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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2011.00482 |
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Table of Contents:
- The resolution of the $G_2$-orbifold $T^7/Γ$, where $Γ$ is a suitably chosen finite group, admits a $1$-parameter family of $G_2$-structures with small torsion $φ^t$, obtained by gluing in Eguchi-Hanson spaces. It was shown by Joyce that $φ^t$ can be perturbed to torsion-free $G_2$-structures $\tildeφ^t$ for small values of $t$. Using norms adapted to the geometry of the manifold we give an alternative proof of the existence of $\tildeφ^t$. This alternative proof produces the estimate $\left|\left| \tildeφ^t-φ^t \right|\right|_{C^0} \leq ct^{5/2}$. This is an improvement over the previously known estimate $\left|\left| \tildeφ^t-φ^t \right|\right|_{C^0} \leq ct^{1/2}$. As part of the proof, we show that Eguchi-Hanson space admits a unique (up to scaling) harmonic form with decay, which is a result of independent interest.