On the spectrum of the Page and the Chen-LeBrun-Weber metrics

TL;DR

This paper establishes bounds on the first non-zero eigenvalue of the scalar Laplacian for the Page and Chen-LeBrun-Weber Einstein metrics without explicit metric knowledge, aiding stability analysis and spectrum understanding.

Contribution

It introduces a novel method to bound the spectrum of these metrics without explicit metric data, and computes the invariant spectrum part.

Findings

Bounds are close to actual eigenvalues based on numerical evidence

Method applies to metrics without explicit forms

Bounds facilitate stability analysis

Abstract

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the calculation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue.