Regularity of viscosity solutions of the ${\sigma }_k$-Yamabe-type Problem for $k>n\mathrm{/}2$

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 $k>n\mathrm{/}2$, we prove that all Lipschitz viscosity solutions are smooth away from a closed set of measure zero. For the general $k>n\mathrm{/}2$ case, under certain assumptions, we prove the existence of a Lipschitz viscosity solution that is smooth away from a closed set of measure zero.