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We investigate the local geometry of a class of Kähler submanifolds M⊂ 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 z ̄ _j-components) of the second fundamental form α, which we call the pluri-mean curvature. We show that these Kähler 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.