On the eigenvalues of the Laplacian on fibred manifolds

TL;DR

This paper develops comparison theorems for Laplacian eigenvalues on fibred Riemannian manifolds, extending classical inequalities and establishing eigenvalue bounds for fiber bundles with positive Ricci curvature.

Contribution

It generalizes key eigenvalue inequalities to fiber bundles and provides new comparison results for eigenvalues under curvature bounds.

Findings

Eigenvalues of fiber bundles with positive Ricci curvature match those of the base.

Eigenvalues of certain fibrations are bounded below by those of Euclidean disk bundles.

Generalization of Lichnerowicz and Faber-Krahn inequalities to fibred manifolds.

Abstract

We prove various comparison theorems of the $i$-th eigenvalue $\lambda_i$ 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 $\lambda_1$ comparison theorem under a lower Ricci curvature bound. By applying these, it is shown that $\lambda_1,\cdots,\lambda_k$ of a fiber bundle given by a Riemannian submersion with totally geodesic fibers of sufficiently positive Ricci curvature are respectively equal to $\lambda_1,\cdots,\lambda_k$ of its base, and $\lambda_1$ of a (possibly singular) fibration with Euclidean subsets as fibers is no less than $\lambda_1$ of the disk bundle obtained by replacing each fiber with a Euclidean disk of the same dimension and volume.