## Abstract

We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7- manifolds .M;g/ whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger-Fukaya-Gromov theory of collapse with bounded curvature. The construction starts with a complete noncompact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M ! B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics g_ on M that collapses with bounded curvature as _ ! 0 to the original Calabi-Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry. We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact G2-metrics were known.

Original language | English |
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Pages (from-to) | 3323-3416 |

Number of pages | 94 |

Journal | Duke Mathematical Journal |

Volume | 170 |

Issue number | 15 |

DOIs | |

Publication status | Published - 15 Oct 2021 |

### Bibliographical note

Funding Information:Foscolo’s work was partially supported by a Royal Society University Research Fellowship and National Science Foundation (NSF) grant DMS-1608143. Haskins’s and Nordström’s work was partially supported by the Simons Collaboration “Special Holonomy in Geometry, Analysis and Physics” (grants #488620 and #488631). Research at MSRI was partially supported by NSF grant DMS-1440140.

Publisher Copyright:

© 2021 Duke University Press. All rights reserved.

## Keywords

- math.DG
- hep-th
- 53C10, 53C25, 53C29, 53C80

## ASJC Scopus subject areas

- Geometry and Topology
- Mathematical Physics