Metrics of constant negative scalar-Weyl curvature

TL;DR

This paper proves that every closed manifold admits a metric with constant negative scalar-Weyl curvature, extending previous work on scalar curvature and showing no topological obstructions for certain negatively curved metrics.

Contribution

It generalizes Aubin's construction to include scalar-Weyl curvature, demonstrating the existence of such metrics on all closed manifolds.

Findings

Every closed manifold admits a metric with constant negative scalar-Weyl curvature.

No topological obstructions exist for metrics with -pinched Weyl curvature and negative scalar curvature.

Extension of Aubin's construction to scalar-Weyl curvature case.

Abstract

Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, t\in\mathbb{R}$. In particular, there are no topological obstructions for metrics with $\varepsilon$-pinched Weyl curvature and negative scalar curvature.