Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-K?hler manifolds

摘要

Let L be a Lagrangian submanifold of a pseudo- or para-K?hler 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-K?hler 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-K?hler manifolds, similar to Oh’s stability criterion for minimal Lagrangian manifolds in K?hler-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)×? ? ?×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)×? ? ?×H~1(r_n) of n hyperbolas in para-complex vector space Dn 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.