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 $\mathbb{R}^3 \times (\mathbb{R}^4 / {\lbrace \pm 1 \rbrace})$. 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.

中文翻译：

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通过粘接Eguchi–Hanson空间族的新构造的紧凑型无扭转$ G_2 $流形

我们给出了新的紧致黎曼$ 7 $流形与完整$ G_2 $的构造。假设$ M $是无扭转的$ G_2 $-流形（可以使完整的$ G_2 $子集具有完整性），使得$ M $承认保留$ G_2 $结构的对合$ \ iota $。那么$ M / {\ langle \ iota \ rangle} $是一个$ G_2 $-歧义，其中奇异集$ L $是$ M $的一个关联子流形，其中奇异点在本地的形式为$ \ mathbb {R} ^ 3 \ times（\ mathbb {R} ^ 4 / {\ lbrace \ pm 1 \ rbrace}）$。我们通过粘在一个Eguchi–Hanson空间家族中来解决这个难题，其特点是在$ L $上封闭且共封闭的1形式$ \ lambda $消失了。分析上的大部分困难在于构造具有足够小的扭转量的适当的闭合$ G_2 $结构，以能够应用第一作者的一般存在性定理。特别是，构造涉及在非紧致的Eguchi–Hanson空间上求解一类椭圆方程，由奇异集$ L $参数化。我们还给出了主定理的两种概括，并讨论了从该构造中产生示例的几种方法。