A classical result of Loewner and Nirenberg asserts that on every bounded smooth Euclidean domain there exists a unique smooth complete conformally flat metric of constant negative scalar curvature. In 2018, in a joint work with Maria del Mar Gonzalez, Yanyan Li, we showed the existence and uniqueness of a locally Lipschitz solution for fully nonlinear versions of the Loewner-Nirenberg problem, still on Euclidean domains. In this talk, I discuss my recent joint work with Jonah A.J. Duncan on generalisations on Riemannian manifolds with boundary.
