  In the homogeneous case, the only curvature behavior which is still far from
being understood is Ricci negative. In this paper, we study which nilpotent Lie
algebras admit a Ricci negative solvable extension. Different unexpected
behaviors were found. On the other hand, given a nilpotent Lie algebra, we
consider the space of all the derivations such that the corresponding solvable
extension has a metric with negative Ricci curvature. Using the nice convexity
properties of the moment map for the variety of nilpotent Lie algebras, we
obtain a useful characterization of such derivations and some applications.
