On Scalar and Ricci Curvatures
2021
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.
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