Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 3

Iterated collapsing phenomenon on $G_2$-manifolds

Pages: 971 – 1036

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n3.a5


Yang Li (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


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.

Received 10 February 2021

Received revised 5 February 2022

Accepted 5 February 2022

Published 24 July 2022