Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kahler manifolds

Let L be a Lagrangian submanifold of a pseudo-or para-Kahler manifold with nondegenerate induced metric which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation formula of the volume of L with respect to Hamiltonian variations and apply this formula to several cases. We observe that a minimal Lagrangian submanifold L in a Ricci-flat pseudo-or para-Kahler manifold is H-stable, i.e. its second variation is definite and L is in particular a local extremizer of the volume with respect to Hamiltonian variations. We also give a stability criterion for spacelike minimal Lagrangian submanifolds in para-Kahler manifolds, similar to Oh's stability criterion for minimal Lagrangian manifolds in Kahler-Einstein manifolds (cf. [20]). Finally, we determine the H-stability of a series of examples of H-minimal Lagrangian submanifolds: the product S-1 (r(1)) X . . . X S 1 (r(n)) of n circles of arbitrary radii in complex space C n is H-unstable with respect to any indefinite flat Hermitian metric, while the product H-1 (r(1)) X . . . X H-1 (r(n)) of n hyperbolas in para-complex vector space D-n is H-stable for n = 1, 2 and H-unstable for n >= 3. Recently, minimal Lagrangian surfaces in the space of geodesics of space forms have been characterized ([4], [11]); on the other hand, a class of H-minimal Lagrangian surfaces in the tangent bundle of a Riemannian, oriented surface has been identified in [6]. We discuss the H-stability of all these examples.