We study conditions for which the mapping torus of a 6-manifold endowed with an $SU(3)$-structure is a locally conformal calibrated $G_2$-manifold, that is, a 7-manifold endowed with a $G_2$-structure $varphi$ such that $d varphi = - theta wedge varp
hi$ for a closed non-vanishing 1-form $theta$. Moreover, we show that if $(M, varphi)$ is a compact locally conformal calibrated $G_2$-manifold with $mathcal{L}_{theta^{#}} varphi =0$, where ${theta^{#}}$ is the dual of $theta$ with respect to the Riemannian metric $g_{varphi}$ induced by $varphi$, then $M$ is a fiber bundle over $S^1$ with a coupled $SU(3)$-manifold as fiber.
A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a Kahler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold $M$ of compl
etely solvable type endowed with an invariant complex structure $J$ admits a symplectic form taming J if and only if $M$ is a complex torus. This result generalizes the one obtained in [7] for nilmanifolds.
We prove that the Calabi-Yau equation on the Kodaira-Thurston manifold has a unique solution for every $S^1$-invariant initial datum.
We study the existence of left invariant closed $G_2$-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these $G_2$-structures, we show long time existence and uniqueness of solution for
the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on $R^7$ in a similar way as in [23] we prove that the underlying metrics $g(t)$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as $t$ goes to infinity.
