Closed $G_2$-eigenforms and exact $G_2$-structures

Abstract

A study is made of left-invariant ${\mathrm{G}}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit a left-invariant closed ${\mathrm{G}}_2$-eigenform for the Laplacian and that any compact solvmanifold $\Gamma \backslash G$ arising from $G$ does not admit an (invariant) exact ${\mathrm{G}}_2$-structure. We also classify the seven-dimensional Lie algebras $\mathfrak{g}$ with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact ${\text{G}}_2$-structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras $\mathfrak{h}$ admitting an exact $\text{SL}(3,\mathbb{C})$-structure $\rho$ or a half-flat $\mathrm{SU}(3)$-structure $(\omega ,\rho )$ with exact $\rho$, respectively.