Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds

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 gϵ on M that collapses with bounded curvature as ϵ→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.