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.