Define a suitable G2-structure around zeros of the harmonic 1-form in the Joyce–Karigiannis gluing construction

Construct a G2-structure in a neighborhood of a zero of the harmonic 1-form λ on the associative submanifold L = fix(Γ) within the Joyce–Karigiannis framework for resolving the orbifold M/Γ by gluing Eguchi–Hanson spaces, so that the construction can be carried out even when λ vanishes.

Background

In the Joyce–Karigiannis construction of compact torsion-free G2-manifolds, one starts from a G2-orbifold M/Γ, with fixed locus L = fix(Γ) an associative 3-manifold, and resolves singularities by gluing Eguchi–Hanson spaces over points of L. The gluing requires a harmonic 1-form λ on L which sets the gluing scale, and the construction currently works only when λ is nowhere zero.

The paper notes that extending the construction to cases where λ has zeros would require defining a suitable G2-structure in a neighborhood of each zero. This is identified explicitly as an open problem and is referenced to Joyce–Karigiannis (2021).