Iterated collapsing phenomenon on $G_2$-manifolds

TL;DR

This paper introduces a novel collapsing mechanism for $G_2$-metrics involving circle bundles over K3 fibrations, with formal solutions and discussions on compactification and $Spin(7)$ analogues.

Contribution

It presents a new collapsing mechanism for $G_2$-metrics and develops formal solutions in a local smooth setting, expanding understanding of geometric structures.

Findings

Existence of formal power series solutions for the collapsing mechanism.

Description of the collapsing structure involving circle bundles over K3 fibrations.

Discussion on the compactification problem and $Spin(7)$ analogues.

Abstract

We propose a new collapsing mechanism for $G_2$-metrics, with the generic region admitting a circle bundle structure over a K3 fibration over a Riemann surface. The adiabatic description involves a weighted version of the maximal submanifold equation. In a local smooth setting we prove the existence of formal power series solutions, and the problem of compactification is discussed at a heuristic level. The $Spin(7)$ analogue is also discussed.