Regularity of Singular Solutions to Sigma_k-Yamabe Problems

We prove some regularity results for singular solutions of $sigma_k$-Yamabe problem, where the singular set is a compact hypersurface in a Riemannian manifold. This problem is a fully nonlinear version of the singular Yamabe problem, which is an equation of semilinear type. Apart from their importance in conformal geometry, the blow-up solutions along a hypersurface or the boundary of a manifold have also received much attention in the study of AdS/CFT correspondence in physics. We study the problem in the case of negative cone. In this case, the main difficulty is the lack of $C^2$ estimate, so we have to rely on the maximum principle and method of sub- and super-solutions. Combining our result with the theory of 'Edge Differential Operators' developed by R. Mazzeo we obtain a polyhomogeneous expansion of any singular solutions in terms of the distance to the singular set.