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Transactions of the American Mathematical Society

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The spectrum of a Riemannian manifold with a unit Killing vector field


Author: David D. Bleecker
Journal: Trans. Amer. Math. Soc. 275 (1983), 409-416
MSC: Primary 53C20; Secondary 58G30
DOI: https://doi.org/10.1090/S0002-9947-1983-0678360-0
MathSciNet review: 678360
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Abstract: Let $ (P,g)$ be a compact, connected, $ {C^\infty }$ Riemannian $ (n + 1)$-manifold $ (n \geqslant 1)$ with a unit Killing vector field with dual $ 1$-form $ \eta $. For $ t > 0$, let $ {g_{t}} = {t^{ - 1}}g + (t^{n}-t^{-1})\eta \otimes \eta$, a family of metrics of fixed volume element on $ P$. Let $ {\lambda _1}(t)$ be the first nonzero eigenvalue of the Laplace operator on $ {C^\infty }(P)$ of the metric $ {g_t}$. We prove that if $ d\eta $ is nowhere zero, then $ {\lambda _1}(t) \to \infty$ as $ t \to \infty $. Using this construction, we find that, for every dimension greater than two, there are infinitely many topologically distinct compact manifolds for which $ {\lambda _1}$ is unbounded on the space of fixed-volume metrics.


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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, Gauge theory and variational principles, Global Analysis, vol. I, Addison-Wesley, Reading, Mass., 1981. MR 643361 (83h:53049)
  • [3] A. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. MR 0104925 (21:3675)
  • [4] J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), 1645-1648. MR 0292357 (45:1444)
  • [5] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I, Interscience, New York, 1963. MR 0152974 (27:2945)
  • [6] H. Muto and H. Urakawa, On the least positive eigenvalue of the Laplacian for compact homogeneous spaces, Osaka J. Math. 17 (1980), 471-484. MR 587767 (81m:58083)
  • [7] S. Tanno, The first eigenvalue of the Laplacian on spheres, Tôhoku Math. J. 31 (1979), 179-185. MR 538918 (80g:58050)
  • [8] 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)
  • [9] P. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa 7 (1980), 55-63. MR 577325 (81m:58084)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678360-0
Keywords: Laplacian, Riemannian manifold, Killing vector field, spectrum
Article copyright: © Copyright 1983 American Mathematical Society

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