Abstract
In this paper, we establish a kind of splitting theorem for the eigenvalues of a specific family of operators on the base of a warped product. As a consequence, we prove a density theorem for a set of warping functions that makes the spectrum of the Laplacian a warped-simple spectrum. This is then used to study the generic situation of the eigenvalues of the Laplacian on a class of compact G-manifolds. In particular, we give a partial answer to a question posed in 1990 by Steven Zelditch about the generic situation of multiplicity of the eigenvalues of the Laplacian on principal bundles.
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Notes
See definition given in Sect. 3.
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Acknowledgements
The authors would like to express their sincere thanks to Department of Mathematics at Lehigh University, where part of this work was carried out. They are grateful to Huai-Dong Cao and Mary Ann for their warm hospitality and constant encouragement. The first author is also grateful to Gérard Besson for his time and inspiring and helpful discussions, as well as to Institut Fourier in Grenoble France for a good atmosphere while this work was done.
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Marcus A. M. Marrocos: Partially supported by Grant 2016/10009-3, São Paulo Research Foundation (FAPESP). José N. V. Gomes: Partially supported by Grant 202234/2017-7, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), of the Ministry of Science, Technology and Innovation of Brazil.
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Marrocos, M.A.M., Gomes, J.N.V. Generic Spectrum of Warped Products and G-Manifolds. J Geom Anal 29, 3124–3134 (2019). https://doi.org/10.1007/s12220-018-00106-x
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DOI: https://doi.org/10.1007/s12220-018-00106-x