We show that on a compact Riemannian manifold with boundary there exists u ∈ C ∞ (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.
Abstract. We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to MongeAmpère equations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.