arXiv subjects

Gerard Besson

Publications and source records attributed to Gerard Besson.

2 recordsLinked to original sources

arXiv2020

On Scalar and Ricci Curvatures

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.

math.DG↗

arXiv2005

Rigidity of amalgamated product in negative curvature

Let $Γ$ be the fundamental group of a compact n-dimensional riemannian manifold X of sectional curvature bounded above by -1. We suppose that $Γ$ is a free product of its subgroup A and B over the amalgamated subgroup C. We prove that the critical exponent $δ(C)$ of C satisfies $δ(C) \geq n-2$. The equality happens if and only if there exist an embedded compact hypersurface Y in X, totally geodesic, of constant sectional curvature -1, with fundamental group C and which separates X in two connected components whose fundamental groups are A and B. Similar results hold if $Γ$ is an HNN extension, or more generally if $Γ$ acts on a simplicial tree without fixed point.

math.DG↗