Pure and Applied Mathematics Quarterly
Volume 9 (2013)
A Dajczer-Rodriguez type cylinder theorem for real Kähler submanifolds
Pages: 563 – 577
In 1991, Dajczer and Rodriguez proved in  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.
real Kähler submanifolds, relative nullity index, partially holomorphic extension, cylinder theorem, complex ruled submanifolds
2010 Mathematics Subject Classification
53C40, 53C42, 53C55