Admissibility, Topology, and the ${\sigma }_k$-Loewner--Nirenberg problem

The ${\sigma }_k$-Loewner–Nirenberg problem is a fully nonlinear generalization of the classical Loewner–Nirenberg problem of constructing complete conformal metrics with constant negative scalar curvature in the interior of domains. For the ${\sigma }_k$-version, once $k>1$, one must impose a negative admissibility condition to ensure ellipticity of the equation. In this note, we exhibit topological obstructions to admissibility and illustrate this with examples.