaces [31], Alexandrov spaces [35], [42], cones [8] and warped products of Riemannian manifolds [24].”
Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case (2012, background): “Also, from the studies made in [27], [33], [25] and [16] we also know that finite dimensional Alexandrov spaces with curvature bounded from below are RCD(K,∞) spaces as well.”
Ricci Curvature on Alexandrov spaces and Rigidity Theorems (2009, methodology): “In [31], Petrunin proved Δu nλ · vol, in particular, the singular part of Δu is non-positive.”
The globalization theorem for the Curvature-Dimension condition (2016, background): “It was subsequently also shown [57, 63] that Finsler manifolds and Alexandrov spaces satisfy the Curvature-Dimension condition.”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “The final statement follows from the compatibility between Alexandrov spaces and RCD-spaces [45,56].”
Localization and Tensorization Properties of the Curvature-Dimension Condition for Metric Measure Spaces (2010, background): “dition CD(K,N) include • Riemannian manifolds and weighted Riemannian spaces [OV00], [CMS01], [RS05], [Stu05] • Finsler spaces [Oht] • Alexandrov spaces of generalized nonnegative sectional curvature [Pet09] • Finite or inﬁnite dimensional Gaussian spaces [Stu06a], [LV09]. Slightly modiﬁed versions ”