AI 中文总结
本文研究紧致7维流形上极值Ricci-pinched $G_2$-结构的性质，给出其等价刻画，证明自同构群有限，并构造保持该条件的扰动。
AI 中文摘要
我们研究了紧致$7$维流形上极值Ricci-pinched（ERP）$G_2$-结构的性质。我们利用无迹$\tast$-Ricci张量、挠率的Hodge拉普拉斯算子、Ricci特征值、拉普拉斯流以及$27$维Weyl曲率分量给出了ERP条件的刻画。我们还证明了紧致ERP $G_2$-结构的自同构群是有限的。随后，我们识别了保持ERP条件的ERP $G_2$-结构的扰动，包括一般情形和具体例子。
英文摘要
We study properties of Extremally Ricci-Pinched (ERP) $G_2$-Structures on compact $7$-manifolds. We provide characterizations of the ERP condition in terms of the traceless $\ast$-Ricci tensor, the Hodge Laplacian of the torsion, the Ricci eigenvalues, the Laplacian flow, and the $27$-dimensional Weyl curvature component. We also prove that the automorphism group of a compact ERP $G_2$-Structure is finite. We then identify perturbations of ERP $G_2$-Structures, both in general and for specific examples, that preserve the ERP condition.
Comments17 pages
