# Scaling Inequalities for Spherical and Hyperbolic Eigenvalues

## Publication Date

2023

## Description

Neumann and Dirichlet eigenvalues of the Laplacian on spherical and hyperbolic domains are shown to satisfy scaling inequalities or monotonicities analogous to the $(\text{length})^{-2}$ scaling relation in Euclidean space.

For a cap of aperture $\Theta$ on the sphere $\mathbb{S}^2$, normalizing the $k$-th eigenvalue by the square of the Euclidean radius of the boundary circle yields that $\mu_k(\Theta) \sin^2 \Theta$ is strictly decreasing, while normalizing by the stereographic radius squared gives that $\mu_k(\Theta) 4 \tan^2 \Theta/2$ is strictly increasing. For the second Neumann eigenvalue, normalizing instead by the cap area establishes the stronger result that $\mu_2(\Theta) 4 \sin^2 \Theta/2$ 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.

## Journal

Journal of Spectral Theory

## Volume

13

## Issue

1

## First Page

263

## Last Page

296

## Department

Mathematics

## Link to Published Version

https://ems.press/journals/jst/articles/11709728

## DOI

10.4171/JST/447

## Recommended Citation

Langford, Jeffrey J. and Laugesen, Richard S.. "Scaling Inequalities for Spherical and Hyperbolic Eigenvalues." (2023) : 263-296.