Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915310281424896 |
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| author | Batista, Márcio Cavalcante, Marcos P. Mendes, Abraão Nunes, Ivaldo |
| author_facet | Batista, Márcio Cavalcante, Marcos P. Mendes, Abraão Nunes, Ivaldo |
| contents | In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator Batista, Márcio Cavalcante, Marcos P. Mendes, Abraão Nunes, Ivaldo Differential Geometry Primary 53C24, 58J50, Secondary 49Q05, 53A10 In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$. |
| title | Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator |
| topic | Differential Geometry Primary 53C24, 58J50, Secondary 49Q05, 53A10 |
| url | https://arxiv.org/abs/2505.22439 |