Kaehler submanifolds of hyperbolic space

Abstract

We present several local and global results on isometric immersions of Kaehler manifolds $M^{2n}$ into hyperbolic space $\Hy^{2n+p}$. For instance, a classification is given in the case of dimension $n\ge 4$ and codimension $p=2$. Moreover, as corollaries of general results, we conclude that there are no isometric immersion in codimension $p\le n-2$ if the Kaehler manifold is of dimension $n\ge 4$ and either has a point of positive holomorphic sectional curvature or is compact.