Harmonic flow of $\mathrm{Spin}(7)$-structures

Abstract

We formulate and study the isometric flow of Spin(7)-structures on compact 8-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of H-structures. We then specialise to H = Spin(7) ⊂ SO(8), obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an ε-regularity theorem. Moreover, we prove Cheeger-Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-I singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric Spin(7)-structures, based on squares of spinors, which may be of independent interest.