Homogeneous Lagrangian submanifolds

Abstract

We characterize isometric actions on compact Kähler manifolds admitting a Lagrangian orbit, describing under which condition the Lagrangian orbit is unique. We furthermore give the complete classification of simple groups acting on the complex projective space with a Lagrangian orbit, and we give the explicit list of these orbits.

A Hamiltonian Stable Minimal Lagrangian Submanifold of Projective Space with Nonparallel Second Fundamental Form

“…New examples of compact homogeneous minimal Lagrangian submanifolds of CP n were found in [2]. The result of the present paper also suggests further investigations of the H-stability of the examples with nonparallel second fundamental form.…”

“…From now on we will regard the representation space as the space of complex homogeneous polynomials of degree 3 in two variables z 1 , z 2 . As proven in [14] and [2], P(S 3 C 2 ) ∼ = CP 3 is almost homogeneous under the complexified action of SL(2, C) and the open orbit is a Stein manifold. From Theorem 1 in [2] it follows that CP 3 admits a unique SU(2)-homogeneous Lagrangian submanifold L which is also minimal with respect to the Kähler structure given by the Fubini-Study metric g FS .…”

“…The induced metric on L Fix an orthonormal basis with respect to the opposite of the Killing form B on su (2). Recall that if we see su(2) as a matrix group, B has the following form:…”

Hamiltonian minimality of normal bundles over the isoparametric submanifolds

“…We note that there are some known H-minimal Lagrangian submanifolds in CP m−1 . For instance, any compact, extrinsically homogeneous Lagrangian submanifolds in CP m−1 are H-minimal, and Bedulli and Gori [4] give the complete classification of Lagrangian orbits which are obtained by a simple Lie group of isometries acting on CP m−1 . On the other hand, Anciaux and Castro [2] gave examples of H-minimal Lagrangian immersions of manifolds with various topologies by taking a product of a Lagrangian surface and Legendrian immersions in odd-dimensional unit spheres.…”

Construction of Lagrangian submanifolds in ℂℙn

“…D. Joyce [2002] gave many examples of minimal Lagrangian submanifolds with symmetries in ‫ރ‬ n . L. Bedulli and A. Gori [2008] studied homogeneous Lagrangian submanifolds in ‫ސރ‬ n . R. Chiang [2004] gave many Lagrangian submanifolds in ‫ސރ‬ n with interesting topological feature.…”