zbMATH — the first resource for mathematics

Lagrangian submanifolds attaining equality in the improved Chen’s inequality. (English) Zbl 1130.53016
For a Riemannian manifold \(M\) Chen introduced an invariant \(\delta_M(p)=\tau(p)-(\text{inf\,}K)(p):M\mapsto\mathbb{R}\), where \((\text{inf\,}K)(p)\) is the infimum of the sectional curvature of \(M\) at \(p\), and \(\tau(p)\) is the scalar curvature. It is known that \[ \delta_M\leq{(n+ 2)(n+1)\over 2}{c\over 2}+ {n^2\over 2} {2n-3\over 2n+3}\| H\|^2, \] if \(M\) is a complex space form of constant holomorphic sectional curvature \(4c\) and \(H\) is the mean curvature for the \(n\)-dimensional Lagrangian submanifold of \(M\).
In the paper the non-minimal 3-dimensional Lagrangian submanifolds in \(\mathbb{C} P^3(4)\) attaining at all points equality in [J. Bolton, C. Scharlach, L. Vrancken and L. M. Woodward, Sendai: Tôhoku University. Tôhoku Math. Publ. 20, 23–31 (2001; Zbl 1064.53039)] are considered. It is shown how such submanifolds may be obtained starting from a minimal Lagrangian surface in \(\mathbb{CP}^2(4)\).

53B25 Local submanifolds
53B20 Local Riemannian geometry
Zbl 1064.53039