Curvature inequalities for Lagrangian submanifolds
Résumé
Let M-n be an n-dimensional Lagrangian submanifold of a complex space form (M-n) over tilde (4c) of constant holomorphic sectional curvature 4c. We prove a pointwise inequality
delta(n(1),...,n(k)) <= a(n,k,n(1),...n(h))parallel to H parallel to(2) + b(n,k,n(1),...n(k))c,
with on the left-hand side any delta-invariant of the Riemannian manifold M-n 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]