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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 $$\mathbb{CP}^2(4)$$.

##### MSC:
 53B25 Local submanifolds 53B20 Local Riemannian geometry
Zbl 1064.53039