A version of the singular Yamabe problem in smooth domains in a closed
manifold yields complete conformal metrics with negative constant scalar
curvatures. In this paper, we study the blow-up phenomena of Ricci curvatures
of these metrics on domains whose boundary is close to a certain limit set of a
lower dimension. We will characterize the blow-up set according to the Yamabe
invariant of the underlying manifold. In particular, we will prove that all
points in the lower dimension part of the limit set belong to the blow-up set
on manifolds not conformally equivalent to the standard sphere and that all but
one point in the lower dimension part of the limit set belong to the blow-up
set on manifolds conformally equivalent to the standard sphere. In certain
cases, the blow-up set can be the entire manifold. We will demonstrate by
examples that these results are optimal