  We consider the problem of finding on a given Euclidean domain $\Omega$ of
dimension $n \geq 3$ 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.
