On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

TL;DR

This paper establishes that geodesic balls maximize the first eigenvalue of certain convolution operators, including Riesz transforms, on spherical and hyperbolic spaces, extending classical inequalities to these geometries.

Contribution

It proves an analogue of the Rayleigh-Faber-Krahn inequality for Riesz transforms on spheres and hyperbolic spaces, and explores extremal properties of the second eigenvalue.

Findings

Geodesic balls maximize the first eigenvalue of convolution operators on $ ext{S}^n$ and $ ext{H}^n$.

Established a Hong-Krahn-Szeg"o type inequality for the second eigenvalue on hyperbolic space.

Extended classical eigenvalue inequalities to non-Euclidean geometries.

Abstract

In this note we prove an analogue of the Rayleigh-Faber-Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere $\mathbb{S}^{n}$ and on the real hyperbolic space $\mathbb{H}^{n}$. It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on $\mathbb{H}^{n}$ and prove the Hong-Krahn-Szeg\"{o} type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.