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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

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

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Author(s):
Anciaux, Henri [1] ; Georgiou, Nikos [2]
Total Authors: 2
Affiliation:
[1] Univ Sao Paulo, IME, BR-05508090 Sao Paulo - Brazil
[2] Univ Cyprus, Dept Math & Stat, CY-1678 Nicosia - Cyprus
Total Affiliations: 2
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