A Dajczer-Rodriguez Type Cylinder Theorem for Real Kahler Submanifolds

TL;DR

This paper extends a classical theorem on minimal real Kahler submanifolds, showing that higher codimension submanifolds with certain properties are either partially holomorphic, cylindrical, or twisted cylinders, with conditions for being complex ruled.

Contribution

It generalizes the Dajczer-Rodriguez theorem from codimension 2 to codimension 4 for complete real Kahler submanifolds, introducing the concept of twisted cylinders.

Findings

Submanifolds are either partially holomorphic, cylindrical, or twisted cylinders.

The complex relative nullity foliation is contained in a larger holomorphic foliation.

Conditions for when such submanifolds are complex ruled are examined.

Abstract

In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kahler submanifolds of codimension 4. The conclusion is that such the submanifold, if with complex dimension > 4, would be either partially holomorphic, or a cylinder, or a twisted cylinder in the sense that the complex relative nullity foliation is contained in a strictly larger holomorphic foliation, whose leaves are cylinders. We also examine the question of when such a submanifold is complex ruled.