Joseph Duthie | Searcharxiv

arXiv2026

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

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 \[ \Lambda(x,t)=\left(|\mathrm{Riem}(x,t)|_{g(t)}^2+|T(x,t)|_{g(t)}^4+|\nabla T(x,t)|_{g(t)}^2\right)^{1/2} \] implies bounds on all covariant derivatives of the Riemann curvature tensor $\mathrm{Riem}$ and the torsion tensor $T$. We show further that $\Lambda(x,t)$ 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.

math.DG↗

arXiv2025

Explicit solutions to the gradient flow of Spin(7)-structures

We study the gradient flow of Spin($7$)-structures and construct the first explicit solutions, in the homogeneous setting. As an intermediate step, we obtain formulae expressing the Spin($7$)-torsion tensor and gradient flow in terms of the Spin($7$)-torsion forms, which makes explicit computations more tractable. We use these formulae to find explicit solutions to the gradient flow of Spin($7$)-structures, obtaining a shrinking soliton on $\mathrm{SU}(3)$ as well as another explicit solution on a certain $T^7$-bundle over $S^1$. We also find an explicit solution to the coupled Ricci-harmonic flow of Spin($7$)-structures. Finally, we consider the question of stability of solitons for the renormalised gradient flow, and show that the soliton on $\mathrm{SU}(3)$ admits stable directions, unstable directions, and zero modes.

math.DG↗