Full Laplace spectrum of distance spheres in symmetric spaces of rank one

TL;DR

This paper explicitly computes the full Laplace spectrum of distance spheres in rank one symmetric spaces using Lie theory, providing a unified formula and analyzing stability and bifurcation properties.

Contribution

It introduces a unified Lie-theoretic approach to compute spectra of distance spheres in rank one symmetric spaces and analyzes their stability and bifurcation behavior.

Findings

Unified formula for the Laplace spectrum of distance spheres

Identification of resonant radii in the compact case

Proof of stability and local rigidity in the noncompact case

Abstract

We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a single unified formula for all cases. As an application, we find all resonant radii for distance spheres in the compact case, i.e., radii where there is bifurcation of embedded constant mean curvature spheres, and show that distance spheres are stable and locally rigid in the noncompact case.