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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.07880 |
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| _version_ | 1866915645840424960 |
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| author | Sung, Chanyoung |
| author_facet | Sung, Chanyoung |
| contents | We prove various comparison theorems of the $i$-th eigenvalue $λ_i$ of the Laplacian on fibred Riemannian manifolds by using fiberwise spherical and Euclidean (or hyperbolic) symmetrization. In particular we generalize the Lichnerowicz inequality and the Faber-Krahn inequality to fiber bundles, and prove a counterpart to Cheng's $λ_1$ comparison theorem under a lower Ricci curvature bound.
By applying these, it is shown that $λ_1,\cdots,λ_k$ of a fiber bundle given by a Riemannian submersion with totally geodesic fibers of sufficiently positive Ricci curvature are respectively equal to $λ_1,\cdots,λ_k$ of its base, and $λ_1$ of a (possibly singular) fibration with Euclidean subsets as fibers is no less than $λ_1$ of the disk bundle obtained by replacing each fiber with a Euclidean disk of the same dimension and volume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the eigenvalues of the Laplacian on fibred manifolds Sung, Chanyoung Differential Geometry 58E99, 35P15, 49R05 We prove various comparison theorems of the $i$-th eigenvalue $λ_i$ of the Laplacian on fibred Riemannian manifolds by using fiberwise spherical and Euclidean (or hyperbolic) symmetrization. In particular we generalize the Lichnerowicz inequality and the Faber-Krahn inequality to fiber bundles, and prove a counterpart to Cheng's $λ_1$ comparison theorem under a lower Ricci curvature bound. By applying these, it is shown that $λ_1,\cdots,λ_k$ of a fiber bundle given by a Riemannian submersion with totally geodesic fibers of sufficiently positive Ricci curvature are respectively equal to $λ_1,\cdots,λ_k$ of its base, and $λ_1$ of a (possibly singular) fibration with Euclidean subsets as fibers is no less than $λ_1$ of the disk bundle obtained by replacing each fiber with a Euclidean disk of the same dimension and volume. |
| title | On the eigenvalues of the Laplacian on fibred manifolds |
| topic | Differential Geometry 58E99, 35P15, 49R05 |
| url | https://arxiv.org/abs/2408.07880 |