摘要
The author gives an optimum estimate of the first eigenvalue of a compact Riemannian manifold. It is shown that let M be a compact Riemannian manifold, then the first eigenvalue λ<sub>1</sub> of the Laplace operator of M satisfies α<sub>1</sub>+max{0,-（n-1）K}≥π<sup>2</sup>/d<sup>2</sup> where d is the diameter of M and （n-1）K is the negative lower bound of the Ricci curvature of M.
The author gives an optimum estimate of the first eigenvalue of a compact Riemannian manifold. It is shown that let M be a compact Riemannian manifold, then the first eigenvalue λ_1 of the Laplace operator of M satisfies α_1+max{0,-(n-1)K}≥π~2/d^2 where d is the diameter of M and (n-1)K is the negative lower bound of the Ricci curvature of M.
