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

TL;DR

This paper investigates the geometry of $G_2$-moduli spaces, deriving energy formulas for paths, and demonstrates their incompleteness along degenerating families of $G_2$-manifolds, with implications for the structure of these moduli spaces.

Contribution

It provides a new energy formula for paths in $G_2$-moduli spaces and shows their incompleteness in certain degenerating families, advancing understanding of $G_2$-geometry.

Findings

Derived a formula for the energy of paths in $G_2$-moduli spaces.

Proved that moduli spaces of certain $G_2$-manifolds are incomplete.

Identified conditions under which paths have finite energy and length.

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.