Kaehler submanifolds of the real hyperbolic space

The local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic space ${\mathbb{H}}^{2n+p}$ with low codimension $2\le p\le n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere ${\mathbb{S}}^{2n+p}$, $2\le p\le n-1$, since Florit, Hui and Zheng have shown that the codimension has to be $p=n-1$ and then that any submanifold is just part of an extrinsic product of two-dimensional umbilical spheres in ${\mathbb{S}}^{3n-1}\subset {\mathbb{R}}^{3n}$. The main result of this paper is a version for Kaehler manifolds isometrically immersed into the hyperbolic ambient space of the result for spherical submanifolds. Besides, we generalize several results obtained by Dajczer and Vlachos.