On Scalar and Ricci Curvatures

Gérard Besson (1) , Sylvestre Gallot (1)

1 IF - Institut Fourier

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.

Mots clés: scalar curvature; Whitehead manifold; metric measured space; volume entropy; Ricci curvature; Bishop-Gromov's Inequality; Whitehead 3-manifolds; infinite connected sums; Ricci flow; synthetic Ricci curvature; metric spaces; Bishop-Gromov inequality; Gromov-hyperbolic spaces; hyperbolic groups; Busemann spaces

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Théorie des groupes [math.GR]

Gérard Besson, Sylvestre Gallot. On Scalar and Ricci Curvatures. Symmetry, Integrability and Geometry : Methods and Applications, 2021, ⟨10.3842/SIGMA.2021.046⟩. ⟨hal-02966630v2⟩