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 $(M^4,g)$ with scalar-flat and $\pi_2(M^4) \neq 0$, we show that the universal Riemannian cover $(\widetilde{M},\tilde{g})$ is homothetic to the standard product $\mathbb{H}^2 \times \mathbb{S}^2$. This affirmatively answers a question of N. H. Noronha.