Spectral constant rigidity of warped product metrics
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916323673505792 |
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| author | Chai, Xiaoxiang Pyo, Juncheol Wan, Xueyuan |
| author_facet | Chai, Xiaoxiang Pyo, Juncheol Wan, Xueyuan |
| contents | A theorem of Llarull says that if a smooth metric $g$ on the $n$-sphere $\mathbb{S}^n$ is bounded below by the standard round metric and the scalar curvature $R_g$ of $g$ is bounded below by $n (n - 1)$, then the metric $g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound $R_g \geq n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature $R_g$. We utilize two methods: spinor and spacetime harmonic function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13329 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral constant rigidity of warped product metrics Chai, Xiaoxiang Pyo, Juncheol Wan, Xueyuan Differential Geometry 53C24, 53C27, 58C40 A theorem of Llarull says that if a smooth metric $g$ on the $n$-sphere $\mathbb{S}^n$ is bounded below by the standard round metric and the scalar curvature $R_g$ of $g$ is bounded below by $n (n - 1)$, then the metric $g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound $R_g \geq n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature $R_g$. We utilize two methods: spinor and spacetime harmonic function. |
| title | Spectral constant rigidity of warped product metrics |
| topic | Differential Geometry 53C24, 53C27, 58C40 |
| url | https://arxiv.org/abs/2310.13329 |