On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

TL;DR

This paper explores the geometry of compact Lagrangian submanifolds in complex hyperquadrics, classifies homogeneous examples, and analyzes their Hamiltonian stability using isoparametric hypersurfaces and moment map techniques.

Contribution

It provides a classification of compact homogeneous Lagrangian submanifolds in complex hyperquadrics and determines their Hamiltonian stability.

Findings

Classification of compact homogeneous Lagrangian submanifolds

Determination of Hamiltonian stability for certain minimal Lagrangian submanifolds

Connection between isoparametric hypersurfaces and Lagrangian geometry

Abstract

The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In this paper we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with $g(=1,2,3)$ distinct principal curvatures.