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Mathematische Annalen

, Volume 331, Issue 2, pp 445–460 | Cite as

Estimates on Eigenvalues of Laplacian

  • Qing-Ming ChengEmail author
  • Hongcang Yang
Article

Abstract.

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere S n (1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere S N (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 S n (1).

Keywords

Manifold Riemannian Manifold Unit Sphere Asymptotical Formula Connected Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and EngineeringSaga UniversitySagaJapan
  2. 2.Academy of Mathematics and Systematical SciencesBeijingChina

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