Mathematische Annalen

, Volume 331, Issue 2, pp 445–460

Estimates on Eigenvalues of Laplacian


DOI: 10.1007/s00208-004-0589-z

Cite this article as:
Cheng, QM. & Yang, H. Math. Ann. (2005) 331: 445. doi:10.1007/s00208-004-0589-z


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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© 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