Scaling inequalities for spherical and hyperbolic eigenvalues
Jeffrey J. Langford, Richard Snyder Laugesen · Journal of Spectral Theory · 2023
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 \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.