Complete Noncompact Spin(7) Manifolds From Self-Dual Einstein 4-Orbifolds

We present an analytic construction of complete noncompact 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 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 hyperkahler 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(7)-manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7)-metrics on the same smooth 8-manifold.