Curvature inequalities for Lagrangian submanifolds: The final solution

Let MnMn be an n  -dimensional Lagrangian submanifold of a complex space form M˜n(4c) of constant holomorphic sectional curvature 4c. We prove a pointwise inequalityδ(n1,…,nk)⩽a(n,k,n1,…,nk)‖H‖2+b(n,k,n1,…,nk)c,δ(n1,…,nk)⩽a(n,k,n1,…,nk)‖H‖2+b(n,k,n1,…,nk)c, with on the left-hand side any delta-invariant of the Riemannian manifold MnMn and on the right-hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right-hand side are optimal in the sense that there exist non-minimal examples satisfying equality at least one point. We also characterize those Lagrangian submanifolds satisfying equality at any of their points. Our results correct and extend those given in [6].