An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature (Q1650521)

In [Bull. Transilv. Univ. Braşov, Ser. B (N.S.) 14(49), 85--93 (2007; Zbl 1195.53083)] \textit{S. Decu} et al. introduced the normalized \(\delta \)-Casorati curvatures \(\widehat{\delta } _{C}(n-1)\) and \(\delta _{C}(n-1)\) for an \(n\)-dimensional Riemannian submanifold of a Riemannian manifold. They also proved that the normalized \( \delta \)-Casorati curvature \(\delta _{C}(n-1)\) satisfies the inequality \[ \delta _{C}(n-1)\geq \rho -c.\tag{i1} \] In the paper under review, by investigating a string of suitable constrained extremum problems on a Lagrangian submanifold, the author improves the inequality (i1) in the case of Lagrangian submanifolds of complex space forms. The following result is obtained (Theorem \(3.4.\)): \[ \delta _{C}(n-1)\geq \rho -c+\frac{n}{n+3}||H||^{2}.\tag{i2} \] An obstruction to minimal Lagrangian isometric immersion in \(\mathbb{C}^{n}\) is derived. The equality case of the improved inequality is investigated and the classification of Casorati ideal submanifolds is given. The paper ends with four examples \ showing that the inequality (i2) is the best possible.