Regularity of viscosity solutions of the σk-Yamabe-type problem for k>n/2

We study the regularity of Lipschitz viscosity solutions to the σk 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.