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

TL;DR

This paper derives explicit solutions for the gradient flow of Spin(7)-structures, including solitons and stability analysis, advancing understanding of geometric flows in special holonomy manifolds.

Contribution

It provides the first explicit solutions to the gradient flow of Spin(7)-structures in homogeneous settings, with formulae for torsion tensors and flow equations.

Findings

Constructed explicit shrinking soliton on SU(3)

Found explicit solution on T^7-bundle over S^1

Analyzed stability of the SU(3) soliton

Abstract

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.