Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kähler manifolds
-
Henri Anciaux
and
Nikos Georgiou
Abstract
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. [20]). Finally, we determine the H-stability of a series of examples of H-minimal Lagrangian submanifolds: the product S1(r1) x ··· x S1(rn) of n circles of arbitrary radii in complex space Cn is H-unstable with respect to any indefinite flat Hermitian metric, while the product ℍ1(r1) x ···x ℍ1(rn) 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.
© 2014 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the type number of a hypersurface in the 6-dimensional sphere
- On the equality case in Ehrhart’s volume conjecture
- Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kähler manifolds
- The holomorphic symplectic structures on hyper-Kähler manifolds of type A∞
- Bifurcation values and stability of algebras of bounded polynomials
- Families of bitangent planes of space curves and minimal non-fibration families
- On deformations of G2-structures by Killing vector fields
- Cohomology of 3-points configuration spaces of complex projective spaces
- Slope stability of Fano fourfolds with respect to smooth surfaces
- On quartics with lines of the second kind
Articles in the same Issue
- Frontmatter
- On the type number of a hypersurface in the 6-dimensional sphere
- On the equality case in Ehrhart’s volume conjecture
- Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kähler manifolds
- The holomorphic symplectic structures on hyper-Kähler manifolds of type A∞
- Bifurcation values and stability of algebras of bounded polynomials
- Families of bitangent planes of space curves and minimal non-fibration families
- On deformations of G2-structures by Killing vector fields
- Cohomology of 3-points configuration spaces of complex projective spaces
- Slope stability of Fano fourfolds with respect to smooth surfaces
- On quartics with lines of the second kind
