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 $\tau$ and Ricci curvature Ric, respectively: $\tau \ge \frac{23}{8}$ implies either $\tau =6$ with it totally geodesic, or $\tau =\frac{23}{8}$ with it congruent to the embedded Berger sphere of Dillen–Verstraelen–Vrancken; whereas $\mathrm{R}\mathrm{i}\mathrm{c}\ge \frac{1}{8}$ implies that the submanifold is either totally geodesic, or has constant sectional curvature $K=\frac{1}{16}$, or is congruent to the embedded Berger sphere of Dillen-Verstraelen-Vrancken.