On Scalar and Ricci Curvatures

Abstract

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.