On the spectrum of the Page and the Chen-LeBrun-Weber metrics
arXiv:1206.5489
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.
arXiv comment15 pages, v2 substantially rewritten, section on linear stability added; v3 updated to reflect referee's comments, v4 final version to appear in Ann. Glob. Anal. Geom