Infinitely many new families of complete cohomogeneity one G$_2$-manifolds: G$_2$ analogues of the Taub–NUT and Eguchi–Hanson spaces
Infinitely many new families of complete cohomogeneity one G$_2$-manifolds: G$_2$ analogues of the Taub–NUT and Eguchi–Hanson spaces
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完全同质的无限多个新族一 G$_2$-流形：Taub–NUT 和 Eguchi–Hanson 空间的 G$_2$ 类似物
DOI：
10.4171/jems/1051
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Johannes Nordstrom
中科院分区：
文献类型：
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作者：
Lorenzo Foscolo;M. Haskins;Johannes Nordstrom
我们构造了无穷多个具有控制几何的单连通完全非紧g_2流形的新的1参数族。每个族的一般成员都具有所谓的渐近局部圆锥（ALC）几何。然而，在两个特殊参数值处，渐近几何的性质发生了变化：在一个特殊参数值处，我们得到了每个族中具有渐近圆锥（AC）几何的唯一成员；在接近其他特殊参数值时，度量族分解为AC Calabi-Yau 3-fold。特别值得注意的是，我们的无限多的AC g_2 -流形的新微分同构类型：先前由Bryant和Salamon在1989年构造的三个例子提供了唯一已知的单连通AC g_2 -流形。
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.
DOI：
10.1515/9781400859306
发表时间：
1988
期刊：
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影响因子：
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作者：
M. Atiyah;N. Hitchin
通讯作者：
M. Atiyah;N. Hitchin