On energy gap phenomena of the Whitney sphere and related problems

In this paper, we study Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c=0\ or \ 1)$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is trace-free second fundamental form. We obtain some Simons' type integral inequalities rigidity theorems for such submanifolds. Moreover $\mathbb{C}^n$ $\nabla^*\nabla^*T=0$ introduce a flow method related to them.