## Abstract

Neumann and Dirichlet eigenvalues of the Laplacian on spherical and hyperbolic domains are shown to satisfy scaling inequalities or monotonicities analogous to the (length)^{-2} scaling relation in Euclidean space. For a cap of aperture Θ on the sphere S^{2}, normalizing the k-th eigenvalue by the square of the Euclidean radius of the boundary circle yields that μ_{k}(Θ) sin^{2} ‚ is strictly decreasing, while normalizing by the stereographic radius squared gives that μ_{k}(Θ)4 tan^{2} Θ/2 is strictly increasing. For the second Neumann eigenvalue, normalizing instead by the cap area establishes the stronger result that μ_{2}(Θ)4 sin^{2} Θ/2 is strictly increasing. Monotonicities of this kind are somewhat surprising, since the Neumann eigenvalues themselves can vary non-monotonically. Cheng and Bandle-type inequalities are deduced by assuming either fixed radius or fixed area and comparing eigenvalues of disks having different curvatures.

Original language | English (US) |
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Pages (from-to) | 263-296 |

Number of pages | 34 |

Journal | Journal of Spectral Theory |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - 2023 |

## Keywords

- Cheng’s inequality
- Dirichlet
- Laplace–Beltrami operator
- Laplacian
- Neumann
- geodesic disk
- spherical cap
- vibrating membrane

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Geometry and Topology