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Register a new userExistence of torsion-free $G_2$-structures on resolutions of $G_2$-orbifolds using weighted Holder norms
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Daniel Platt
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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.
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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 $G$.
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.
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 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.
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 of the description of the eigenspace to the eigenvalue $lambda = 12$ of the Laplacian acting on 2-forms. A similar result concerning nearly parallel $G_2$-structures on the round sphere $S^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.
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