  A contact stationary Legendrian submanifold (briefly, CSL submanifold) is a
stationary point of the volume functional of Legendrian submanifolds in a
Sasakian manifold. Much effort has been paid in the last two decades to
construct examples of such manifolds, mainly by geometers using various
geometric methods. But we have rare knowledge about their geometric properties
till now. Recently, Y. Luo (\cite{ Luo2, Luo1}) proved that a closed CSL
surface in $\mathbb{S}^5$ with the square length of its second fundamental form
belonging to $[0,2]$ must be totally geodesic or be a flat minimal Legendrian
torus, which generalizes a related gap theorem of minimal Legendrian surface
due to Yamaguchi et al. (\cite{YKM}). In this paper, we will study the general
dimensional case of this result.
