Infinitely many new families of complete cohomogeneity one G2-manifolds: G2analogues of the Taub-NUT and Eguchi-Hanson spaces

Lorenzo Foscolo, Mark Haskins, Johannes Nordström

Research output: Contribution to journalArticlepeer-review

Abstract

We construct infinitely many new 1-parameter families of simply connected complete non-compact G2-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 G2-manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G2-manifolds. We also 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 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 AC G2-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.

Original languageEnglish
Pages (from-to)2153-2220
Number of pages68
JournalJournal of the European Mathematical Society
Volume23
Issue number7
Early online date8 Mar 2021
DOIs
Publication statusPublished - 2021

Keywords

  • Cohomogeneity one metrics
  • Collapsed Riemannian manifolds
  • Differential geometry
  • Einstein and Ricci-flat metrics
  • Non-compact Gholonomy manifolds
  • Special and exceptional holonomy

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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