, Volume 9, Issue 1, pp 73-92

On the negative case of the Singular Yamabe Problem

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Abstract

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 g 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 (with or without boundary) of dimension d, there exists such a metric if and only if \(d > \frac{{n - 2}}{2}\) . In this paper, we consider a more general class of closed sets and show the existence of a complete conformai metric ĝ with constant negative scalar curvature which 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 \(\frac{{n - 2}}{2}\) then there cannot exist a negative Singular Yamabe metric ĝ on M∖Γ.

Communicated by Steven Krantz