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An alternative proof of the existence of torsion-free $G_2$-structures on resolutions of $G_2$-orbifolds considered in arXiv:1707.09325 is given. The proof uses weighted Holder norms which are adapted to the geometry of the manifold. This leads to better control of the torsion-free $G_2$-structure and a simplification over the original proof.
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{
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
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 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
In some other context, the question was raised how many nearly Kahler structures exist on the sphere $S^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence