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arXiv · 2408.07880

On the eigenvalues of the Laplacian on fibred manifolds

Chanyoung Sung
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Abstract

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.

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Chanyoung Sung. 2025-12-01. On the eigenvalues of the Laplacian on fibred manifolds. https://arxiv.org/abs/2408.07880

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