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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


The first eigenvalue of a closed manifold with positive Ricci curvature

Author: Jun Ling
Journal: Proc. Amer. Math. Soc. 134 (2006), 3071-3079
MSC (2000): Primary 58J50, 35P15; Secondary 53C21
Published electronically: May 1, 2006
MathSciNet review: 2231634
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a new estimate on the lower bound for the first positive eigenvalue of the Laplacian on a closed manifold with positive Ricci curvature in terms of the lower bound of the Ricci curvature and the largest interior radius of the nodal domains of eigenfunctions of the eigenvalue.

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  • 1. 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 (86g:58140)
  • 2. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391 (16,426a)
  • 3. Peter Li, Lecture notes on geometric analysis, Lecture Notes Series, vol. 6, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1993. MR 1320504 (96m:58269)
  • 4. Peter Li and Shing Tung Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309–318. MR 701919 (84k:58225)
  • 5. Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR 573435 (81i:58050)
  • 6. André Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, III. Dunod, Paris, 1958 (French). MR 0124009 (23 #A1329)
  • 7. Jun Ling, A lower bound for the gap between the first two eigenvalues of Schrödinger operators on convex domains in 𝑆ⁿ or 𝑅ⁿ, Michigan Math. J. 40 (1993), no. 2, 259–270. MR 1226831 (94h:35185),
  • 8. Jun Ling,
    A bound for the first fundamental gap.
    Ph.D. Dissertation, State University of New York at Buffalo.
  • 9. Jun Ling,
    Estimates on the lower bound of the first gap,
    Preprint, 2004.
  • 10. R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; Preface translated from the Chinese by Kaising Tso. MR 1333601 (97d:53001)
  • 11. DaGang Yang, Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature, Pacific J. Math. 190 (1999), no. 2, 383–398. MR 1722898 (2001b:53039),
  • 12. Jia Qing Zhong and Hong Cang Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27 (1984), no. 12, 1265–1273. MR 794292 (87a:58162)

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Additional Information

Jun Ling
Affiliation: Department of Mathematics, Utah Valley State College, Orem, Utah 84058

PII: S 0002-9939(06)08332-8
Keywords: Eigenvalue, lower bound, closed Riemannian manifold
Received by editor(s): October 15, 2004
Received by editor(s) in revised form: April 28, 2005
Published electronically: May 1, 2006
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2006 American Mathematical Society