Eigenvalue estimates on domains in complete noncompact Riemannian manifolds

Abstract

In this paper, we obtain universal inequalities for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian and the clamped plate problem on a bounded domain in an n-dimensional (n ≥ 3) noncompact simply connected complete Riemannian manifold with sectional curvature Sec satisfying -K 2 ≤ Sec ≤ -k 2 , where K ≥ k ≥ 0 are constants.When M is ‫ވ‬ n (-1) (n ≥ 3), these inequalities become ones previously found by Cheng and Yang.