  We investigate the local geometry of a class of K\"ahler submanifolds $M
\subset \R^n$ which generalize surfaces of constant mean curvature. The role of
the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar
z_j$-components) of the second fundamental form $\alpha$, which we call the
pluri-mean curvature. We show that these K\"ahler 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.
