Aloff-Wallach Spaces: Volumes, Curvatures, Injectivity Radii

Aloff-Wallach空间上Yamabe问题齐次解的分岔与局部刚性

Bifurcation and Local Rigidity of Homogeneous Solutions to the Yamabe Problem on Aloff-Wallach Spaces

本文通过构造Aloff-Wallach空间上的Yamabe问题单参数解族并考察Morse指数变化确定分岔时刻，同时定义并区分了分岔点与局部刚性点。

Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighborhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed b…

以一般Aloff–Wallach空间为主轨道的余齐性一 $\mathrm{Spin}(7)$ 度量

Cohomogeneity one $\mathrm{Spin}(7)$ metrics with generic Aloff--Wallach spaces as principal orbits

基于Reidegeld的初值分析，证明了以一般Aloff–Wallach空间为主轨道、CP^2为奇异轨道的向前完备余齐性一Spin(7)度量的存在性，构造了三个连续单参数族非紧Spin(7)度量，其中两个AC度量共享相同渐近锥，展现了几何过渡现象。

$\bar{\nu}$-不变量的$G_2$结构在阿洛夫-沃拉奇空间上

The Crowley-Nordström invariants of $G_2$-Structures on Aloff--Wallach Spaces

研究基于Goette公式计算阿洛夫-沃拉奇空间上近平行$G_2$结构的$\bar{\nu}$-不变量，发现两种结构的$\bar{\nu}$值分别为±41，并比较了3-Sasakian结构下的不变量。

Spin(7) metrics of cohomogeneity one with Aloff--Wallach spaces as principal orbits

In this article, we construct two continuous 1-parameter family of non-compact $\mathrm{Spin}(7)$ metrics with both chiralities, with the principal orbit an Aloff--Wallach space $N_{k,l}$ and the singular orbit $\mathbb{CP}^2$. For generic $N_{k,l}$, metrics constructed are locally asymptotically conical (ALC). For $N_{1,1}$, we construct two continuous 1-parameter families with geometric transition from asymptotically conical (AC) metrics to ALC metrics.

Gauge theory on Aloff-Wallach spaces

For gauge groups $U(1)$ and $SO(3)$ we classify invariant $G_2$-instantons for homogeneous coclosed $G_2$-structures on Aloff-Wallach spaces $X_{k,l}$. As a consequence, we give examples where $G_2$-instantons can be used to distinguish between different strictly nearly parallel $G_2$-structures on the same Aloff-Wallach space. In addition to this, we find that while certain $G_2$-instantons exist for the strictly nearly parallel $G_2$-structure on $X_{1,1}$, no such $G_2$-instantons exist for the tri-Sasakian one. As a further consequence of the classification, we produce examples of some oth…