Flows of Geometric Structures, especially G2-Structures

Spiro Karigiannis (University of Waterloo)

These lectures will provide an introduction to flows of geometric structures, with a specific focus on G2-structures. We begin by discussing the Ricci flow, which is the classical example of a flow of geometric structures. We use the Ricci flow to introduce the main ideas which arise in the study of geometric flows, including: short-time existence, characterization of blow-up time, singularities and their relation to solitons, Shi-type estimates, long-time existence, convergence, and stability.

We discuss foundational aspects of parabolic PDEs, using the heat equation for differential forms on compact oriented Riemannian manifolds as a motivating example. We show that the Ricci flow is not parabolic, but can nevertheless be viewed as a “heat-type” flow. The failure of the Ricci flow to be parabolic is due precisely to diffeomorphism invariance. This will be examined in detail, and the classical DeTurck trick will be explained. We will also briefly explore the Kähler-Ricci flow, discussing its preservation of Kählerity and its use in providing an alternative proof of the Calabi conjecture.

We then focus specifically on G2-structures. The lectures will include an introduction to the differential geometry G2-structures, and will survey some known results on various flows of G2-structures such as the Laplacian flow, the coflow and modified coflow, the isometric flow, and the Dirichlet gradient flow. Finally, the series will delve into the classification of second-order differential invariants of G2-structures, based on Riemann curvature and the covariant derivative of the torsion tensor, and discuss the relationships these invariants have under the G2-Bianchi identity. The lectures will culminate with a presentation of a recent result of Dwivedi-Gianniotis-Karigiannis, exhibiting a large class of geometric flows of G2-structures admitting a DeTurck trick for establishing short-time existence and uniqueness.

These lectures are designed to provide a foundational understanding of the flows of geometric structures, with an emphasis on G2-structures, and will be accompanied by extensive lecture notes. Henrique Sá Earp will also contribute discussions on abstract formulations involving principal bundles and universal sections.