Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces, I

Abstract

The image of the Gauss map of any oriented isoparametric hypersurface in the standard unit sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n(\mathrm{C})$. 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 $2n / g$. We obtain the Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces of classical type with $g = 4$. Combining with our results in On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres and Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces II, we completely determine the Hamiltonian stability of the Gauss images of all homogeneous isoparametric hypersurfaces.