Dimensional Reduction of Geometric Flows of Spin(7)-Structures

This project takes a look when an abstract flow of Spin(7)-Structures, as originally studied by Karigiannis, preserves a product metric on a manifold of the form S^1 x M^7. In this case, we get an induced flow of G2-Structures on the 7-Manifold. For geometric flows of Spin(7)-Structures that are quadratic in the torsion, which encompasses most of the interesting cases, it is possible to establish sufficient conditions for such a Spin(7) flow to preserve a product ansatz using the analytic theory for abstract flows of G2-Structures developed recently by Dwivedi, Panagiotis, and Karigiannis. I then take a look at the dimensional reduction of Dwivedi's gradient flow of the torsion functional for Spin(7)-Structures, which is the first flow of Spin(7)-Structures that nontrivially modifies the metric. 

Forthcoming, manuscript will be available here.