# Classification of $\delta(2,n-2)$-ideal Lagrangian submanifolds in $n$-dimensional complex space forms

## TL;DR

This paper classifies Lagrangian submanifolds in complex space forms that achieve equality in a specific curvature inequality, providing a complete characterization of these ideal submanifolds.

## Contribution

It provides a full classification of $ ext{delta}(2,n-2)$-ideal Lagrangian submanifolds in complex space forms for dimensions n ≥ 5, satisfying the equality case everywhere.

## Key findings

- Classification of $ ext{delta}(2,n-2)$-ideal Lagrangian submanifolds.
- Characterization of submanifolds achieving equality in the curvature inequality.
- Extension of previous inequality results to a complete classification.

## Abstract

It was proven in [B.-Y. Chen, F. Dillen, J. Van der Veken and L. Vrancken, Curvature inequalities for Lagrangian submanifolds: the final solution, Differ. Geom. Appl. 31 (2013), 808-819] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{n}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: \begin{align*} \delta(2,n-2) \leq \frac{n^2(n-2)}{4(n-1)} H^2 + 2(n-2) c, \end{align*} where $H^2$ is the squared mean curvature and $\delta(2,n-2)$ is a $\delta$-invariant on $M$. In this paper we classify Lagrangian submanifolds of complex space forms $\tilde M^{n}(4c)$, $n \geq 5$, which satisfy the equality case of this inequality at every point.