Kaehler submanifolds of the real hyperbolic space

TL;DR

This paper investigates the classification of Kaehler submanifolds in hyperbolic space, extending known results from spherical cases and generalizing previous findings by Dajczer and Vlachos.

Contribution

It provides a classification result for Kaehler submanifolds in hyperbolic space, paralleling spherical case results, and generalizes earlier work by Dajczer and Vlachos.

Findings

Classification of Kaehler submanifolds in hyperbolic space under low codimension.

Extension of spherical submanifold results to hyperbolic ambient spaces.

Generalization of previous theorems by Dajczer and Vlachos.

Abstract

The local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic space $\mathbb{H}^{2n+p}$ with low codimension $2\leq p\leq n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere $\mathbb{S}^{2n+p}$, $2\leq p\leq n-1$, 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.