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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.16832 |
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| _version_ | 1866909999075164160 |
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| author | Dwivedi, Shubham Singhal, Ragini |
| author_facet | Dwivedi, Shubham Singhal, Ragini |
| contents | We find explicit solutions and singularities of the Ricci-harmonic flow of $\mathrm{G_2}$-structures, the Ricci-like flows of $\mathrm{G_2}$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and of the negative gradient flow of an energy functional of $\mathrm{G_2}$-structures, on $7$-dimensional contact Calabi-Yau manifolds and the $7$-dimensional Heisenberg group. We prove that the natural co-closed $\mathrm{G_2}$-structure on a contact Calabi-Yau manifold as the initial condition leads to an ancient solution of the Ricci-harmonic flow with a finite time Type I singularity, and it gives an immortal solution to the Ricci-like flows with an infinite time singularity which are Type III if the transversal Calabi-Yau distribution is flat, and Type IIb otherwise. The same ansatz gives ancient solution to the negative gradient flow of $\mathrm{G_2}$-structures. These are the first examples of Type I singularities of the Ricci-harmonic flow and Type IIb and Type III singularities of the Ricci-like flows. We also obtain similar solutions for all the three flows on the $7$-dimensional Heisenberg group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solutions and singularities of the Ricci-harmonic flow and Ricci-like flows of $\mathrm{G_2}$-structures Dwivedi, Shubham Singhal, Ragini Differential Geometry We find explicit solutions and singularities of the Ricci-harmonic flow of $\mathrm{G_2}$-structures, the Ricci-like flows of $\mathrm{G_2}$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and of the negative gradient flow of an energy functional of $\mathrm{G_2}$-structures, on $7$-dimensional contact Calabi-Yau manifolds and the $7$-dimensional Heisenberg group. We prove that the natural co-closed $\mathrm{G_2}$-structure on a contact Calabi-Yau manifold as the initial condition leads to an ancient solution of the Ricci-harmonic flow with a finite time Type I singularity, and it gives an immortal solution to the Ricci-like flows with an infinite time singularity which are Type III if the transversal Calabi-Yau distribution is flat, and Type IIb otherwise. The same ansatz gives ancient solution to the negative gradient flow of $\mathrm{G_2}$-structures. These are the first examples of Type I singularities of the Ricci-harmonic flow and Type IIb and Type III singularities of the Ricci-like flows. We also obtain similar solutions for all the three flows on the $7$-dimensional Heisenberg group. |
| title | Solutions and singularities of the Ricci-harmonic flow and Ricci-like flows of $\mathrm{G_2}$-structures |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2601.16832 |