Abstract
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions, respectively all but one, coincide.
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Acknowledgements
This work was supported by the Tsinghua University-KU Leuven Bilateral Scientific Cooperation Fund and a collaboration project funded by National Natural Science Foundation of China and the Research Foundation Flanders (Grant No. 11961131001). The first author was supported by National Natural Science Foundation of China (Grant Nos. 11831005 and 11671224). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11831005 and 11671223). The third author was supported by the Excellence of Science Project of the Belgian Government (Grant No. G0H4518N). The third author and the fourth author were supported by the KU Leuven Research Fund (Grant No. 3E160361). The fifth author was supported by National Natural Science Foundation of China (Grant No. 11571185) and the Fundamental Research Funds for the Central Universities, and she expresses her deep gratitude to the Mathematical Sciences Institute at the Australian National University for its hospitality and to Professor Ben Andrews for his encouragement and help during her stay in MSI of ANU as a visiting fellow, while part of this work was completed. The authors thank the referees for carefully reading this paper and providing some helpful suggestions.
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Li, H., Ma, H., Van der Veken, J. et al. Minimal Lagrangian submanifolds of the complex hyperquadric. Sci. China Math. 63, 1441–1462 (2020). https://doi.org/10.1007/s11425-019-9551-2
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DOI: https://doi.org/10.1007/s11425-019-9551-2
Keywords
- minimal Lagrangian submanifolds
- the complex hyperquadric
- constant sectional curvature
- Gauss map
- isoparametric hypersurface