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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.14182 |
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| _version_ | 1866916698409402368 |
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| author | G., Hector Barrantes Dávila, Jorge |
| author_facet | G., Hector Barrantes Dávila, Jorge |
| contents | We study Yamabe-type equations on the product of two spheres $(S^n \times S^n, G_δ)$, where $G_δ$ is a family of Riemannian metrics parametrized by $δ> 0$. Using bifurcation theory and isoparametric functions, we establish the existence of degenerate solutions that are invariant under the diagonal action of $O(n+1)$ and depend non-trivially on both factors. Our analysis relies on the properties of Gegenbauer polynomials and a careful application of local bifurcation techniques for simple eigenvalues. These results extend previous studies by demonstrating the emergence of solutions that do not solely depend on a single factor, thereby providing new insights into the structure of solutions for Yamabe-type problems on product manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degenerate Solutions of Yamabe-Type Equations on Products of Spheres G., Hector Barrantes Dávila, Jorge Differential Geometry We study Yamabe-type equations on the product of two spheres $(S^n \times S^n, G_δ)$, where $G_δ$ is a family of Riemannian metrics parametrized by $δ> 0$. Using bifurcation theory and isoparametric functions, we establish the existence of degenerate solutions that are invariant under the diagonal action of $O(n+1)$ and depend non-trivially on both factors. Our analysis relies on the properties of Gegenbauer polynomials and a careful application of local bifurcation techniques for simple eigenvalues. These results extend previous studies by demonstrating the emergence of solutions that do not solely depend on a single factor, thereby providing new insights into the structure of solutions for Yamabe-type problems on product manifolds. |
| title | Degenerate Solutions of Yamabe-Type Equations on Products of Spheres |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.14182 |