On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds
Resource
arXiv: 0902.3338
Journal
American Journal of Mathematics
Pages
1067-1092
Date Issued
2009
Date
2009
Author(s)
Abstract
Let (M, ω) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with ω. For instance, g could be Kähler, with Kähler form ω. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional Volg under Hamiltonian deformations, computing Volg (L) using g|L. It is called Hamiltonian stable if in addition the second variation of Volg under Hamiltonian deformations is nonnegative. Our main result is that if L is a compact, Hamiltonian stationary Lagrangian in Cn which is Hamiltonian rigid, then for any M, ω, g as above there exist compact Hamiltonian stationary Lagrangians L' in M contained in a small ball about some p ∈ M and locally modelled on tL for small t > 0, identifying M near p with Cn near 0. If L is Hamiltonian stable, we can take L' to be Hamiltonian stable. Applying this to known examples L in Cn shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to Tn, and to (S1 × Sn-1)/Z2, and with other topologies, in every compact symplectic 2n-manifold (M, ω) with compatible metric g. © 2011 by The JohnsHopkinsUniversity Press.
Type
journal article
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