Properties of hypersurface singular sets of solutions to the $ \sigma_k $ -Yamabe equation in the negative cone

We consider conformally flat Lipschitz viscosity solutions to the $ \sigma_k $-Yamabe equation in the negative cone which admit smooth hypersurface singularities. Under natural regularity assumptions (that are satisfied by solutions to the $ \sigma_k $-Loewner-Nirenberg problem on annuli, for example), we first prove that the trace and normal derivatives of such a solution along the hypersurface satisfy a certain PDE. For $ k = 2 $, we also show that the hypersurface is minimal with respect to the Lipschitz solution and address some questions related to the formal expansion of the solution near the hypersurface.