Estimates for eigenvalues on Riemannian manifolds

In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n  -dimensional Euclidean space RnRn, the conjecture of Pólya is well known: the k  th eigenvalue λkλk of the Dirichlet eigenvalue problem of Laplacian satisfiesλk⩾4π2(ωnvolΩ)2nk2n,for k=1,2,…. Li and Yau [P. Li, S.T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983) 309–318] (cf. Lieb [E. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, in: Proc. Sympos. Pure Math., vol. 36, 1980, pp. 241–252]) have given a partial solution for the conjecture of Pólya, that is, they have proved1k∑i=1kλi⩾nn+24π2(ωnvolΩ)2nk2n,for k=1,2,…, which is sharp in the sense of average. In this paper, we consider a general setting for complete Riemannian manifolds. We establish an analog of the Li and Yau's inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a complete Riemannian manifold. Furthermore, we obtain a universal inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a hyperbolic space Hn(−1)Hn(−1). From it, we prove that when the bounded domain Ω   tends to Hn(−1)Hn(−1), all eigenvalues tend to (n−1)24.