An energy gap phenomenon for the Whitney sphere

Abstract: For an immersed Lagrangian submanifold, let $\check{A}$ be the Lagrangian trace-free second fundamental form. In this note we consider the equation $\nabla^*T=0$ on Lagrangian surfaces immersed in $\mathbb{C}^2$, where $T=-2\nabla^*(\check{A}\lrcorner\omega)$, and we prove a gap theorem for the Whitney sphere as a solution to this equation.