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تسجيل مستخدم جديدOn $G_2$-structures, special metrics and related flows
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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.
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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 $G$.
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.
We describe the Special Kahler structure on the base of the so-called Hitchin system in terms of the geometry of the space of spectral curves. It yields a simple formula for the Kahler potential. This extends to the case of a singular spectral curve
and we show that this defines the Special Kahler structure on certain natural integrable subsystems. Examples include the extreme case where the metric is flat.
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.
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 be
tter control of the torsion-free $G_2$-structure and a simplification over the original proof.
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