Abstract. The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety Nn(C) of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GLn(C)-orbits in Nn(C) have a critical point of the squared norm of the moment map. In this paper, we give a classification result of such distinguished orbits for n = 7. The set N7(C)/GL7(C) is formed by 148 nilpotent Lie algebras and 6 one-parameter families of pairwise non-isomorphic nilpotent Lie algebras. We have applied to each Lie algebra one of three main techniques to decide whether it has a distinguished orbit or not.

Abstract. The aim of this paper is to classify Ricci soliton metrics on 7-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in Transformation Groups, Volume 17, Number 3 (2012), 639-656. To this end, we use the classification of 7-dimensional real nilpotent Lie algebras given by Ming-Peng Gong in his Ph.D thesis and some techniques from the results of Michael Jablonski in [J1, J2] and of Yuri Nikolayevsky in [Nk2]. Of the 9 one-parameter families and 140 isolated 7-dimensional indecomposable real nilpotent Lie algebras, we have 99 nilsoliton metrics given in an explicit form and 7 one-parameter families admitting nilsoliton metrics.Our classification is the result of a case-by-case analysis, so many illustrative examples are carefully developed to explain how to obtain the main result.

The aim of this paper is to study self-similar solutions to the symplectic cuvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention in the family of symplectic Two-and Three-step nilpotent Lie algebras admitting a minimal compatible metric and we give a complete classification of these algebras together with their respective metric. Such classification is given by using our generalization of Nikolayevsky's nice basis criterium, which will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds, for the convenience of the reader.By computing the Chern-Ricci operator P in each case, we show that the above distinguished metrics define a soliton almost Kähler structure.Many illustrative examples are carefully developed.2010 Mathematics Subject Classification. Primary 57N16 Secondary 22E25; 22E45.

Abstract. Let G be a real reductive Lie group and let τ ∶ G → GL(V ) be a real reductive representation of G with (restricted) moment map mg ∶ V ∖ {0} → g. In this work, we introduce the notion of nice space of a real reductive representation to study the problem of how to determine if a G-orbit is distinguished (i.e. it contains a critical point of the norm squared of mg). We give an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in our particular case and we use it to give an easy-to-check sufficient condition for a G-orbit of a element in a nice space to be distinguished. In the case where G is algebraic and τ is a rational representation, the above condition is also necessary (making heavy use of recent results of M. Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterium. We also provide useful characterizations of nice spaces in terms of the weights of τ . Finally, some applications to ternary forms and minimal metrics on nilmanifolds are presented.

Let G be a Lie group of even dimension and let (g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure (g, J) where J is abelian then the Lie algebra of G is unimodular and (G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple (G, g, J) is an anti-Kähler manifold.Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor θ on its Lie algebra and prove that such structure is anti-Kähler if and only if θ is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).2010 Mathematics Subject Classification.