Estimates on Eigenvalues of Laplacian

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 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 Weyls 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).Mathematics Subject Classification (2000): 35P15, 58G25, 53C42Research 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.