Infinitely Many New Families Of Complete Cohomogeneity One G(2)-Manifolds: G(2) Analogues Of The Taub-Nut And Eguchi-Hanson Spaces

We construct infinitely many new 1-parameter families of simply connected complete non-compact G(2)-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G(2)-manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G(2)-manifolds.We also construct a closely related conically singular G(2)-holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G(2)-cone over the standard nearly Kahler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G(2)-space is the natural G(2) analogue of the Taub-NUT metric in 4-dimensional hyperKahler geometry and that our new AC G(2)-metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperKahler manifold. Like the Taub-NUT and Eguchi-Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one.