Abstract:
We will discuss the Laplacian flow for closed \(G_2\) structures. This flow was introduced by R. Bryant in 1992 to study the geometry of \(G_2\) structures, inspired by Hamilton's Ricci flow in studying the generic Riemannian structures and the Kähler Ricci flow in studying the Kähler structures. The primary goal is to understand the conditions under which the Laplacian flow can converge to a torsion free \(G_2\) structure, and thus Ricci flat metric with holonomy \(G_2\). I will start with the background of \(G_2\) structure and the motivation of introducing the Laplacian flow, and then describe my recent results on this flow. This is based on joint work with Jason D. Lotay (UCL).