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

TL;DR

This paper introduces a new analytic method to construct complete non-compact G2-manifolds starting from asymptotically conical Calabi-Yau 3-folds, producing families of metrics with controlled asymptotic geometry and expanding known examples.

Contribution

The authors develop a novel technique to generate infinite families of non-compact G2-manifolds from Calabi-Yau bases, significantly broadening the known landscape of such manifolds.

Findings

Constructed infinitely many diffeomorphism types of G2-manifolds.

Proved existence of high-dimensional continuous families of G2-metrics.

Produced G2-metrics with asymptotically locally conical geometry.

Abstract

We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics on M that collapses to the original Calabi-Yau metric on the base B as the parameter converges to 0. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics, and are the natural higher-dimensional analogues of the ALF metrics that are well known in 4-dimensional hyperk\"ahler 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.