Minimal Lagrangian submanifolds of the complex hyperquadric

Abstract: In this talk, I will discuss some classification results about minimal Lagrangian submanifolds of the complex hyperquadric. 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 curvature and all minimal Lagrangian submanifolds for which all, respectively all but one, local angle functions coincide. This is joint work with Haizhong Li, Hui Ma, Joeri Van der Veken and Luc Vrancken.