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.
Categories: math.DG