Complete noncompact Spin(7) manifolds from self-dual Einstein $4$–orbifolds

Abstract

We present an analytic construction of complete noncompact $8$–dimensional Ricci-flat manifolds with holonomy $Spin\left(7\right)$. The construction relies on the study of the adiabatic limit of metrics with holonomy $Spin\left(7\right)$ on principal Seifert circle bundles over asymptotically conical $G_2$–orbifolds. 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 $G_2$–metrics arising from self-dual Einstein $4$–orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete noncompact $Spin\left(7\right)$–manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC $Spin\left(7\right)$–metrics on the same smooth $8$–manifold.