Closed $text{SL}(3,mathbb{C})$-structures on nilmanifolds

In this paper we consider closed $text{SL}(3,mathbb{C})$-structures which are either mean convex or tamed by a symplectic form. These notions were introduced by Donaldson in relation to $text{G}_2$-manifolds with boundary. In particular, we classify nilmanifolds which carry an invariant mean convex closed $text{SL}(3,mathbb{C})$-structure and those which admit an invariant mean convex half-flat $text{SU}(3)$-structure. We also prove that, if a solvmanifold admits an invariant tamed closed $text{SL}(3,mathbb{C})$-structure, then it also has an invariant symplectic half-flat $text{SU}(3)$-structure.