On energy gap phenomena of the Whitney spheres in \(\mathbb{C}^n\) or \(\mathbb{CP}^n\) (Q6063484)

The second author [Math. Z. 297, No. 3--4, 1601--1611 (2021; Zbl 1465.53087)] and the first author and \textit{J. Yin} [Pac. J. Math. 320, No. 2, 299--318 (2022; Zbl 1509.53068)] proved several rigidity theorems for Lagrangian surfaces satisying \(\nabla^*T=0\) or \(\nabla^*\nabla^*T=0\) in \(\mathbb C^2\) after investigating Lagrangian submanifolds in \(\mathbb C^n\) satisfying the same equations using the Lagrangian trace-free second fundamental form \(\tilde h\) and \(T=\nabla^*\tilde h\). As a byproduct, they obtained a new characterization of Whitney spheres in \(\mathbb C^2\). In this paper, an higher dimensional result valid in \(\mathbb C^n\) is obtained. Theorem. Assume that \(M^n \hookrightarrow \mathbb C^n\) is a complete Lagrangian submanifold satisfying \(\nabla^*T=0\). Then there exists a constant \(\epsilon_0 >0\) such that if \[ \int_M|\tilde{h}|^nd\mu\leq \epsilon_0 \text{ and } \displaystyle \lim_{R\rightarrow \infty} \frac {1}{R^2}\int_{M_R}|h|^2d\mu=0, \] where \(M_R\) denotes the geodesic ball in \(M^n\) with radius \(R\), then \(M^n\) is either a Lagrangian subspace or a Whitney sphere. Similar results for a submanifold satisfying \(\nabla^*\nabla^*T=0\) and for submanifolds in \(\mathbb CP^n\) are proved as well.