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

الأصل · EN

We classify Hopf hypersurfaces of non-flat complex space forms CPᵐ(4) and CHᵐ(-4), denoted jointly by CQᵐ(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 CPᵐ 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 CQᵐ(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 CQᵐ(4c) is mass-symmetric and we determine its Chen-type.