scholarly journals Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma $-finite measure

2015 ◽  
Vol 367 (7) ◽  
pp. 4661-4701 ◽  
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Andrea Mondino ◽  
Tapio Rajala
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Finite Measure ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals On Scalar and Ricci Curvatures

2021 ◽  
Author(s):  
Gerard Besson ◽  
◽  
Sylvestre Gallot ◽  
◽  
◽  
...  
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Riemannian Metric ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Volume Entropy ◽  
Bounded Below

The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look for weak forms of the notion of ''lower bounds of the Ricci curvature'' on non necessarily smooth metric measure spaces. We describe recent results some of which are already posted in [arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop-Gromov's inequality.

scholarly journals H1and BMO for certain locally doubling metric measure spaces of finite measure

2010 ◽  
Vol 118 (1) ◽  
pp. 13-41 ◽  
Author(s):  
Andrea Carbonaro ◽  
Giancarlo Mauceri ◽  
Stefano Meda
Keyword(s):  
Finite Measure ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals On interpolation and curvature via Wasserstein geodesics

2017 ◽  
Vol 10 (2) ◽  
pp. 125-167 ◽  
Author(s):  
Martin Kell
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Similar Condition ◽  
Interpolation Inequality ◽  
Metric Measure Spaces ◽  
Main Ingredient ◽  
Curvature Condition ◽  
Finsler Manifolds ◽  
Measure Spaces ◽  
Positively Curved

AbstractIn this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel–Brascamp–Lieb inequality for general Riemannian and Finsler manifolds and led Lott–Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli’s recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz–Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

scholarly journals The Measure Preserving Isometry Groups of Metric Measure Spaces

2020 ◽  
Author(s):  
Yifan Guo ◽  
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Measure Space ◽  
Isometry Group ◽  
Compact Riemannian Manifold ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Isometry Groups ◽  
Measure Spaces ◽  
Effective Estimate

Bochner's theorem says that if M is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso(M) is finite. In this article, we show that if (X,d,m) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure preserving isometry group Iso(X,d,m) is finite. We also give an effective estimate on the order of the measure preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Emery Ricci curvature except for small portions.

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