Infinitely many new families of complete cohomogeneity one G_2-manifolds: G_2 analogues of the Taub-NUT and Eguchi-Hanson spaces

TL;DR

This paper constructs infinitely many new complete G_2-manifolds with controlled asymptotic geometries, including AC and ALC types, and introduces a conically singular G_2 space, expanding the known landscape of G_2-geometry.

Contribution

It presents new families of complete G_2-manifolds with diverse asymptotic behaviors and introduces a conically singular G_2 space, extending the known examples beyond Bryant-Salamon.

Findings

Infinitely many new G_2-manifolds with ALC and AC geometries.

A conically singular G_2 space analogous to Taub-NUT.

New diffeomorphism types of AC G_2-manifolds.

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 K\"ahler 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 hyperK\"ahler 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.