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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A metric deformation and the first eigenvalue of Laplacian on $1$-forms
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by Takashi Otofuji PDF
Trans. Amer. Math. Soc. 339 (1993), 205-220 Request permission

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
  • © Copyright 1993 American Mathematical Society
  • 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