A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures.In this talk,I will discuss whether these metrics have negative Ricci curvatures.We will 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 Greens function and the positive mass theorem play essential roles in certain cases.On the other hand,we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space.