Existence and uniqueness to a fully non-linear version of the Loewner-Nirenberg problem

Abstract

We consider the problem of finding on a given Euclidean domain $\Omega$ of dimension $n\ge 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(\lambda (-A))=1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partial \Omega$ is a smooth bounded hypersurface (of codimension one). When $\partial \Omega$ contains a compact smooth submanifold $\Sigma$ of higher codimension with $\partial \Omega \setminus \Sigma$ being compact, we also give a `sharp' condition for the divergence to infinity of the conformal factor near $\Sigma$ in terms of the codimension.