Sur le volume des variétés riemanniennes pincées
Résumé
Our purpose is to study the volume of those riemannian manifoids the metric of which is "close" to a constantly curved one. In even dimension the Gauss-Bonnet formula together with a pinching hypothesis (depending on thé dimension) yield a lower bound for this volume. In the general case we prove but an infinitésimal resuit: its setup is a C°° variation of metrics starting from an Einstein one. We assume that the first variation of the scalar curvature has a constant sign over the manifold and we dérive the sign of the first variation of the volume