On energy gap phenomena of the Whitney spheres in $\mathbb{C}^n$ or $\mathbb{CP}^n$

Abstract: In \cite{Zh} \cite{LY} Zhang, Luo and Yin initiated the study of Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ or ${\rm \nabla^\nabla^}T=0$ in $\mathbb{C}^n$ or $\mathbb{CP}^n$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ${\rm \nabla^*} T=0$ or ${\rm \nabla^\nabla^}T=0$ in $\mathbb{C}^2$ under proper small energy assumption and gave new characterization of the Whitney spheres in $\mathbb{C}^2$. In this paper we extend these results to Lagrangian submanifolds in $\mathbb{C}^n$ of dimension $n\geq3$ and to Lagrangian submanifolds in $\mathbb{CP}^n$.