Existence and Uniqueness to a Fully Nonlinear 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 equations 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.

Keywords

Fully nonlinear Loewner–Nirenberg problem / Singular fully nonlinear Yamabe metrics / Conformal invariance