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 ${\mathrm{G}}_2$-manifolds, that is, Riemannian 7-manifolds $(M,g)$ whose holonomy group is the compact exceptional Lie group ${\mathrm{G}}_2$. 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 $M\to B$ satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete ${\mathrm{G}}_2$-metrics $g_{\mathit{ϵ}}$ on M that collapses with bounded curvature as $\mathit{ϵ}\to 0$ to the original Calabi–Yau metric on the base B. The ${\mathrm{G}}_2$-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 ${\mathrm{G}}_2$-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 ${\mathrm{G}}_2$-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact ${\mathrm{G}}_2$-metrics were known.