Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary

Abstract

We show that on a compact Riemannian manifold with boundary there exists \({u \in C^{\infty}(M)}\) such that, u |∂M ≡ 0 and u solves the σ k -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 σ k -Ricci problem. By adopting results of (Mazzeo and Pacard, Pacific J. Math. 212(1), 169–185 (2003)), 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.