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Transactions of the American Mathematical Society

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A metric deformation and the first eigenvalue of Laplacian on $ 1$-forms


Author: Takashi Otofuji
Journal: Trans. Amer. Math. Soc. 339 (1993), 205-220
MSC: Primary 58G25; Secondary 58E11
DOI: https://doi.org/10.1090/S0002-9947-1993-1124172-X
MathSciNet review: 1124172
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Abstract: We search for a higher-dimensional analogue of Calabi's example of a metric deformation, quoted by Cheeger, which inspired him to prove an inequality between the first eigenvalue of the Laplacian on functions and an isoperimetric constant. We construct an example of a metric deformation on $ {S^n}$, $ {n} \geq 5$, where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on $ 1$-forms tends to zero. This metric deformation makes the sphere in the limit into a manifold with a cone singularity, which is an intermediate point on a path of deformation from an ($ {S^n}$, some metric) to an ( $ {S^{n - 1}} \times {S^1}$, some metric).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1124172-X
Keywords: Metric deformation, Laplacian, $ 1$-form, first eigenvalue, isoperimetric constant
Article copyright: © Copyright 1993 American Mathematical Society

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