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 manifold (M4,g) with scalar-flat and π2(M4)≠0, we show that the universal Riemannian cover (M~,g~) is homothetic to the standard product H2×S2. This affirmatively answers a question of N. H. Noronha.