On the essential minimal volume of Einstein 4-manifolds

Appears in collection : Summer School 2021 - Curvature Constraints and Spaces of Metrics

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.