Improved Estimates for $G_2$-structures on the Generalised Kummer Construction

TL;DR

This paper improves the estimate for torsion-free $G_2$-structures near orbifold resolutions, using adapted norms and harmonic form analysis, leading to a sharper convergence rate in the generalized Kummer construction.

Contribution

It provides an alternative proof with a better estimate for the approximation of torsion-free $G_2$-structures, and establishes the uniqueness of a harmonic form on Eguchi-Hanson space.

Findings

Improved estimate: || ilde{}^t - ^t||_{C^0} \u2264 c t^{5/2}

Existence of a unique harmonic form with decay on Eguchi-Hanson space

Alternative proof method using adapted norms

Abstract

The resolution of the $G_2$-orbifold $T^7/\Gamma$, where $\Gamma$ is a suitably chosen finite group, admits a $1$-parameter family of $G_2$-structures with small torsion $\varphi^t$, obtained by gluing in Eguchi-Hanson spaces. It was shown by Joyce that $\varphi^t$ can be perturbed to torsion-free $G_2$-structures $\tilde{\varphi}^t$ for small values of $t$. Using norms adapted to the geometry of the manifold we give an alternative proof of the existence of $\tilde{\varphi}^t$. This alternative proof produces the estimate $\left|\left| \tilde{\varphi}^t-\varphi^t \right|\right|_{C^0} \leq ct^{5/2}$. This is an improvement over the previously known estimate $\left|\left| \tilde{\varphi}^t-\varphi^t \right|\right|_{C^0} \leq ct^{1/2}$. 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.