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

We consider the problem of finding on a given Euclidean domainof 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 δΩ is a smooth bounded hypersurface(of codimension one).When δΩ contains a compact smooth submanifold ∑ of higher codimension with δΩ\∑ being compact,we also give a‘sharp’condition for the divergence to infinity of the conformal factor near ∑ in terms of the codimension.