Improved Estimates for G₂-structures on the Generalised Kummer Construction

2020/11/01 by Daniel E. Platt, Platt, Daniel
Mathematics · #35J60 (Secondary) #53C29 (Primary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2011.00482

openalex publication_date 2020/11/01 · openalex created_date 2022/02/24 · openalex updated_date 2026/07/29

Abstract

The resolution of the G₂-orbifold T⁷/Γ, where Γ is a suitably chosen finite group, admits a 1-parameter family of G₂-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 G₂-structures φ^t for small values of t. Using norms adapted to the geometry of the manifold we give an alternative proof of the existence of φ^t. This alternative proof produces the estimate || φ^t-φ^t ||_C⁰ ≤ ct^5/2. This is an improvement over the previously known estimate || φ^t-φ^t ||_C⁰ ≤ 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.

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