A gradient flow of Spin(7)-structures

Abstract

We formulate and study the negative gradient flow of an energy functional of Spin(7)-structures on compact $8$-manifolds. The energy functional is the $L^2$-norm of the torsion of the Spin(7)-structure. Our main result is the short-time existence and uniqueness of solutions to the flow. We also explain how this negative gradient flow contains, as the highest order terms, all independent second order differential invariants of Spin(7)-structures which can be made into an admissible $4$-form. We also study solitons of the flow and prove a non-existence result for compact expanding solitons.