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 N n in Sasakian space forms N ˜ 2 n + 1 ( ϵ ) . 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 N n . We also obtain a Simons-type inequality for the same ambient space forms N ˜ 2 n + 1 ( ϵ ) .