Construction of homogeneous Lagrangian submanifolds in $\boldsymbol{CP}^n$ and Hamiltonian stability

Abstract

We apply the concept of castling transform of prehomogeneous vector spaces to produce new examples of minimal homogeneous Lagrangian submanifolds in the complex projective space. Furthermore we verify the Hamiltonian stability of a low dimensional example that can be obtained in this way.2010 Mathematics Subject Classification. 32J27, 53D12, 57S25.

Equivariant deformations of Hamiltonian stationary Lagrangian submanifolds

“…Hamiltonian stationary Lagrangians in Kähler-Einstein manifolds have been extensively studied in the literature, see e.g. [2,3,5,9,10,11,12]. In this subsection, we are interested in the particular case of minimal Lagrangians Σ of these manifolds, which are automatically Hamiltonian stationary, see Subsection 3.2.…”

Geometry of Certain Lagrangian Submanifolds in Hermitian Symmetric Spaces

“…It is a crucial to use Theorem and theory of prehomogeneous vector spaces due to Professors Mikio Sato and Tatsuo Kimura. It is still an open problem to classify compact homogeneous Lagrangian submanifolds in CP n obtained as orbits of non-simple compact Lie groups of P U (n + 1) (See also [19]).…”