On Energy Gap Phenomena of the Whitney Spheres in ℂn or ℂℙn.

Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂn or ℂℙn, where T = ∇ ∗ h ˜ and h ˜ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ2 under proper small energy assumption and gave new characterization of the Whitney spheres in ℂ2. In this paper, the authors extend these results to Lagrangian submanifolds in ℂn of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙn. [ABSTRACT FROM AUTHOR]