Regularity of viscosity solutions of the $\sigma_k$-Yamabe-type Problem for $k>n/2$

We study the regularity of Lipschitz viscosity solutions to the $\sigma_k$ Yamabe problem in the negative cone case. If either $k=n$ or the manifold is conformally flat and $k>n/2$, we prove that all Lipschitz viscosity solutions are smooth away from a closed set of measure zero. For the general $k>n/2$ case, under certain assumptions, we prove the existence of a Lipschitz viscosity solution that is smooth away from a closed set of measure zero.