First eigenvalue, volume growth and Ricci curvature (Q815169)

About the estimate of the first eigenvalue, many results have been established, see for example \textit{M. P. do Carmo} and \textit{D. Zhou} [Trans. Am. Math. Soc. 351, 1391--1401 (1999; Zbl 0910.53039)], \textit{M. E. Gage}, Indiana Univ. Math. J. 29, 897--912 (1980; Zbl 0465.53031)]. In this paper a new relation between Ricci curvature and volume growth is proved in a complete Riemannian manifold. Also the first eigenvalue of complete simply connected Riemannian manifolds with negative constant sectional curvature is calculated.