On the Negative Case of the Singular Yamabe Problem

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3, and let Γ be a nonempty closed subset of M . The negative case of the Singular Yamabe Problem concerns the existence and behavior of a complete metric ĝ on M\Γ that has constant negative scalar curvature and is pointwise conformally related to the smooth metric g. Previous results have shown that when Γ is a smooth submanifold of dimension d there exists such a metric if and only if d > n−2 2 . In this paper, we consider a general class of closed sets and show the existence of a complete conformal metric ĝ with constant negative scalar curvature depends on the dimension of the tangent cone to Γ at every point. Specifically, provided Γ admits a nice tangent cone at p, we show that when the dimension of the tangent cone to Γ at p is less than n−2 2 then there can not exist a negative Singular Yamabe metric ĝ on M\Γ. Subject Classifications: 58G30, 53C21, 35J60, 35B40.