The hypersymplectic flow descended from the $G_2$-Laplacian coflow
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908980894236672 |
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| author | Petcu, Amanda Maria |
| author_facet | Petcu, Amanda Maria |
| contents | A conjecture of Simon Donaldson is that on a compact $4$-manifold $X^4$ one can flow from a hypersymplectic structure to a hyperkähler structure while remaining in the same cohomology class. To this end the hypersymplectic flow was introduced by Fine-Yao. In this paper the notion of a positive triple on $X^4$ is used to describe a hypersymplectic and hyperkähler structure. Given a closed positive triple one can define either a closed $G_2$ structure or a coclosed $G_2$ structure on $\mathbb{T}^3 \times X^4$. The coclosed $G_2$ structure is evolved under the modified $G_2$-Laplacian coflow. The coflow descends to a flow of the positive triple on $X^4$, which is again the Fine-Yao hypersymplectic flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18554 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The hypersymplectic flow descended from the $G_2$-Laplacian coflow Petcu, Amanda Maria Differential Geometry 53C26, 53C25, 53E99 A conjecture of Simon Donaldson is that on a compact $4$-manifold $X^4$ one can flow from a hypersymplectic structure to a hyperkähler structure while remaining in the same cohomology class. To this end the hypersymplectic flow was introduced by Fine-Yao. In this paper the notion of a positive triple on $X^4$ is used to describe a hypersymplectic and hyperkähler structure. Given a closed positive triple one can define either a closed $G_2$ structure or a coclosed $G_2$ structure on $\mathbb{T}^3 \times X^4$. The coclosed $G_2$ structure is evolved under the modified $G_2$-Laplacian coflow. The coflow descends to a flow of the positive triple on $X^4$, which is again the Fine-Yao hypersymplectic flow. |
| title | The hypersymplectic flow descended from the $G_2$-Laplacian coflow |
| topic | Differential Geometry 53C26, 53C25, 53E99 |
| url | https://arxiv.org/abs/2604.18554 |