Complete noncompact 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 G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperk\"ahler 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.