ON THE EXISTENCE OF HAMILTONIAN STATIONARY LAGRANGIAN SUBMANIFOLDS IN SYMPLECTIC MANIFOLDS

Let (M,omega) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with omega. For instance, g could be Kahler, with Kahler form omega. 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 vertical bar 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 C-n which is Hamiltonian rigid, then for any M, omega, g as above there exist compact Hamiltonian stationary Lagrangians L' in M contained in a small ball about some p is an element of M and locally modelled on tL for small t > 0, identifying M near p with C-n near 0. If L is Hamiltonian stable, we can take L' to be Hamiltonian stable. Applying this to known examples L in C-n shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to T-n, and to (S-1 x Sn-1)/Z(2), and with other topologies, in every compact symplectic 2n-manifold (M, omega) with compatible metric g.