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# Pure and Applied Mathematics Quarterly

## Volume 9 (2013)

### Number 3

### A Dajczer-Rodriguez type cylinder theorem for real Kähler submanifolds

Pages: 563 – 577

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n3.a9

#### Authors

#### Abstract

In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kähler submanifold of codimension $2$, if with complex dimension $\gt 2$, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kähler submanifolds of codimension $4$. The conclusion is that such the submanifold, if with complex dimension $\gt 4$, 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.

#### Keywords

real Kähler submanifolds, relative nullity index, partially holomorphic extension, cylinder theorem, complex ruled submanifolds

#### 2010 Mathematics Subject Classification

53C40, 53C42, 53C55