Families of complete non-compact Spin(7) holonomy manifolds

We consider complete non-compact Spin(7)-manifolds which are either asymptotically locally conical (ALC) or asymptotically conical (AC). The thesis consists of two parts. In the first part we develop the deformation theory of AC Spin(7)-manifolds. We show that the moduli space of torsion-free AC Spin(7)-structures on a given 8-manifold M asymptotic to a fixed Spin(7)-cone is an orbifold for generic decay rates in the non-L² regime. Furthermore, we derive a formula for the dimension of the moduli space, which has contributions from the topology of M and from solutions of a first order PDE system on the link of the asymptotic cone. In the second part we prove existence results of cohomogeneity one Spin(7) holonomy metrics for which a generic orbit is isomorphic to the Aloff–Wallach space N(1, −1) ∼= SU(3)/U(1). The unique non-trivial rank 3 vector bundle over the 5-sphere and the universal quotient bundle of CP² each carry a 1-parameter family (up to scale) of such metrics. We show that these families share a common behaviour: a generic member of these families belongs to one of two open intervals, of which one consists of ALC Spin(7) holonomy metrics and the other one of incomplete metrics. These two intervals are separated by a distinguished parameter which gives rise to an AC Spin(7) holonomy metric. Another interesting phenomenon occurs at the other endpoint of the open interval of ALC metrics, where the family collapses to the Bryant–Salamon AC G₂ holonomy metric on Λ²_CP². Notable is the existence of the two AC spaces. These are the first examples of smooth AC Spin(7) holonomy manifolds known to exist since Bryant Salamon’s original example on S₊(S⁴) in 1989. Furthermore, they provide a Spin(7) analogue of the well-known conifold transition in the setting of Calabi–Yau 3-folds