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

Abstract

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

36 pages, 3 figures, to appear in Communications in Analysis and Geometry