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.
Highlights
Over the past 40 years cohomogeneity one Riemannian metrics, i.e. metrics admitting an isometric Lie group action with generic orbit of codimension one, have played a distinguished role in the construction of complete Ricci-flat or Einstein metrics, in the cases of metrics with special or exceptional holonomy
We construct a closely related conically singular G2–holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G2–cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has asymptotically locally conical (ALC) geometry at infinity. We argue that this conically singular ALC G2–space is the natural G2 analogue of the Taub–NUT metric in 4-dimensional hyperKähler geometry and that our new asymptotically conical (AC) G2–metrics are all analogues of the Eguchi–Hanson metric, the simplest ALE hyperKähler manifold
The cohomogeneity one property affords a reduction of the system of nonlinear partial differential equations that characterises an Einstein metric or a holonomy reduction, to a system of nonlinear ordinary differential equations (ODEs)
Summary
Over the past 40 years cohomogeneity one Riemannian metrics, i.e. metrics admitting an isometric Lie group action with generic orbit of codimension one, have played a distinguished role in the construction of complete Ricci-flat or Einstein metrics, in the cases of metrics with special or exceptional holonomy. In each case there are strong restrictions on the values of the constants p and q compatible with smooth extension over a given singular orbit type and in all cases there is (up to scale) a 1-parameter family of smooth solutions defined in a neighbourhood of each type of singular orbit To prove these results we adapt to our first-order ODE systems, the representation-theoretic approach to singular initial value problems for cohomogeneity one Einstein metrics developed by Eschenburg–Wang [32]. Recall that the analysis of the ODEs on the principal orbits showed that, for each fixed p and q, there is a 2-parameter family of local SU(2) × SU(2) × U(1)–invariant solutions Putting this together, a complete AC G2–metric corresponds to an intersection point of two curves in a 2-dimensional manifold.
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