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

Abstract

We classify Hopf hypersurfaces of non-flat complex space forms CP^m(4) and CH^m(-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-Ros (minimal case) and Udagawa (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 mass-symmetric in a naturally defined hyperquadric containing the image of CQ^m(4c) via these embeddings. The classification of such hypersurfaces is done in CQ^2(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.