  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.
