Riemannian metrics with large 𝜆₁

Colbois, B.; Dodziuk, J.

doi:10.1090/S0002-9939-1994-1213857-9

Riemannian metrics with large $\lambda _ 1$
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by B. Colbois and J. Dodziuk PDF

Proc. Amer. Math. Soc. 122 (1994), 905-906 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

Diverging eigenvalues and collapsing Riemannian metrics

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