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

2010/05/19 by Ivko Dimitrić, Ivko Dimitric, Dimitric, Ivko
Mathematics · #53C40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #msc:53C40

paper · pdf · doi:10.48550/arxiv.1005.3572

37 pages

arxiv created 2010/05/19 · openalex publication_date 2010/05/19 · arxiv updated 2010/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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