标题： 关于具有余维二对称性的$G_2$流形 显示英文标题

标题： On $G_2$ manifolds with cohomogeneity two symmetry

摘要： 我们考虑具有余维二的$\mathbb{T}^2\times \mathrm{SU}(2)$对称群的$G_2$流形。 我们给出这些流形的局部特征，并描述其中余维一校准子流形的几何性质，包括正则性和奇异性分析。 我们将这些结果应用于 Foscolo--Haskins--Nordström 最近构造的流形以及拓扑为$S^3\times \mathbb{R}^4$的 Bryant--Salamon 流形。 特别是，我们在其中描述了新的完整$\mathbb{T}^2$-不变的关联子流形的大族。 显示英文摘要

摘要： We consider $G_2$ manifolds with a cohomogeneity two $\mathbb{T}^2\times \mathrm{SU}(2)$ symmetry group. We give a local characterization of these manifolds and we describe the geometry, including regularity and singularity analysis, of cohomogeneity one calibrated submanifolds in them. We apply these results to the manifolds recently constructed by Foscolo--Haskins--Nordstr\"om and to the Bryant--Salamon manifold of topology $S^3\times \mathbb{R}^4$. In particular, we describe new large families of complete $\mathbb{T}^2$-invariant associative submanifolds in them.