Explicit solutions to the gradient flow of Spin(7)-structures
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914166560784384 |
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| author | Duthie, Joseph |
| author_facet | Duthie, Joseph |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17356 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit solutions to the gradient flow of Spin(7)-structures Duthie, Joseph Differential Geometry 53C25, 53C29, 53C30 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. |
| title | Explicit solutions to the gradient flow of Spin(7)-structures |
| topic | Differential Geometry 53C25, 53C29, 53C30 |
| url | https://arxiv.org/abs/2511.17356 |