Scaling inequalities for spherical and hyperbolic eigenvalues

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 $\Theta$ 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.