A new example of a compact ERP G2‐structure

Abstract

We provide the second-known example of an extremally Ricci pinched closed G2-structure on a compact 7-manifold, by finding a lattice in the only unimodular solvable Lie group admitting a left-invariant G2-structure. Furthermore, the Laplacian coflow and its solitons are studied on a 6parameter family of left-invariant coclosed G2-structures on this Lie group. In this way, we obtain a 4-parameter subfamily of expanding solitons. The family is locally pairwise non-equivalent. Contents

New examples of G$_2$-structures with divergence-free torsion

“…so as to make pg A,B,C , r¨, ¨sq a real seven-dimensional Lie algebra. These algebras were first defined in [KL21], and they constitute a generalization of almost abelian Lie algebras. Note that the Jacobi condition is equivalent to the fact that A, B, C are pairwise commuting.…”

“…For all choices of A, B, C we have that g A,B,C is a solvable Lie algebra and, as a consequence of tracelessness, that g A,B,C is unimodular. These last two facts are exploited in [KL21] to determine equivalence clases among G 2 -structures defined over Lie groups whose Lie algebra is isomorphic to g A,B,C , and we will use this result to establish that there are many non-equivalent examples of divergence-free G 2structures defined over g A,B,C , or equivalently, left-invariant divergence-free G 2 -structures defined over simply connected Lie groups G A,B,C such that their Lie algebra equals g A,B,C .…”

“…Under the further restriction that A, B, C P slp4, Rq are linearly independent and that they simultaneously diagonalize over R, in which case we call the triple compatible, all Lie algebras g A,B,C are isomorphic to each other, and the correspoding Lie groups G A,B,C are isomorphic to G J , the only completely soluble unimodular Lie group in the classification appearing in [LN20]. More on this topic as well as their relation to G 2 -structures can be found in [KL21]. We will not be dealing with compatible triples in this article.…”

“…Our research led us to left-invariant G 2 -structures defined on simply connected solvable Lie groups G A,B,C , defined in Section 3, which depend on the choice of a triple A, B, C P slp4, Rq of traceless pairwise-commuting 4 ˆ4 matrices. Recall from [KL21] that these groups admit lattices, so there are compact versions of the examples. Two large families of triples are considered: The diagonal case, in which A, B, C P slp4, Rq are diagonal (which is proven to be equivalent to the symmetric case, see Proposition 4.1), and the antidiagonal case, in which A, B, C P slp4, Rq are antidiagonal.…”

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.…”