Abstract
It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex 2-plane Grassmannian with respect to the natural para-quaternionic Kähler structure. We also identify these twistor spaces with the sets of circles in totally geodesic complex hyperbolic lines in the complex hyperbolic space. As an application, we describe all classical Hopf examples. We also solve the remarkable and long-standing problem of the existence of Hopf real hypersurfaces in the complex hyperbolic space, different from the horosphere, such that the associated principal curvature is 2. We exhibit a method to obtain plenty of them.
Funding Statement
The first author is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2022R1F1A1062694). The second author is supported by JSPS KAKENHI Grant Number JP20K03575. The third author is partially supported by the projects PID2020-116126GB-I00 (MICINN), PY20-01391 (Junta de Andalucía) and the “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M, funded by MCIN/AEI/10.13039/501100011033/.
Citation
Jong Taek CHO. Makoto KIMURA. Miguel ORTEGA. "A twistor construction of Hopf real hypersurfaces in complex hyperbolic space." J. Math. Soc. Japan 75 (3) 1025 - 1053, July, 2023. https://doi.org/10.2969/jmsj/88968896
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