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

Author(s)

Anciaux, Henri ^[1] ; Georgiou, Nikos ^[2]

Affiliation

^[1] Univ Sao Paulo, IME, BR-05508090 Sao Paulo - Brazil

^[2] Univ Cyprus, Dept Math & Stat, CY-1678 Nicosia - Cyprus

• Document type: Journal article
• Source: ADVANCES IN GEOMETRY; v. 14, n. 4, p. 587-612, OCT 2014.
• Web of Science Citations: 2

Abstract

Let L be a Lagrangian submanifold of a pseudo-or para-Kahler 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-Kahler 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-Kahler manifolds, similar to Oh's stability criterion for minimal Lagrangian manifolds in Kahler-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)) X . . . X 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)) 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 ({[}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. (AU)

• FAPESP's process: 10/18752-0 - Lagrangian submanifolds in pseudo-Riemannian geometry
• Grantee: Henri Nicolas Guillaume Anciaux
• Support Opportunities: Regular Research Grants
• FAPESP's process: 10/08669-9 - Normal Congruences and Lagrangian submanifolds in spaces of geodesics
• Grantee: Nikos Georgiou
• Support Opportunities: Scholarships in Brazil - Post-Doctoral