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

摘要

We show that on a compact Riemannian manifold with boundary there exists u ε C~∞(M)such that, u{pipe}?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 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