A splitting theorem for Kähler submanifolds of space-forms

Abstract

Isometric immersions from Kahler manifolds with parallel pluri-mean curvature (ppmc) generalize, in a natural way, the constant mean curvature (cmc) sufaces. The (2,0) part of the complexified second fundamental form is a holomorphic quadratic differential (Q) which plays a central role in the study of the cmc sufaces. Likewise, for ppmc immersions, Q is also a vector bundle valued holomorphic quadratic differencial, significant in the study of the geometry of the immersion. It is well known that those immersions with Q vanishing are extrinsically symmetric ([10] and [11]). In this work we study ppmc immersions with big nullity index of Q.