Hopf hypersurfaces of low type in non-flat complex space forms

We classify Hopf hypersurfaces of non-flat complex space forms CP^m(4) and CHm(−4), denoted jointly by CQ^m(4c), that are of 2-type in the sense of B. Y. Chen, via the embedding into a suitable (pseudo) Euclidean space of Hermitian matrices by projection operators. This complements and extends earlier classifications by Martinez and Ros (the minimal case) and Udagawa (the CMC case), who studied only hypersurfaces of CP^m and assumed them to have constant mean curvature instead of being Hopf. Moreover, we rectify some claims in Udagawa’s paper to give a complete classification of constant-mean-curvature-hypersurfaces of 2-type. We also derive a certain characterization of CMC Hopf hypersurfaces which are of 3-type and masssymmetric in a naturally-defined hyperquadric containing the image of CQ^m(4c) via these embeddings. The classification of such hypersurfaces is done in CQ²(4c), under an additional assumption in the hyperbolic case that the mean curvature is not equal to ±2/3. In the process we show that every standard example of class B in CQ^m(4c) is mass-symmetric and we determine its Chen-type.