Ricci-harmonic flow of $\mathrm{G}_2$ and Spin(7)-structures

Abstract

We introduce and study a new general flow of $\mathrm{G}_2$-structures which we call the Ricci-harmonic flow of $\mathrm{G}_2$-structures. The flow is the coupling of the Ricci flow of underlying metrics and the isometric flow of $\mathrm{G}_2$-structures, but we also provide explicit lower order in the torsion terms. The lower order terms and the flow are obtained by analyzing the second order term in the Taylor series expansion of $\mathrm{G}_2$-structures in normal coordinates. As such, the Ricci-harmonic flow described in the paper can be interpreted as the "heat equation" for $\mathrm{G}_2$-structures. The lower order terms allow us to prove that the stationary points of the Ricci-harmonic flow are exactly torsion-free $\mathrm{G}_2$-structures on compact manifolds. We study various analytic and geometric properties of the flow. We show that the flow has short-time existence and uniqueness on compact manifolds starting with an arbitrary $\mathrm{G}_2$-structure and prove global Shi-type estimates. We also prove a modified local Shi-type estimates for the flow which assume bounds on the initial derivatives of the Riemann curvature tensor and the torsion but give uniform bounds on these quantities for all times. We prove a compactness theorem for the solutions of the flow and use it to prove that the Ricci-harmonic flow exists as long as the velocity of the flow remains bounded. We also study Ricci-harmonic solitons where we prove that there are no compact expanding solitons and the only steady solitons are torsion-free. We derive an analog of Hamilton's identity for gradient Ricci-harmonic solitons and prove some integral identities for the solitons. Finally, we prove a version of the Taylor series expansion for Spin(7)-structures and use it to derive the Ricci-harmonic flow of Spin(7)-structures.