A Universal Bound for the Low Eigenvalues of Neumann Laplacians on Compact Domains in a Hadamard Manifold

 Let M be an n-dimensional simply connected Hadamard manifold with Ricci curvature satisfying and be a bounded domain having smooth boundary. In this paper, we prove that the first n nonzero Neumann eigenvalues of the Laplacian on Ω satisfy , where is a computable constant depending only on and , Ω being the volume of Ω. This result generalizes the corresponding estimate for bounded domains in a Euclidean space obtained recently by M. S. Ashbaugh and R. D. Benguria. (Received 19 May 1998; in revised form 21 September 1998)