Infinitely many new families of complete cohomogeneity one G${}_2$-manifolds: G${}_2$ analogues of the Taub–NUT and Eguchi–Hanson spaces

Abstract

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 Kähler 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 G2-space is the natural G${}_2$ analogue of the Taub–NUT metric in 4-dimensional hyperKähler geometry and that our new AC G${}_2$-metrics are all analogues of the Eguchi–Hanson metric, the simplest ALE hyperKähler 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.