Existence and Uniqueness to a Fully Nonlinear Version of the Loewner–Nirenberg Problem: Dedicated to Celebrate the Sixtieth Anniversary of USTC

© 2018, School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature. We consider the problem of finding on a given Euclidean domain Ω of dimension n≥ 3 a complete conformally flat metric whose Schouten curvature A satisfies some equations of the form f(λ(- 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 ∂Ω is a smooth bounded hypersurface (of codimension one). When ∂Ω contains a compact smooth submanifold Σ of higher codimension with ∂Ω \ Σ being compact, we also give a ‘sharp’ condition for the divergence to infinity of the conformal factor near Σ in terms of the codimension.