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).References
- Colette Anné, Spectre du laplacien et écrasement d’anses, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 2, 271–280 (French, with English summary). MR 911759
- I. Chavel and E. A. Feldman, Spectra of manifolds with small handles, Comment. Math. Helv. 56 (1981), no. 1, 83–102. MR 615617, DOI 10.1007/BF02566200
- Isaac Chavel and A. Feldman, Isoperimetric constants of manifolds with small handles, Math. Z. 184 (1983), no. 4, 435–448. MR 719487, DOI 10.1007/BF01161728
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
- Bruno Colbois and Gilles Courtois, A note on the first nonzero eigenvalue of the Laplacian acting on $p$-forms, Manuscripta Math. 68 (1990), no. 2, 143–160. MR 1063223, DOI 10.1007/BF02568757 —, Convergence de variétés et convergence du spectre du Laplacien, preprint.
- Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287
- Jozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862, DOI 10.1512/iumj.1983.32.32046
- Kenji Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), no. 3, 517–547. MR 874035, DOI 10.1007/BF01389241
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