Aaron Kennon, UC Santa Barbara

Title: Nearly Parallel G₂-Structures from the Perspective of Geometric Flows

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 G₂-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 G₂-structures provide natural critical points of the (rescaled) geometric flows of G₂-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 G₂-structures. We also compare the behaviour of the flows of G₂-structures with the (rescaled) Ricci flow.

This is joint work with Jason Lotay.