On Energy Gap Phenomena of the Whitney Spheres in ℂ ⁿ or ℂℙ ⁿ

Abstract

Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ ⁿ or ℂℙ ⁿ, where $T = {\nabla ^ * }\tilde h$ and ${\tilde h}$ 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 ℂ ⁿ of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙ ⁿ.

Keywords

Lagrangian submanifolds / Whitney spheres / Gap theorem / Conformal maslov form

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