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Minimal Lagrangian submanifolds of the complex hyperquadric

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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Submitted on : Friday, July 15, 2022 - 9:25:11 AM
Last modification on : Saturday, July 16, 2022 - 3:50:56 AM

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Haizhong Li, Hui Ma, Joeri van Der Veken, Luc Vrancken, Xianfeng Wang. Minimal Lagrangian submanifolds of the complex hyperquadric. Science China Mathematics, Science China Press, 2020, 63 (8), pp.1441-1462. ⟨10.1007/s11425-019-9551-2⟩. ⟨hal-03723643⟩

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