  The negative case of the Singular Yamabe Problem concerns the existence and
behavior of complete metrics with constant negative scalar curvature on the
complement of a closed set in a compact Riemannian manifold which are
conformally equivalent to a smooth metric on this compact manifold. When the
closed set is a smooth submanifold, it is known by the results of
Loewner-Nirenberg and Aviles-McOwen that there exists such a complete metric if
and only if $d > (n-2)/2$, and in general the Hausdorff dimension of the set
must be at least $(n-2)/2$. In this paper, we show that the existence of such a
complete conformal metric with constant negative scalar curvature depends on
the tangent structure of the closed set. Specifically, provided the set has a
nice tangent cone at a point, we show that when the dimension of this tangent
cone is less than $(n-2)/2$ there can not exist such a negative Singular Yamabe
metric.
