Estimates on Eigenvalues of Laplacian
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- Cheng, QM. & Yang, H. Math. Ann. (2005) 331: 445. doi:10.1007/s00208-004-0589-z
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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 , Leung , Li  and Harrel II and Stubbe , 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).