Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature

Abstract

We present some new lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature in terms of the diameter and the lower Ricci curvature bound of the manifolds. For compact manifolds with boundary, it is assumed that, with respect to the outward normal, it is of nonnegative second fundamental form for the first Neumann eigenvalue and the mean curvature of the boundary is nonnegative for the first Dirichlet eigenvalue.