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 scalarWeyl curvature, that is R + t|W|, t is an element of R. In particular, there are no topological obstructions for metrics with epsilon-pinched Weyl curvature and negative scalar curvature.