Real hypersurfaces with constant principal curvatures in complex space forms

We present the motivation and current state of the classification problem of real hypersurfaces with constant principal curvatures in complex space forms. In particular, we explain the classification result of real hypersurfaces with constant principal curvatures in nonflat complex space forms and whose Hopf vector field has nontrivial projection onto two eigenspaces of the shape operator. This constitutes the following natural step after Kimura and Berndt?s classifications of Hopf real hypersurfaces with constant principal curvatures in complex space forms.