On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms

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)).