TY - JOUR

T1 - Infinitely many new families of complete cohomogeneity one G2-manifolds

T2 - G2analogues of the Taub-NUT and Eguchi-Hanson spaces

AU - Foscolo, Lorenzo

AU - Haskins, Mark

AU - Nordström, Johannes

N1 - Funding Information:
All three authors would like to thank MSRI and the program organisers for providing an extremely productive environment during the Differential Geometry Program at MSRI in Spring 2016. Research at MSRI was partially supported by NSF grant DMS-1440140.
Funding Information:
LF and MH would like to thank the Riemann Centre for Geometry and Physics at Leibniz Universität Hannover for supporting LF under a Riemann Research Fellowship and for hosting a research visit of MH in Autumn 2015. They would also like to thank the Newton Institute in Cambridge for hosting them during the Metric and Analytic Aspects of Moduli Spaces workshop in Summer 2015. Discussions about this work took place at both institutions.
Funding Information:
Acknowledgments. LF would like to thank the Royal Society for the support of his research under a University Research Fellowship and NSF for the partial support of his work under grant DMS-1608143. MH and JN would like to thank the Simons Foundation for its support of their research under the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics (grants #488620, Mark Haskins, and #488631, Johannes Nordström).

PY - 2021/12/31

Y1 - 2021/12/31

N2 - 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.

AB - 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.

KW - Cohomogeneity one metrics

KW - Collapsed Riemannian manifolds

KW - Differential geometry

KW - Einstein and Ricci-flat metrics

KW - Non-compact Gholonomy manifolds

KW - Special and exceptional holonomy

UR - http://www.scopus.com/inward/record.url?scp=85108434857&partnerID=8YFLogxK

U2 - 10.4171/JEMS/1051

DO - 10.4171/JEMS/1051

M3 - Article

AN - SCOPUS:85108434857

SN - 1435-9855

VL - 23

SP - 2153

EP - 2220

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 7

ER -