Metrics of constant negative scalar-Weyl curvature

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, $R + t {\lvert W \rvert}, t \in \mathbb{R}$. In particular, there are no topological obstructions for metrics with $\varepsilon$-pinched Weyl curvature and negative scalar curvature.

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Received 26 February 2021

Received revised 27 August 2021

Accepted 19 October 2021

Published 13 September 2023