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

Henri Anciaux, Nikos Georgiou

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2 Citations (Scopus)

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) × ⋯ × 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) × ⋯ ×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.

Original languageEnglish
Pages (from-to)587-612
Number of pages26
JournalAdvances in Geometry
Volume14
Issue number4
DOIs
Publication statusPublished - 01 Oct 2014
Externally publishedYes

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