Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918105199935488 |
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| author | Mari, Luciano Ranieri, Marcos Sampaio, Elaine Vitório, Feliciano |
| author_facet | Mari, Luciano Ranieri, Marcos Sampaio, Elaine Vitório, Feliciano |
| contents | We investigate the spectrum of the Laplacian on complete, non-compact manifolds $M^n$ whose Ricci curvature satisfies $\mathrm{Ric} \geq -(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with $\mathrm{H}-1 \in L^1(\infty)$. We prove that if the bottom spectrum attains the maximal value $\frac{(n-1)^2}{4}$ compatible with the curvature bound, then the spectrum of $M$ coincides with that of hyperbolic space $\mathbb{H}^n$, namely, $σ(M) = \left[ \frac{(n-1)^2}{4}, \infty \right)$. The result can be localized to an end $E$ with infinite volume. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_19020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum Mari, Luciano Ranieri, Marcos Sampaio, Elaine Vitório, Feliciano Differential Geometry Spectral Theory Primary 53B30, 35P15, 58J50, Secondary 53C21, 31C12 We investigate the spectrum of the Laplacian on complete, non-compact manifolds $M^n$ whose Ricci curvature satisfies $\mathrm{Ric} \geq -(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with $\mathrm{H}-1 \in L^1(\infty)$. We prove that if the bottom spectrum attains the maximal value $\frac{(n-1)^2}{4}$ compatible with the curvature bound, then the spectrum of $M$ coincides with that of hyperbolic space $\mathbb{H}^n$, namely, $σ(M) = \left[ \frac{(n-1)^2}{4}, \infty \right)$. The result can be localized to an end $E$ with infinite volume. |
| title | Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum |
| topic | Differential Geometry Spectral Theory Primary 53B30, 35P15, 58J50, Secondary 53C21, 31C12 |
| url | https://arxiv.org/abs/2506.19020 |