论文标题
$ g_2 $ - 结构的几何流量在3-Sasakian 7-manifolds上
Geometric Flows of $G_2$-Structures on 3-Sasakian 7-Manifolds
论文作者
论文摘要
7个模型上的3-Sasakian结构可用于定义两个不同的爱因斯坦指标：3-Sasakian公制和挤压的爱因斯坦公制。这两个指标均由几乎平行的$ g_2 $结构引起，这些结构也可以根据3-Sasakian结构表示。正如爱因斯坦指标是RICCI流动以重新进行的关键点一样，几乎平行的$ g_2 $结构提供了$ G_2 $结构的（重新定位）几何流量的自然关键点，称为Laplacian Flow和Laplacian Coflow。我们在3-Sasakian环境中研究了每个流动，并确保它们的行为明显不同，尤其是关于几乎平行的$ G_2 $结构的稳定性。我们还将$ g_2 $ - 结构的流量与（重新缩放的）RICCI流动进行比较。
A 3-Sasakian structure on a 7-manifold may be used to define two distinct Einstein metrics: the 3-Sasakian metric and the squashed Einstein metric. Both metrics are induced by nearly parallel $G_2$-structures which may also be expressed in terms of the 3-Sasakian structure. Just as Einstein metrics are critical points for the Ricci flow up to rescaling, nearly parallel $G_2$-structures provide natural critical points of the (rescaled) geometric flows of $G_2$-structures known as the Laplacian flow and Laplacian coflow. We study each of these flows in the 3-Sasakian setting and see that their behaviour is markedly different, particularly regarding the stability of the nearly parallel $G_2$-structures. We also compare the behaviour of the flows of $G_2$-structures with the (rescaled) Ricci flow.
