Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 3

A Dajczer-Rodriguez type cylinder theorem for real Kähler submanifolds

Pages: 563 – 577

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

Authors

Jinwen Yan (Center for Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China)

Fangyang Zheng (Center for Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China; Department of Mathematics, The Ohio State University, Columbus, Oh., U.S.A.)

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

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Published 13 November 2013