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We consider the problem of finding on a given Euclidean domain $\Omega$ ofdimension $n \geq 3$ a complete conformally flat metric whose Schoutencurvature $A$ satisfies some equation of the form $f(\lambda(-A)) = 1$. Thisgeneralizes a problem considered by Loewner and Nirenberg for the scalarcurvature. We prove the existence and uniqueness of such metric when theboundary $\partial\Omega$ is a smooth bounded hypersurface (of codimensionone). When $\partial\Omega$ contains a compact smooth submanifold $\Sigma$ ofhigher codimension with $\partial\Omega\setminus\Sigma$ being compact, we alsogive a `sharp' condition for the divergence to infinity of the conformal factornear $\Sigma$ in terms of the codimension.

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