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. I will discuss recent joint works with Maria del Mar Gonzalez, Yanyan Li and Jingang Xiong on existence, uniqueness and regularity for fully nonlinear versions of the Loewner-Nirenberg problem. In particular, I will discuss how minimal surface theory is used to deduce the non-differentiability of solutions when the boundary of the domain is disconnected.
