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

Daniel Platt

TL;DR

The paper provides improved control on the deformation from a small-torsion $G_2$-structure $oldsymbol{ ho}^t$ to a torsion-free structure on the generalized Kummer resolution $N_t$, establishing a sharper $C^0$-estimate $ig rbracket ilde{oldsymbol{ ho}}^t-oldsymbol{ ho}^tig rbracket_{C^0} \\le c t^{5/2}$ than Joyce’s prior $t^{1/2}$ bound. It achieves this by introducing geometry-adapted weighted Hölder norms, decomposing forms into a fibrewise-harmonic part supported near the Eguchi–Hanson fibers and a remainder, and proving a robust linear inverse bound for the Laplacian on $N_t$ uniform in the gluing parameter $t$. A key analytic input is the uniqueness of a decaying harmonic 2-form on the Eguchi–Hanson space, which underpins the kernel/obstruction analysis. The results yield a refined perturbative construction of torsion-free $G_2$-structures on the generalized Kummer construction and have potential implications for the study of associative submanifolds and $G_2$-instantons in this setting.

Abstract

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 $\varphi^t$, 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.

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

TL;DR

The paper provides improved control on the deformation from a small-torsion -structure to a torsion-free structure on the generalized Kummer resolution , establishing a sharper -estimate than Joyce’s prior bound. It achieves this by introducing geometry-adapted weighted Hölder norms, decomposing forms into a fibrewise-harmonic part supported near the Eguchi–Hanson fibers and a remainder, and proving a robust linear inverse bound for the Laplacian on uniform in the gluing parameter . A key analytic input is the uniqueness of a decaying harmonic 2-form on the Eguchi–Hanson space, which underpins the kernel/obstruction analysis. The results yield a refined perturbative construction of torsion-free -structures on the generalized Kummer construction and have potential implications for the study of associative submanifolds and -instantons in this setting.

Abstract

The resolution of the -orbifold , where is a suitably chosen finite group, admits a -parameter family of -structures with small torsion , obtained by gluing in Eguchi-Hanson spaces. It was shown by Joyce that can be perturbed to torsion-free -structures for small values of . Using norms adapted to the geometry of the manifold we give an alternative proof of the existence of . This alternative proof produces the estimate . This is an improvement over the previously known estimate . 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.

Paper Structure

This paper contains 19 sections, 37 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Definition of the Eguchi-Hanson Space
  4. $G_2$-structures
  5. Harmonic Forms with Decay on the Eguchi-Hanson Space
  6. Harmonic Forms on $(\mathbb{C}^2\setminus \{0\})/\{ \pm 1\}$
  7. Harmonic Forms on Eguchi-Hanson Space
  8. Torsion-Free $G_2$-Structures on the Generalised Kummer Construction
  9. Resolutions of $T^7/\Gamma$
  10. The Laplacian on $\mathbb{R}^3 \times X_{\text{EH}}$
  11. The Laplacian on $N_t$
  12. The Existence Theorem
  13. Proof of the linear estimate \ref{['proposition:x-y-laplace-estimate-N_t']}
  14. The approximate kernel
  15. Estimates for the composite norms
  16. ...and 4 more sections

Key Result

Theorem 1

Choose $\alpha \in (0,1)$ and $\beta \in (-1,0)$ both close to $0$. Let $N_t$ be the resolution of $T^7/\Gamma$ from equation:resolution-of-t7-gamma and $\varphi^t \in \Omega^3(N_t)$ the $G_2$-structure with small torsion from equation:g2-structure-on-kummer-construction. There exists $c>0$ independ In particular,

Figures (3)

  • Figure 1: Blowup analysis near the associative is reduced to the analysis of the Laplacian on $\mathbb{R}^3 \times {X_{\text{EH}}}$. The figure is taken from Platt2024.
  • Figure 2: Blowup analysis away from the associative is reduced to the analysis of the Laplacian on $T^7/\Gamma$. The figure is taken from Platt2024.
  • Figure 3: Blowup analysis in the neck region is reduced to the analysis of the Laplacian on $\mathbb{R}^3 \times \mathbb{R}^4$. The figure is taken from Platt2024.

Theorems & Definitions (78)

  • Theorem
  • Theorem
  • Proposition 2.1
  • proof
  • Remark 2.3
  • Definition 2.4: Definition 7.2.1 in Joyce2000
  • Proposition 2.6: Example 7.2.2 in Joyce2000
  • Remark 2.8
  • Definition 2.9: Definition 10.1.1 in Joyce2000
  • Definition 2.12
  • ...and 68 more