Shi-type estimates and finite-time singularities of reasonable flows of Spin(7)-structures
Published in , 2026
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 \(\Lambda(x,t) = \left(\lvert \mathrm{Rm}(x,t)\rvert_{g(t)}^2 + \lvert T(x,t)\rvert^4_{g(t)} + \lvert \nabla T(x,t)\rvert_{g(t)}^2 \right )^{1/2}\) implies bounds on all covariant derivatives of the curvature \(\mathrm{Rm}\) and torsion tensor \(T\).We show further that \(\Lambda(x,t)\) must blow up at a finite-time singularity, and establish a lower bound on the blow-up rate. We also prove a compactness theorem for solutions to 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.