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General formula for lower bound of the first eigenvalue on Riemannian manifolds

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Abstract

A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’ s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).

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References

  1. Bérard, P. H.,Spectral Geometry, Direct and Inverse Problem, Lecture Notes in Mathematics, Vol. 1207, Berlin: Springer-Verlag, 1986.

    Google Scholar 

  2. Chavel, I.,Eigenwrlues in Riemannian Geometry, Orlando: Academic Press, 1984.

    Google Scholar 

  3. Yau, S. T., Schoen, R.,Differentia Geometry (in Chinese), Beijing: Science Press, 1988.

    Google Scholar 

  4. Chen, M. F., Wang, F. Y., Application of coupling method to the first eigenvalue on manifold,Science in China, Ser. A, 1994, 37(1): 1.

    MATH  MathSciNet  Google Scholar 

  5. Chen, M. F., Optimal Markovian couplings and applications,Acta Math. Sin., New Ser., 1994, 10(3): 260.

    Article  MATH  Google Scholar 

  6. Wang, F.Y., Application of coupling method to the Neumann eigenvalue problem,Prob. Th. Re. Fields, 1994, 98: 299.

    Article  MATH  Google Scholar 

  7. Wang, F. Y., Spectral gap for diffusion processes on noncompact manifolds,Chin. Sci. Bull., 1995, 40(14): 1145.

    MATH  Google Scholar 

  8. Lichnemwicz, A.,Gémétrie des Groupes des Transformations, Paris: Dunod, 1958.

    Google Scholar 

  9. Zhong, J. Q., Yang, H. C., On estimate. of the first eigenvalue of a compact Riemannian manifold,Sci. Sin., Ser. A, 1984, 27: 1251.

    MathSciNet  Google Scholar 

  10. Cai, K. R., Estimate on lower bound of the first eigenvalue of a compact Riemannian manifold,Chin. Ann. Math., Ser. B, 1991, 12(3): 267.

    MATH  Google Scholar 

  11. Yang, H. C., Estimate of the first eigenvalue on a compact Riemannian manifold with negative lower bound of Ricci curvature,Science in China, Ser. A, 1989. 32(7): 689.

    Google Scholar 

  12. Jia, F., Estimate of the first eigenvalue on a compact Riemannian manifold with negative lower bound of Ricci curvature,Chin. Ann. Math. (in Chinese), 1991, 12(4): 496.

    Google Scholar 

  13. Kendall, W. S., Nonnegative Ricci curvature and the Brownian coupling property,Stochastics, 1986, 19: 111.

    MATH  MathSciNet  Google Scholar 

  14. Cranston, M., Gradient estimates on manifolds using coupling,J. Funct. Anal., 1991, 99: 110.

    Article  MATH  MathSciNet  Google Scholar 

  15. Cheeger, J., Ebin, D. G.,Comparison Theorems in Riemannian Geometry, Amsterdam: North-Holland, 1975.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, 100875, Beijing, China

    Mufa Chen & Fengyu Wang

Authors
  1. Mufa Chen
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  2. Fengyu Wang
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Additional information

Project supported in part by the National Natural Science Foundation of China, Qiu Shi Science and Technologies Foundation and the Doctoral Program Foundation of the State Education Commission of China.

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Chen, M., Wang, F. General formula for lower bound of the first eigenvalue on Riemannian manifolds. Sci. China Ser. A-Math. 40, 384–394 (1997). https://doi.org/10.1007/BF02911438

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  • DOI: https://doi.org/10.1007/BF02911438

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