On the essential minimal volume of Einstein 4-manifolds ; Volume minimal essentiel des 4 dimensions d'Einstein ; On the essential minimal volume of Einstein 4-manifolds: Curvature Constraints and Spaces of Metrics ; Volume minimal essentiel des 4 dimensions d'Einstein: Contraintes de courbures et espaces métriques

Summer School 2021. Given a positive epsilon, a closed Einstein 4-manifold admits a natural thick-thin decomposition. I will explain how, for any delta, one can modify the Einstein metric to a bounded sectional curvature metric so that the thick part has volume linearly bounded by the Euler characteristic and the thin part has injectivity radius less than delta. I will also discuss relations to conjectural obstructions to collapsing with bounded sectional curvature or to the existence of Einstein metrics.