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Estimates on Eigenvalues of Laplacian

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In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sn(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere SN(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sn(1).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan

    Qing-Ming Cheng

  2. Academy of Mathematics and Systematical Sciences, CAS, Beijing, 100080, China

    Hongcang Yang

Authors
  1. Qing-Ming Cheng
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  2. Hongcang Yang
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Correspondence to Qing-Ming Cheng.

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Mathematics Subject Classification (2000): 35P15, 58G25, 53C42

Research was partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.

Research was partially Supported by SF of CAS, Chinese NSF and NSF of USA.

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Cheng, QM., Yang, H. Estimates on Eigenvalues of Laplacian. Math. Ann. 331, 445–460 (2005). https://doi.org/10.1007/s00208-004-0589-z

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