## Abstract

We construct infinitely many new 1-parameter families of simply connected complete noncompact 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 Kaehler 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 hyperKaehler geometry and that our new AC G_2-metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperKaehler 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.

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 Kaehler 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 hyperKaehler geometry and that our new AC G_2-metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperKaehler 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.

Original language | English |
---|---|

Number of pages | 53 |

Journal | Journal of the European Mathematical Society |

Publication status | Acceptance date - 26 Nov 2019 |