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

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 estimat...

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Bibliographic Details
Published in:Annals of global analysis and geometry Vol. 46; no. 1; pp. 87 - 101
Main Authors: Hall, Stuart J., Murphy, Thomas
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.06.2014
Springer Nature B.V
Subjects:
Analysis
Approximation
Differential Geometry
Eigenvalues
Explicit knowledge
Geometry
Global Analysis and Analysis on Manifolds
Laplace transforms
Mathematical models
Mathematical Physics
Mathematics
Mathematics and Statistics
Studies
Theorems
Toric-Kähler metric
Ricci flow
Einstein metric
Linear stability
Spectrum of Laplacian
ISSN:0232-704X, 1572-9060
Online Access:Get full text
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Summary: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 estimation 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.
Bibliography:SourceType-Scholarly Journals-1
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ISSN:0232-704X
1572-9060
DOI:10.1007/s10455-014-9412-6