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arXiv · 2405.02943

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

Thibault Langlais
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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.

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Thibault Langlais. 2024-05-05. On the incompleteness of $G_2$-moduli spaces along degenerating families of $G_2$-manifolds. https://doi.org/10.1093/imrn%2Frnaf069

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