scholarly journals Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case

2013 ◽  
pp. 63-115 ◽  
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Giuseppe Savaré
Keyword(s):  
Heat Flow ◽  
Ricci Curvature ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Compact Case ◽  
Bounded Below

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 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
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