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Riemannian metrics with large first eigenvalue on forms of degree $ p$

Authors: G. Gentile and V. Pagliara
Journal: Proc. Amer. Math. Soc. 123 (1995), 3855-3858
MSC: Primary 58G25; Secondary 35P15, 53C20
MathSciNet review: 1277111
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Abstract: Let (M, g) be a compact, connected, $ {C^\infty }$ Riemannian manifold of n dimensions. Denote by $ {\lambda _{1,p}}(M,g)$ the first nonzero eigenvalue of the Laplace operator acting on differential forms of degree p. We prove that for $ n \geq 4$ and $ 2 \leq p \leq n - 2$, there exists a family of metrics $ {g_t}$ of volume one, such that $ {\lambda _{1,p}}(M,{g_t}) \to \infty $ as $ t \to \infty $.

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  • [1] M. Berger, Sur les premières valeurs propres des variétés riemanniennes, Compositio Math. 26 (1973), 129-149. MR 0316913 (47:5461)
  • [2] D. Bleecker, The spectrum of a Riemannian manifold with a unit Killing vector field, Trans. Amer. Math. Soc. 275 (1983), 409-416. MR 678360 (84c:53037)
  • [3] B. Colbois and J. Dodziuk, Riemannian metrics with large $ {\lambda _4}$, Proc. Amer. Math. Soc. 122 (1994), 905-906. MR 1213857 (95a:58130)
  • [4] J. Hersch, Quatre propriétés isopérimétriques des membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A 270 (1970), 139-144. MR 0292357 (45:1444)
  • [5] J. McGowan, The p-spectrum of the Laplacian on compact hyperbolic three manifolds, Math. Ann. 279 (1993), 729-745. MR 1245416 (94g:58239)
  • [6] S. Tanno, Geometric expressions of eigen 1-forms of the Laplacian on spheres, Spectral Riemannian Manifolds, Kaigai, Kyoto, 1983, pp. 115-128.
  • [7] H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan 31 (1979), 209-226. MR 519046 (80e:58046)
  • [8] Y. Xu, Diverging eigenvalues and collapsing Riemannian metrics, Institute for Advanced Study, October 1992.
  • [9] P. Yang and S.-T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 55-63. MR 577325 (81m:58084)

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Article copyright: © Copyright 1995 American Mathematical Society

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