Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds

Abstract

We develop a powerful new analytic method to construct complete non-compact Ricciflat 7-manifolds, more specifically G2-manifolds, i.e. 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 non-compact 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 non-compact 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 gǫ on M that collapses with bounded curvature as ǫ → 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 ALF metrics that are well known in 4-dimensional hyperkähler geometry.We give two illustrations of the strength of our method. Firstly we use it to construct infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Secondly we use it to prove the existence of continuous families of complete non-compact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete non-compact G2-metrics were known.

Complete non-compact Spin(7) manifolds from self-dual Einstein 4-orbifolds

“…In [31], in collaboration with Haskins and Nordström, we developed a similar construction of highly collapsed ALC G 2 -holonomy metrics on suitable principal circle bundles over smooth AC Calabi-Yau 3-folds. The construction of [31] allowed us to exploit recent progress on the existence of Calabi-Yau cone metrics [22,35,39] and AC Calabi-Yau metrics [23,42,85] to produce infinitely many complete non-compact G 2 -manifolds and complete G 2 -metrics depending on an arbitrarily large number of parameters. Only a handful of complete non-compact G 2 -manifolds was previously known.…”

“…The existence of an analogous construction of Spin (7)-metrics from AC G 2 -manifolds as in Theorem A is therefore not in itself surprising. The fact that such a construction can be used to produce significant results in Spin(7)-geometry, however, is a priori much less clear: the naive generalisation of [31] to the Spin(7)-setting using only smooth manifolds would be a theorem that currently applies to only one example! The simultaneous extension of [31] to the orbifold setting is the crucial new ingredient that makes Theorem A useful.…”

“…The fact that such a construction can be used to produce significant results in Spin(7)-geometry, however, is a priori much less clear: the naive generalisation of [31] to the Spin(7)-setting using only smooth manifolds would be a theorem that currently applies to only one example! The simultaneous extension of [31] to the orbifold setting is the crucial new ingredient that makes Theorem A useful. Indeed, in contrast to the Calabi-Yau case, our current knowledge of smooth AC G 2 -manifolds is extremely limited: in 1989 Bryant-Salamon [17] constructed three (explicit) examples of AC G 2 -metrics; only very recently [32, Theorem C] an infinite family of new simply connected examples has been found.…”

“…The analytic framework introduced in this paper to work on AC orbifolds can also be exploited to extend the construction of complete ALC G 2 -metrics in [31] to the orbifold setting. In [31] examples of ALC G 2 -metrics arose from AC Calabi-Yau metrics on crepant resolutions of Calabi-Yau cones.…”

“…The analytic framework introduced in this paper to work on AC orbifolds can also be exploited to extend the construction of complete ALC G 2 -metrics in [31] to the orbifold setting. In [31] examples of ALC G 2 -metrics arose from AC Calabi-Yau metrics on crepant resolutions of Calabi-Yau cones. Often it is natural to consider only partial resolutions of Calabi-Yau cones, which replace the singularity of the cone with simpler (albeit non-isolated) orbifold singularities.…”