论文标题
二次封闭G2结构
Quadratic closed G2-structures
论文作者
论文摘要
本文研究封闭了满足二次条件的G2结构，这是由科比引入的二阶PDE系统，涉及$λ$λ。 $λ= 1/6 $），并且封闭的G2结构是拉普拉斯操作员的特征形式的条件（$λ= 0 $）。在本文的工作之前，二次系统的解决方案仅以$λ= 1/6，$ -1/8，$和2/5，$而闻名。 在本文中，我们提供了许多ERP G2结构的新示例，包括完整的不均匀ERP G2结构的第一个示例，以及紧凑的ERP G2结构的新示例。我们还提供了均匀的ERP G2结构的分类。我们提供了第一个二次封闭式G2结构的示例，价格为$λ= -1，$ 1/3，$和$ 3/4，$，以及无限的许多新示例，$λ= -1/8 $和$ 2/5. $ $ $我们的结构涉及封闭的G2结构的特殊Torsion概念，这是一个新概念，这是一个新的概念，可能是可应用的。 在本文的最后一部分中，我们为Laplacian流量提供了两个不均匀的完全稳定梯度孤子，这是第一个已知的示例。
This article studies closed G2-structures satisfying the quadratic condition, a second-order PDE system introduced by Bryant involving a parameter $λ.$ For certain special values of $λ$ the quadratic condition is equivalent to the Einstein condition for the metric induced by the closed G2-structure (for $λ= 1/2$), the extremally Ricci-pinched (ERP) condition (for $λ=1/6$), and the condition that the closed G2-structure be an eigenform for the Laplace operator (for $λ= 0$). Prior to the work in this article, solutions to the quadratic system were known only for $λ= 1/6,$ $-1/8,$ and $2/5,$ and for these values only a handful of solutions were known. In this article, we produce infinitely many new examples of ERP G2-structures, including the first example of a complete inhomogeneous ERP G2-structure and a new example of a compact ERP G2-structure. We also give a classification of homogeneous ERP G2-structures. We provide the first examples of quadratic closed G2-structures for $λ= -1,$ $1/3,$ and $3/4,$ as well as infinitely many new examples for $λ= -1/8$ and $2/5.$ Our constructions involve the notion of special torsion for closed G2-structures, a new concept that is likely to have wider applicability. In the final section of the article, we provide two large families of inhomogeneous complete steady gradient solitons for the Laplacian flow, the first known such examples.
