Shi-type estimates and finite-time singularities of reasonable flows of Spin(7)-structures
Published:
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
\[\Lambda(x,t) = \left(|\mathrm{Rm}(x,t)|_{g(t)}^2 + |T(x,t)|^4_{g(t)} + |\nabla T(x,t)|_{g(t)}^2 \right )^{1/2}\]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.