Henri Anciaux, Nikos Georgiou
October 8, 2014
Let L be a Lagrangian submanifold of a pseudo- or para-Kähler 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-Kähler 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-Kähler manifolds, similar to Oh’s stability criterion for minimal Lagrangian manifolds in Kähler-Einstein manifolds (cf. ). 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 Cn 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 (, ); on the other hand, a class of H-minimal Lagrangian surfaces in the tangent bundle of a Riemannian, oriented surface has been identified in . We discuss the H-stability of all these examples.