Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces

摘要

The image of the Gauss map of any oriented isoparametric hypersurface of theunit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in thecomplex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gaussimage of a compact oriented isoparametric hypersurface with $g$ distinctconstant principal curvatures in $S^{n+1}(1)$ is a compact monotone and cyclicembedded Lagrangian submanifold with minimal Maslov number $2n/g$. The mainresult of this paper is to determine completely the Hamiltonian stability ofall compact minimal Lagrangian submanifolds embedded in complex hyperquadricswhich are obtained as the images of the Gauss map of homogeneous isoparametrichypersurfaces in the unit spheres, by harmonic analysis on homogeneous spacesand fibrations on homogeneous isoparametric hypersurfaces. In addition, thediscussions on the exceptional Riemannian symmetric space $(E_6, U(1)\cdotSpin(10))$ and the corresponding Gauss image have their own interest.