Geometric Flows of $G_2$-Structures on 3-Sasakian 7-Manifolds

Abstract

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.

23 pages and 3 figures; v2: minor changes, version accepted for publication in Journal of Geometry and Physics

References in corpus (2)

• Stability and instability of Ricci solitons
• Flows of ${\mathrm{G}}_2$-structures on contact Calabi--Yau $7$-manifolds

Cited by in corpus (2)

• Calibrated Geometry in Hyperkahler Cones, 3-Sasakian Manifolds, and Twistor Spaces
• Revisiting 3-Sasakian and $G_2$-structures