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

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An Erratum to this article was published on 12 February 2015

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 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.

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Acknowledgments

SH would like to thank his doctoral advisor Simon Donaldson for introducing him to many of the ideas we have used in this paper. We would like to thank Robert Haslhofer for his interest and comments on a previous version of this paper and the anonymous referees for numerous suggestions for improvements. We would also like to thank Steve Zelditch for his assistance. TM is supported by an A.R.C. grant. We acknowledge the support of a Dennison research grant from the University of Buckingham which funded a research visit by TM.

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  1. Thomas Murphy

    Present address: Department of Mathematics, McMaster University, 1280 Main St. W., Hamilton, ON, L8S 4K1, Canada

Authors and Affiliations

  1. Department of Applied Computing, University of Buckingham, Hunter St., Buckingham,  MK18 1G, UK

    Stuart J. Hall

  2. Départment de Mathématique, Université Libre de Bruxelles, Boulevard du Triomphe, 1050,  Brussels, Belgium

    Thomas Murphy

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  1. Stuart J. Hall
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Correspondence to Stuart J. Hall.

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Hall, S.J., Murphy, T. On the spectrum of the Page and the Chen–LeBrun–Weber metrics. Ann Glob Anal Geom 46, 87–101 (2014). https://doi.org/10.1007/s10455-014-9412-6

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