Abstract
In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold Nn in Sasakian space forms e N 2n+1(e). We prove that a minimal Legendrian submanifolds in a Sasakian space form is isometric to a standard sphere S n if the Ricci curvature satisfies an extrinsic condition which includes a gradient of a function, the constant holomorphic sectional curvature of the ambient space and a dimension of Nn. We also obtain a Simons-type inequality for the same ambient space forms (N) over tilde (2n+1(epsilon)).