A gradient flow of Spin(7)-structures
Shubham Dwivedi
TL;DR
This work introduces and analyzes a negative gradient flow for Spin(7)-structures on compact 8-manifolds, defined via the energy E(\Phi)=\frac{1}{2}\int_M |T_{\Phi}|^2 \mathrm{vol}_{\Phi}, where T_{\Phi} is the torsion. The main result establishes short-time existence and uniqueness of smooth solutions to the flow equation $\partial_t \Phi = (-\mathrm{Ric}+2\mathcal{L}_{T_8}g+T*T-|T|^2g+2\mathrm{div}\,T)\diamond \Phi$, achieved by a detailed principal-symbol analysis and a modified DeTurck trick to overcome diffeomorphism-invariance-induced degeneracy. The paper also isolates the role of lower-order terms in the dynamics, clarifies the parabolicity obstruction, and develops a robust framework for studying solitons of the Spin(7) flow, including a non-existence result for compact expanding solitons and a characterization of steady solitons as torsion-free Spin(7) structures. Overall, the results provide a rigorous analytic foundation for evolving Spin(7)-structures toward torsion-free configurations and offer insight into the behavior of the flow near singularities via soliton models.
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.