Scalar Curvature in Dimension 4

Abstract

We prove that every locally conformally flat metric on a closed, oriented hyperbolic 4-manifold with scalar curvature bounded below by -12 satisfies Schoen's conjecture. We also classify all closed Riemannian 4-manifolds of positive scalar curvature that arise as total spaces of fibre bundles. For a closed locally conformally flat 4-manifold with scalar curvature zero and nontrivial second homotopy group, we show that its universal Riemannian cover is homothetic to the standard product of the hyperbolic plane and the round 2-sphere. This affirmatively answers a question of N. H. Noronha.