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

Abstract

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular we observe that a minimal Lagrangian submanifold L in a Ricci-flat pseudo- or para-Kähler manifold is H-stable, i.e. its second variation is definite and L therefore a local extremizer of the volume with respect to Hamiltonian variations. We also give a stability criterion for spacelike minimal Lagrangian submanifolds in para-Kähler manifolds, similar to Oh's stability criterion for minimal Lagrangian manifolds in Kähler-Einstein manifolds. Finally, we determine the H-stability of a series of examples of H-minimal Lagrangian submanifolds: the product 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 of n hyperbolas in para-complex vector space D^n is H-stable for n=1,2 and H-unstable for n > 2. Recently, minimal Lagrangian surfaces in the space of geodesics of three-dimensional space forms have been characterized; on the other hand, a class of H-minimal Lagrangian surfaces in the tangent bundle of a Riemannian, oriented surface has been identified. We discuss the H-stability of all these examples.