On the incompleteness of $G_2$-moduli spaces along degenerating families of $G_2$-manifolds

Abstract

We derive a formula for the energy of a path in the moduli space of a compact $G_2$-manifold with vanishing first Betti number for the volume-normalised $L^2$-metric. This allows us to give simple sufficient conditions for a path of torsion-free $G_2$-structures to have finite energy and length. We deduce that the compact $G_2$-manifolds produced by the generalised Kummer construction have incomplete moduli spaces. Under some assumptions, we also state a necessary condition for the limit of a path of torsion-free $G_2$-structures to be at infinite distance in the moduli space.

arXiv commentv2. Minor typos fixed, statements of Lemma 3 and Corollary 5 clarified, and remarks added at the end of Section 4. To appear in IMRN

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Cited by in corpus (1)

• Geometry and periods of $G_2$-moduli spaces