On Ricci negative derivations (Q2127860)

The present paper is a contribution to the problem of negative Ricci curvature in the homogeneous setting. A question considered is the following: Given a nilpotent Lie algebra \(\mathfrak{n}\), which are the solvable Lie algebras with nilradical \(\mathfrak{n}\) admitting a metric with \(\mathrm{Ric}<0\)? Lauret and Will have conjectured that given a nilpotent Lie algebra the space of all digonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature, coincides with an open convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. The author proves the validity of this conjecture in dimensions \(\le 5\), as well as for Heisenberg Lie algebras and standard filiform Lie algebras. A related conjecture has been also posed by Nikolayevsky and Nikonorov.