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15 October 2021 Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds
Lorenzo Foscolo, Mark Haskins, Johannes Nordström
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Duke Math. J. 170(15): 3323-3416 (15 October 2021). DOI: 10.1215/00127094-2020-0092

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 MB satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gϵ on M that collapses with bounded curvature as ϵ0 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.

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Lorenzo Foscolo. Mark Haskins. Johannes Nordström. "Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds." Duke Math. J. 170 (15) 3323 - 3416, 15 October 2021. https://doi.org/10.1215/00127094-2020-0092

Information

Received: 9 August 2019; Revised: 3 September 2020; Published: 15 October 2021
First available in Project Euclid: 28 September 2021

Digital Object Identifier: 10.1215/00127094-2020-0092

Subjects:
Primary: 53C29
Secondary: 53C25

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 15 • 15 October 2021
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