Lagrangian submanifolds in complex space forms satisfying an improved equality involving $\delta (2,2)$

Abstract

It was proved in [8,9] that every Lagrangian submanifold $M$ of a complex space form ${\tilde{M}}^5(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: {align}\tag{A}δ(2,2)\leq \text{\small$254$} H^{2}+8c,{align} where $H^2$ is the squared mean curvature and $\delta (2,2)$ is a $\delta$-invariant on $M$ introduced by the first author. This optimal inequality improves a special case of an earlier inequality obtained in [B.-Y. Chen, Japan. J. Math. 26 (2000), 105-127]. The main purpose of this paper is to classify Lagrangian submanifolds of ${\tilde{M}}^5(4c)$ satisfying the equality case of the improved inequality (A).

arXiv comment25 pages; appeared in Publ. Math. Debrecen, 82 (2013), 193-217