Lorenzo Foscolo, Mark Haskins, Johannes Nordström

Duke Math. J. 170 (15), 3323-3416, (15 October 2021) DOI: 10.1215/00127094-2020-0092
KEYWORDS: Differential geometry, Riemannian geometry, Einstein metrics, exceptional holonomy, applications to physics, geometric analysis, partial differential equations, 53C29, 53C25

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 ${g}_{\mathit{\u03f5}}$ on *M* that collapses with bounded curvature as $\mathit{\u03f5}\to 0$ 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.