Mathematical Research Letters

Volume 30 (2023)

Number 2

Metrics of constant negative scalar-Weyl curvature

Pages: 319 – 340



Giovanni Catino (Dipartimento di Matematica, Politecnico di Milano, Italy)


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.

Received 26 February 2021

Received revised 27 August 2021

Accepted 19 October 2021

Published 13 September 2023