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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.04638 |
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| _version_ | 1866908577511243776 |
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| author | Hudecek, Stepan |
| author_facet | Hudecek, Stepan |
| contents | This paper studies the Poisson equation for the $G_2$-Laplacian on 3-forms on the 7-sphere that are invariant under a transitive group action. We establish the existence and uniqueness of $G$-invariant solutions for $G=SU(4),\: Spin(7),\: (Sp(2)\times Sp(1))/\mathbb{Z}_2$. In the case $G=Sp(2)\times U(1)/\mathbb{Z}_2$, we show that the operator does not preserve the set of positive 3-forms. The paper also discusses the eigenvalue problem for the $G_2$-Laplacian. We classify $G$-invariant solutions for the above choices of $G$ and determine which of these solutions are nearly parallel $G_2$-structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $G_2$-Poisson equation on homogeneous spheres Hudecek, Stepan Differential Geometry This paper studies the Poisson equation for the $G_2$-Laplacian on 3-forms on the 7-sphere that are invariant under a transitive group action. We establish the existence and uniqueness of $G$-invariant solutions for $G=SU(4),\: Spin(7),\: (Sp(2)\times Sp(1))/\mathbb{Z}_2$. In the case $G=Sp(2)\times U(1)/\mathbb{Z}_2$, we show that the operator does not preserve the set of positive 3-forms. The paper also discusses the eigenvalue problem for the $G_2$-Laplacian. We classify $G$-invariant solutions for the above choices of $G$ and determine which of these solutions are nearly parallel $G_2$-structures. |
| title | $G_2$-Poisson equation on homogeneous spheres |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2510.04638 |