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

TL;DR

This paper investigates the behavior of geometric flows of $G_2$-structures on 3-Sasakian 7-manifolds, revealing differences in stability and comparing with Ricci flow, to understand their geometric evolution.

Contribution

It analyzes the Laplacian flow and coflow of $G_2$-structures in the 3-Sasakian setting, highlighting their distinct behaviors and stability properties.

Findings

Laplacian flow and coflow exhibit markedly different behaviors.

Nearly parallel $G_2$-structures show different stability characteristics.

Comparison with Ricci flow reveals unique flow dynamics.

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.