Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces

Abstract

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant principal curvatures in $S^{n+1}(1)$ is a compact monotone and cyclic embedded Lagrangian submanifold with minimal Maslov number $2n/g$. The main result of this paper is to determine completely the Hamiltonian stability of all compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as the images of the Gauss map of homogeneous isoparametric hypersurfaces in the unit spheres, by harmonic analysis on homogeneous spaces and fibrations on homogeneous isoparametric hypersurfaces. In addition, the discussions on the exceptional Riemannian symmetric space $(E_6, U(1)\cdot Spin(10))$ and the corresponding Gauss image have their own interest.