Iterated collapsing phenomenon on $G_2$-manifolds

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.

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Received 10 February 2021

Received revised 5 February 2022

Accepted 5 February 2022

Published 24 July 2022