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 ${\nabla }^{\ast }T=0$ or ${\nabla }^{\ast }{\nabla }^{\ast }T=0$ in ${\mathbb{C}}^n$ or ${\mathbb{CP}}^n$, where $T={\nabla }^{\ast }\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ${\nabla }^{\ast }T=0$ or ${\nabla }^{\ast }{\nabla }^{\ast }T=0$ 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 $n\ge 3$ and to Lagrangian submanifolds in ${\mathbb{CP}}^n$.

16 pages. References added. arXiv admin note: substantial text overlap with: arXiv:2006.08191