Quadratic closed G2-structures

Closed $\mathrm{G}_2$-eigenforms and exact $\mathrm{G}_2$-structures

“…In general, quadratic closed G 2 -structures are exactly the closed G 2 -structures for which the exterior derivative dτ of the associated torsion two-form τ depends quadratically on τ . These structures have been studied by Ball [Ba1,Ba2] and include many other interesting closed G 2 -structures. For example, the case λ = 1 6 corresponds to 1 so-called extremally Ricci-pinched (ERP) closed G 2 -structures, and the case λ = 1 2 is equivalent to the induced metric being Einstein.…”

“…By Lauret's work [L], homogeneous λ-quadratically closed G 2 -structures on homogeneous manifolds can only exist for λ ∈ {0, 1 6 , 1 2 }. Homogeneous ERP closed G 2 -structures were classified in [Ba2] using the classification of left-invariant such structures on Lie groups in [LN2]. Moreover, [FFM] shows that no solvable Lie group can admit a left-invariant closed Einstein G 2 -structure.…”

Exact G$_2$-structures on compact quotients of Lie groups

“…Currently, many examples of compact manifolds admitting closed G 2 -structures are available, see [6,16,17,19,20] for examples admitting holonomy G 2 metrics, [9] for an example obtained resolving the singularities of an orbifold, and [1,4,5,7,8,12,18] for examples on compact quotients of Lie groups. However, it is still not known whether exact G 2 -structures may occur on compact 7-manifolds.…”

Closed G$_2$-structures on unimodular Lie algebras with non-trivial center

“…In the non-compact setting, examples of Laplacian solitons of any type are known, see e.g. [6,22,23,25,31,32,33,39]. In particular, the steady solitons in [6] and the shrinking soliton in [25] are inhomogeneous and of gradient type, i.e., X is a gradient vector field.…”

“…[6,22,23,25,31,32,33,39]. In particular, the steady solitons in [6] and the shrinking soliton in [25] are inhomogeneous and of gradient type, i.e., X is a gradient vector field. As for the known homogeneous examples, they consist of simply connected Lie groups G endowed with a left-invariant closed G 2 -structure satisfying the equation (1.1) with respect to a vector field X defined by a one-parameter group of automorphisms induced by a derivation D of the Lie algebra g = Lie(G).…”