Complete noncompact g2-manifolds from asymptotically conical calabi-yau 3-folds

Journal Article (Journal Article)

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.

Full Text

Duke Authors

Cited Authors


Published Date

  • October 15, 2021

Published In

Volume / Issue

  • 170 / 15

Start / End Page

  • 3323 - 3416

International Standard Serial Number (ISSN)

  • 0012-7094

Digital Object Identifier (DOI)

  • 10.1215/00127094-2020-0092

Citation Source

  • Scopus