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

TL;DR

This paper extends previous rigidity results for Lagrangian submanifolds satisfying certain energy conditions from complex dimension 2 to higher dimensions and complex projective spaces, characterizing Whitney spheres.

Contribution

It generalizes rigidity theorems and characterizations of Whitney spheres to Lagrangian submanifolds in higher-dimensional complex spaces and projective spaces.

Findings

Rigidity theorems for higher-dimensional Lagrangian submanifolds.

Characterization of Whitney spheres in $ ext{dim} ext{geq} 3$ cases.

Extension of energy gap phenomena to complex projective spaces.

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 $\mathbb{CP}^n$.