Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 046, 42 pages      arXiv:2010.08207
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

On Scalar and Ricci Curvatures

Gerard Besson and Sylvestre Gallot
CNRS-Université Grenoble Alpes, Institut Fourier, CS 40700, 38058 Grenoble cedex 09, France

Received October 19, 2020, in final form April 05, 2021; Published online May 01, 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.

Key words: scalar curvature; Ricci curvature; Whitehead 3-manifolds; infinite connected sums; Ricci flow; synthetic Ricci curvature; metric spaces; Bishop-Gromov inequality; Gromov-hyperbolic spaces; hyperbolic groups; Busemann spaces; CAT(0)-spaces.

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