论文标题
改进了广义Kummer Construction $ g_2 $结构的估计
Improved Estimates for $G_2$-structures on the Generalised Kummer Construction
论文作者
论文摘要
$ g_2 $ -orbifold $ t^7/γ$的分辨率，其中$γ$是一个适当选择的有限群体，承认$ 1 $ - 参数$ g_2 $结构，带有小质量$φ^t $，是通过在eguchi-hanson Space中获得的。乔伊斯（Joyce）表明，对于$ t $的小值，$φ^t $可能会扰动$ g_2 $ - 结构$ \tildeφ^t $。使用适合歧管几何形状的规范，我们给出了$ \tildeφ^t $存在的替代证明。此替代证明会产生估计$ \ left | \ left | \tildeφ^t-φ^t \ right | \ right | _ {c^0} \ leq ct^{5/2} $。这是对先前已知的估计$ \ left | \ left |的改进。 \tildeφ^t-φ^t \ right | \ right | _ {c^0} \ leq ct^{1/2} $。作为证据的一部分，我们表明，Eguchi-Hanson Space以衰减的形式承认了一种独特的（缩放）谐波形式，这是独立兴趣的结果。
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.
