Kahler submanifolds with parallel pluri-mean curvature

We investigate the local geometry of a class of Kahler submanifolds M (subset of)R^n which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the (1,1)-part (i.e., the dzidzj-components) of the second fundamental form (alpha), which we call the pluri-mean curvature. We show that these Kahler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic.