Lagrangian submanifolds of the complex hyperbolic space

Abstract

In previous papers [2, 3], B. Y. Chen introduced a Riemannian invariant $\delta_M$ for a Riemannian $n$-manifold $M^n$. He proved in [3] that every submanifold $M^n$ in the complex hyperbolic $m$-space $CH^{m}(-4)$ satisfies the sharp inequality: $\delta_{M}\leq(n^{2}(n-2)/2(n-1))H^{2}-2(n+1)(n-2)$, where $H^2$ is the squared mean curvature. In this paper, we study Lagrangian submanifolds in $CH^{n}(-4)$ which satisfy the equality case of the inequality.