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.
Similar content being viewed by others
References
Cartan E. Sur quelques familles remarquables d’hypersurfaces. Oeuvres Complètes, 1939, 2: 1481–1492
Castro I, Urbano F. Minimal Lagrangian surfaces in S2 × S2. Comm Anal Geom, 2017, 15: 217–248
Chen B-Y. Riemannian geometry of Lagrangian submanifolds. Taiwanese J Math, 2001, 5: 681–723
Chen B-Y. Pseudo-Riemannian Geometry, δ-Invariants and Applications. Hackensack: World Scientific, 2011
Chen B-Y, Ogiue K. On totally real submanifolds. Trans Amer Math Soc, 1974, 193: 257–266
do Carmo M, Dajczer M. Rotation hypersurfaces in spaces of constant curvature. Trans Amer Math Soc, 1983, 277: 685–709
Ejiri N. Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms. Proc Amer Math Soc, 1982, 84: 243–246
Ge J, Tang Z. Geometry of isoparametric hypersurfaces in Riemannian manifolds. Asian J Math, 2014, 18: 117–125
Hiepko S. Eine innere Kennzeichnung der verzerrten Produkte. Math Ann, 1979, 241: 209–215
Li H, Ma H, Wei G. A class of minimal Lagrangian submanifolds in complex hyperquadrics. Geom Dedicata, 2012, 158: 137–148
Ma H, Ohnita Y. On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres. Math Z, 2009, 261: 749–785
Ma H, Ohnita Y. Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. I. J Differential Geom, 2014, 97: 275–348
Ma H, Ohnita Y. Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces II. Tohoku Math J (2), 2015, 67: 195–246
Münzner H F. Isoparametrische Hyperffächen in Sphären. Math Ann, 1980, 251: 57–71
Münzner H F. Isoparametrische Hyperflächen in Sphären. II. Über die Zerlegung der Sphäre in Ballbündel. Math Ann, 1981, 256: 215–232
Nölker S. Isometric immersions of warped products. Differential Geom Appl, 1996, 6: 1–30
Oh Y-G. Second variation and stabilities of minimal Lagrangian submanifolds in Kaähler manifolds. Invent Math, 1990, 101: 501–519
Oh Y-G. Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math Z, 1993, 212: 175–192
Palmer B. Buckling eigenvalues, Gauss maps and Lagrangian submanifolds. Differential Geom Appl, 1994, 4: 391–403
Palmer B. Hamiltonian minimality and Hamiltonian stability of Gauss maps. Differential Geom Appl, 1997, 7: 51–58
Qian C, Tang Z. Recent progress in isoparametric functions and isoparametric hypersurfaces. In: Real and Complex Submanifolds. Springer Proceedings in Mathematics Statistics, vol. 106. Tokyo: Springer, 2014, 65–76
Reckziegel H. Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion. In: Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics, vol. 1156. Berlin: Springer, 1985, 264–279
Reckziegel H. On the geometry of the complex quadric. In: Geometry and Topology of Submanifolds, vol. 8. River Edge: World Scientific, 1996, 302–315
Siffert A. A new structural approach to isoparametric hypersurfaces in spheres. Ann Global Anal Geom, 2017, 52: 425–456
Smyth B. Differential geometry of complex hypersurfaces. Ann of Math (2), 1967, 85: 246–266
Tang Z, Yan W. Isoparametric theory and its applications. ArXiv:1709.07235, 2017
Torralbo F, Urbano F. Minimal surfaces in \(\mathbb{S}^{2}\times \mathbb{S}^{2}\). J Geom Anal, 2015, 25: 1132–1156
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-019-9551-2
Keywords
- minimal Lagrangian submanifolds
- the complex hyperquadric
- constant sectional curvature
- Gauss map
- isoparametric hypersurface