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.

arXiv comment23 pages and 3 figures; v2: minor changes, version accepted for publication in Journal of Geometry and Physics

References in corpus (8)

• Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
• Stability and instability of Ricci solitons
• Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry
• Short-time behaviour of a modified Laplacian coflow of G2-structures
• 3-Sasakian manifolds in dimension seven, their spinors and G_2 structures
• New invariants of G_2-structures
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• 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