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 is in particular a local extremizer of the volume with respect to Hamiltonian variations. We also give a stability criterion for spacelike minimal Lagrangian

Second variation for L-minimal Legendrian submanifolds in pseudo-Sasakian manifolds

“…We can group the Hessian term and the Ricci term by means of the pseudo-Riemannian Bochner formula 4 in the Appendix of [3]:…”

Lagrangian immersions in the product of Lorentzian two manifolds

“…Furthermore, the normal curvature of Φ vanishes. It is important to mention that the Boschner formula holds also for pseudo-Riemannian metrics [4].…”

“…For the case where γ is a timelike geodesic, a similar argument shows thatf is H-unstable. We emphasize here that by using different methods in [4], it was first proven that the Gauss mapf is H-stable.…”

“…We recall that the H-stability of a H-minimal surface S in a pseudoRiemannian manifold (M, G) is given by the monotonicity of the second variation formula of the volume V (S) under Hamiltonian deformations (see [4] and [10]). For a smooth compactly supported function u ∈ C ∞ c (S) the second variation δ 2 V (S)(X) formula in the direction of the Hamiltonian vector field X = J∇u is:…”

“…If the second variation of the volume functional of a H-minimal submanifold is monotone for any Hamiltonian compactly supported variation, it is said to be Hamiltonian stable (or H-stable). In [10] and [11], the second variation formula of a H-minimal submanifold has been derived in the case of a Kähler manifold, while for the pseudo-Kähler case it has been given in [4]. The next theorem, in Section 4, investigates the H-stability of projected rank one Hamiltonian G ǫ -minimal surfaces in Σ 1 × Σ 2 : …”