On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Journal article

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 Vol_g under Hamiltonian deformations, computing Vol_g (L) using g|L. It is called Hamiltonian stable if in addition the second variation of Vol_g 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.

Authors: Joyce, D; Lee, Y; Schoen, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Johns Hopkins University Press
• Journal: American Journal of Mathematics
• Volume: 133
• Issue: 4
• Pages: 1067-1092
• Publication date: 2011-08-01
• DOI: 10.1353/ajm.2011.0030
• EISSN: 1080-6377
• ISSN: 0002-9327
• Keywords: math.SG; math.DG