Differential geometry of Lagrangian submanifolds and Hamiltonian variational problems

Abstract

In this article we shall provide a survey on our recent works ((M-O1),(M-O2)) and their environs on differential geometry of Lagrangian submanifolds in specific symplectic Kahler manifolds, such as complex pro- jective spaces, complex space forms, Hermitian symmetric spaces and Kahler C-spaces. We shall discuss (1) Hamiltonian minimality and Hamiltonian sta- bility of Lagrangian submanifolds in Hamiltonian volume minimizing problem, (2) classification problem of homogeneous Lagrangian submanifolds from the viewpoint of Lagrangian orbits and moment maps, (3) tightness problem of Lagrangian submanifolds. Moreover we shall give attention to Lagrangian sub- manifolds in complex hyperquadrics, which are compact Hermitian symmetric spaces of rank 2. The relationship between certain minimal Lagrangian sub- manifold in complex hyperquadrics and isoparametric hypersurfaces in spheres will be emphasized. Recently we gave a complete classification of compact ho- mogeneous Lagrangian submanifolds in complex hyperquadrics and we deter- mined the Hamiltonian stability of ALL compact minimal Lagrangian subman- ifolds embedded in complex hyperquadrics which are obtained as the Gauss images of homogeneous isoparametric hypersurfaces in spheres.