Riemannian metrics with large $\lambda _ 1$
HTML articles powered by AMS MathViewer
- by B. Colbois and J. Dodziuk
- Proc. Amer. Math. Soc. 122 (1994), 905-906
- DOI: https://doi.org/10.1090/S0002-9939-1994-1213857-9
- PDF | Request permission
Abstract:
We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian.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, DOI 10.24033/asens.1533
- M. Berger, Sur les premières valeurs propres des variétés riemanniennes, Compositio Math. 26 (1973), 129–149 (French). MR 316913
- David D. Bleecker, The spectrum of a Riemannian manifold with a unit Killing vector field, Trans. Amer. Math. Soc. 275 (1983), no. 1, 409–416. MR 678360, DOI 10.1090/S0002-9947-1983-0678360-0
- Yves Colin de Verdière, Sur la multiplicité de la première valeur propre non nulle du laplacien, Comment. Math. Helv. 61 (1986), no. 2, 254–270 (French). MR 856089, DOI 10.1007/BF02621914
- Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
- Hideo Mutô, The first eigenvalue of the Laplacian on even-dimensional spheres, Tohoku Math. J. (2) 32 (1980), no. 3, 427–432. MR 590038, DOI 10.2748/tmj/1178229601 Y. Xu, Diverging eigenvalues and collapsing Riemannian metrics, preprint, Institute for Advanced Study, 1992.
- Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR 577325
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 905-906
- MSC: Primary 58G30; Secondary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-1994-1213857-9
- MathSciNet review: 1213857