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

TL;DR

This paper classifies nilmanifolds with invariant mean convex closed $ ext{SL}(3,b{C})$-structures and explores conditions under which solvmanifolds admit tamed structures, linking these to symplectic half-flat $ ext{SU}(3)$-structures.

Contribution

It provides a classification of nilmanifolds with invariant mean convex closed $ ext{SL}(3,b{C})$-structures and establishes a connection between tamed $ ext{SL}(3,b{C})$-structures and symplectic half-flat $ ext{SU}(3)$-structures on solvmanifolds.

Findings

Nilmanifolds with invariant mean convex closed $ ext{SL}(3,b{C})$-structures are classified.

Existence of invariant mean convex half-flat $ ext{SU}(3)$-structures on certain nilmanifolds is established.

Solvmanifolds with invariant tamed closed $ ext{SL}(3,b{C})$-structures also admit invariant symplectic half-flat $ ext{SU}(3)$-structures.

Abstract

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.