Complete non-compact Spin(7) manifolds from self-dual Einstein 4-orbifolds

Abstract

We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperkähler metrics.We apply our construction to asymptotically conical G2-metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete non-compact Spin(7)-manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7)-metrics on the same smooth 8-manifold.

Spin(7) metrics from Kähler Geometry

“…Currently the most effective method of constructing non-compact Spin(7) holonomy metrics involves evolving cocalibrated G 2 -structures on some homogeneous spaces via the Hitchin flow [20,Theorem 7]. Recently a new technique was developed by Foscolo, which involves constructing Spin(7) metrics on 'small' circle bundles over the anti-self-dual orbibundle of self-dual Einstein 4-orbifolds [13]. Our result provides a new way constructing more examples.…”

$S^1$-quotient of $Spin(7)$-structures

“…The above equations have also been described as a Gibbons-Hawking type ansatz for Spin(7)-manifolds in [13], where the author studies adiabatic limits of the equations to produce new complete non-compact Spin(7) manifolds. Remark 3.5.…”

“…There are by now many known examples of holonomy G 2 and Spin(7) metrics cf. [20], [7], [2], [14], [13], yet very few explicitly known ones. In [1], Apostolov and Salamon studied the S 1 -reduction of G 2 manifolds and investigated the situation when the quotient is a Kähler manifold.…”

“…Motivated by their work, in this article we shall carry out the analogous construction in the Spin (7) setting, but more generally we shall look at S 1 -invariant Spin(7)-structures which are not necessarily torsion-free. The situation when N is a Spin (7) manifold has also been studied by Foscolo in [13]. One motivation for studying the non torsion-free cases lies in the fact that they also have interesting geometric properties, for instance, balanced Spin(7)-structures admit harmonic spinors [18] and compact locally conformally parallel ones are fibred by nearly parallel G 2 manifolds [19].…”