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On the eigenvalues of the Laplacian on fibred manifolds

Chanyoung Sung

TL;DR

This work develops a framework for comparing Laplacian eigenvalues on fibred manifolds using fiberwise spherical and Euclidean/Hyperschiorical symmetrizations, extending classical inequalities to bundle settings.By introducing a canonical fiber-form form $\Omega_G$ and a warped variation of Riemannian submersions, the authors derive lower bounds for $\lambda_1$ and, under suitable hypotheses, exact identifications of several $\lambda_i$ with those of the base space, including cases with totally geodesic fibers and positive Ricci curvature.The paper provides explicit computations for a range of fiber bundles (sphere bundles, homogeneous CROSS, projective bundles) and yields Dirichlet-variant results (Faber-Krahn type) via fiberwise rearrangements, accompanied by open questions on extensions and higher-order eigenvalues.Overall, the results offer a robust method to transfer spectral data from base manifolds to total spaces in fibred geometries and illuminate the spectral geometry of many nontrivial fiber bundles.

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.

On the eigenvalues of the Laplacian on fibred manifolds

TL;DR

This work develops a framework for comparing Laplacian eigenvalues on fibred manifolds using fiberwise spherical and Euclidean/Hyperschiorical symmetrizations, extending classical inequalities to bundle settings.By introducing a canonical fiber-form form $\Omega_G$ and a warped variation of Riemannian submersions, the authors derive lower bounds for $\lambda_1$ and, under suitable hypotheses, exact identifications of several $\lambda_i$ with those of the base space, including cases with totally geodesic fibers and positive Ricci curvature.The paper provides explicit computations for a range of fiber bundles (sphere bundles, homogeneous CROSS, projective bundles) and yields Dirichlet-variant results (Faber-Krahn type) via fiberwise rearrangements, accompanied by open questions on extensions and higher-order eigenvalues.Overall, the results offer a robust method to transfer spectral data from base manifolds to total spaces in fibred geometries and illuminate the spectral geometry of many nontrivial fiber bundles.

Abstract

We prove various comparison theorems of the -th eigenvalue 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 comparison theorem under a lower Ricci curvature bound. By applying these, it is shown that of a fiber bundle given by a Riemannian submersion with totally geodesic fibers of sufficiently positive Ricci curvature are respectively equal to of its base, and of a (possibly singular) fibration with Euclidean subsets as fibers is no less than of the disk bundle obtained by replacing each fiber with a Euclidean disk of the same dimension and volume.
Paper Structure (20 sections, 31 theorems, 237 equations)

This paper contains 20 sections, 31 theorems, 237 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Preliminaries
  3. Conventions and notations
  4. Weak derivative and Sobolev space
  5. Rayleigh quotient
  6. Fiberwise symmetrization
  7. $\lambda_i$ comparison on fiber bundles
  8. Introducing $\Omega_G$ on a principal $G$-bundle
  9. Warped variation of Riemannian submersion
  10. Proof of Theorem \ref{['Main-theorem']}
  11. Proof of Theorem \ref{['Main-Corollary']}
  12. Examples of exact computation of $\lambda_i$
  13. Sphere bundles
  14. Homogeneous CROSS
  15. Projective bundles
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(F,g_{_F})$ be a smooth closed Riemannian $m$-manifold for $m\geq 2$ with $Ric_{g_{_F}} \geq m-1$, $(B,g_{_B})$ be a smooth closed Riemannian $n$-manifold, and $\check{\rho}$ be a smooth positive function on $B$. Suppose that $\pi: (X,g_{_X})\rightarrow (B,g_{_B})$ is a smooth Riemannian submer

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 48 more