Sharp scalar and Ricci curvature pinching theorems for Lagrangian submanifolds in the homogeneous nearly Kähler Six-sphere

For connected closed Lagrangian submanifolds of the homogeneous nearly Kähler sphere $\mathbb S^6(1)$, we first establish a new integral inequality. Then, as applications we prove sharp pinching theorems for the scalar curvature $τ$ and Ricci curvature Ric, respectively: $τ\ge \tfrac{23}{8}$ implies