A new construction of compact $G_2$-manifolds by gluing families of
Eguchi-Hanson spaces

We give a new construction of compact Riemannian 7-manifolds with holonomy
$G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a
proper subgroup of $G_2$) such that $M$ admits an involution $\iota$ preserving
the $G_2$-structure. Then $M/{\langle \iota \rangle}$ is a $G_2$-orbifold, with
singular set $L$ an associative submanifold of $M$, where the singularities are
locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve
this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a
nonvanishing closed and coclosed $1$-form $\lambda$ on $L$.
Much of the analytic difficulty lies in constructing appropriate closed
$G_2$-structures with sufficiently small torsion to be able to apply the
general existence theorem of the first author. In particular, the construction
involves solving a family of elliptic equations on the noncompact Eguchi-Hanson
space, parametrized by the singular set $L$. We also present two
generalizations of the main theorem, and we discuss several methods of
producing examples from this construction.