2012/05/08 by Henri Anciaux, Nikos Georgiou, Anciaux, Henri +1
Mathematics · #49Q05 #53A07 #53D12 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1205.1749
openalex publication_date 2012/05/08 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Let L be a Lagrangian submanifold of a pseudo- or para-K "ahler manifold\nwhich is H-minimal, i.e. a critical point of the volume functional restricted\nto Hamiltonian variations. We derive the second variation of the volume of L\nwith respect to Hamiltonian variations. We apply this formula to several cases.\n In particular we observe that a minimal Lagrangian submanifold L in a\nRicci-flat pseudo- or para-K "ahler manifold is H-stable, i.e. its second\nvariation is definite and L therefore a local extremizer of the volume with\nrespect to Hamiltonian variations. We also give a stability criterion for\nspacelike minimal Lagrangian submanifolds in para-K "ahler manifolds, similar\nto Oh's stability criterion for minimal Lagrangian manifolds in\nK "ahler-Einstein manifolds.\n Finally, we determine the H-stability of a series of examples of H-minimal\nLagrangian submanifolds: the product of n circles of arbitrary radii in complex\nspace Cn is H-unstable with respect to any indefinite flat Hermitian metric,\nwhile the product of n hyperbolas in para-complex vector space Dn is H-stable\nfor n=1,2 and H-unstable for n > 2. Recently, minimal Lagrangian surfaces in\nthe space of geodesics of three-dimensional space forms have been\ncharacterized; on the other hand, a class of H-minimal Lagrangian surfaces in\nthe tangent bundle of a Riemannian, oriented surface has been identified. We\ndiscuss the H-stability of all these examples.\n