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Register a new userExact $G_2$-structures on unimodular Lie algebras
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We consider seven-dimensional unimodular Lie algebras $mathfrak{g}$ admitting exact $G_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(mathfrak{g})$. We discuss some examples, both in the case when $b_2(mathfrak{g}) eq0$, and in the case when the Lie algebra $mathfrak{g}$ is (2,3)-trivial, i.e., when both $b_2(mathfrak{g})$ and $b_3(mathfrak{g})$ vanish. These examples are solvable, as $b_3(mathfrak{g})=0$, but they are not strongly unimodular, a necessary condition for the existence of lattices on the simply connected Lie group corresponding to $mathfrak{g}$. More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact $G_2$-structure. From this, it follows that there are no compact examples of the form $(Gammabackslash G,varphi)$, where $G$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, $Gammasubset G$ is a co-compact discrete subgroup, and $varphi$ is an exact $G_2$-structure on $Gammabackslash G$ induced by a left-invariant one on $G$.
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We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed with a closed G$_2$-structure. Using this result, we classify all unimodular Lie algebras with non-trivial center admitting closed G$_2$-structures, up to isomorphism, and we show that six of them arise as the contactization of a symplectic Lie algebra. Finally, we prove that every semi-algebraic soliton on the contactization of a symplectic Lie algebra must be expanding, and we determine all unimodular Lie algebras with center of dimension at least two that admit semi-algebraic solitons, up to isomorphism.
We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify 6-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kahler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedulli and L. Vezzoni and of the anomaly flow of D. H. Phong, S. Picard and X. Zhang on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kahler metrics are fixed points.
We show that the compact quotient $Gammabackslashmathrm{G}$ of a seven-dimensional simply connected Lie group $mathrm{G}$ by a co-compact discrete subgroup $Gammasubsetmathrm{G}$ does not admit any exact $mathrm{G}_2$-structure which is induced by a left-invariant one on $mathrm{G}$.
We review results about $G_2$-structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some examples in detail.
We show obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra $mathfrak g$ of dimension seven with non-trivial center. In particular, we prove that if there exist a Lie algebra epimorphism from $mathfrak g$ to a six-dimensional Lie algebra $mathfrak h$, with kernel contained in the center of $mathfrak g$, then any coclosed $G_2$-structure on $mathfrak g$ induces a closed and stable three form on $mathfrak h$ that defines an almost complex structure on $mathfrak h$. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed $G_2$-structures. We also prove that each one of these Lie algebras has a coclosed $G_2$-structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed $G_2$-structures. The existence of contact metric structures is also studied.
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