  A version of the singular Yamabe problem in bounded domains yields complete
conformal metrics with negative constant scalar curvatures. In this paper, we
study whether these metrics have negative Ricci curvatures. Affirmatively, we
prove that these metrics indeed have negative Ricci curvatures in bounded
convex domains in the Euclidean space. On the other hand, we provide a general
construction of domains in compact manifolds and demonstrate that the
negativity of Ricci curvatures does not hold if the boundary is close to
certain sets of low dimension. The expansion of the Green's function and the
positive mass theorem play essential roles in certain cases.
