REGULARITY FOR THE SINGULAR YAMABE PROBLEM

The singular Yamabe problem concerns the existence and nature of complete metrics with constant scalar curvature on the complement of a closed subset, assumed here to be a smooth submanifold, in a compact Riemannian manifold and which are conformal to a metric smooth across the submanifold. Regularity in the form of smooth asymptotic expansions is shown to always hold when the scalar curvature R of the noncompact metric is negative. Leading asymptotic behaviour is shown to exist for 'admissible' solutions when R greater-than-or-equal-to 0. A complete smooth expansion exists if this leading coefficient is smooth, and a complete distributional expansion always exists when R = 0. Solutions to the problem are completely classified when R = 0. The existence of 'inadmissible' periodic solutions when R > 0 is also proved.