Scalar Curvature in Dimension 4

TL;DR

This paper proves Schoen's conjecture for certain hyperbolic 4-manifolds, classifies positive scalar curvature 4-manifolds as fiber bundles, and describes the universal cover of specific locally conformally flat 4-manifolds.

Contribution

It establishes new results on scalar curvature bounds, classifies certain 4-manifolds, and answers a question about the universal cover of specific conformally flat manifolds.

Findings

Proves Schoen's conjecture for hyperbolic 4-manifolds with scalar curvature ≥ -12.

Classifies closed Riemannian 4-manifolds with positive scalar curvature as fiber bundles.

Shows universal cover of certain zero scalar curvature conformally flat 4-manifolds is a product of hyperbolic plane and sphere.

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.