Shi-type estimates and finite-time singularities of reasonable flows of Spin(7)-structures

Abstract

This paper establishes foundational analytic and geometric results for a broad class of reasonable flows of Spin($7$)-structures. We first prove Shi-type derivative estimates, showing that a uniform bound on the quantity \[ Λ(x,t)=\left(|\mathrm{Riem}(x,t)|_{g(t)}^2+|T(x,t)|_{g(t)}^4+|\nabla T(x,t)|_{g(t)}^2\right)^{1/2} \] implies bounds on all covariant derivatives of the Riemann curvature tensor $\mathrm{Riem}$ and the torsion tensor $T$. We show further that $Λ(x,t)$ must blow up at any finite-time singularity, and we establish a lower bound on the blow-up rate. We also prove a compactness theorem for solutions of such flows and apply these results to the analysis of finite-time singularities. These results provide a general analytic framework for studying flows of Spin($7$)-structures; once a proposed flow is shown to satisfy the reasonable condition, our estimates, compactness theorems, and singularity analysis apply.