Minimal Lagrangian submanifolds of the complex hyperquadric

TL;DR

This paper develops a structural framework using almost product structures to analyze Lagrangian submanifolds of the complex hyperquadric, introducing local angle functions and classifying certain minimal cases.

Contribution

It introduces a new approach with local angle functions for studying Lagrangian submanifolds in complex hyperquadrics and classifies minimal cases with constant curvature and specific angle function conditions.

Findings

Local angle functions are constant for Gauss maps of isoparametric hypersurfaces.

Classified all minimal Lagrangian submanifolds with constant sectional curvature.

Identified minimal Lagrangian submanifolds with uniform or nearly uniform local angle functions.

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, respectively all but one, local angle functions coincide.