Existence of Complete Conformal Metrics of Negative Ricci Curvature on Manifolds with Boundary

We show that on a compact Riemannian manifold with boundary there exists u ∈ C(M) such that, u|∂M ≡ 0 and u solves the σk-Ricci problem. In the case k = n the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving the σk-Ricci problem. By adopting results of [14], we show an interesting relationship between the complete metrics we construct and the existence of Poincaré-Einstein metrics. Finally we give a brief discussion of the corresponding questions in the case of positive curvature.