On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Abstract

Let $(M,\omega)$ be a compact symplectic $2n$-manifold, and $g$ a Riemannian metric on $M$ compatible with $\omega$. For instance, $g$ could be K\"ahler, with K\"ahler form $\omega$. Consider compact Lagrangian submanifolds $L$ of $M$. We call $L$ {\it Hamiltonian stationary}, or {\it H-minimal}, if it is a critical point of the volume functional $\mathop{\rm Vol}\nolimits_g$ under Hamiltonian deformations, computing $\mathop{\rm Vol}\nolimits_g(L)$ using $g\vert_L$. It is called {\it Hamiltonian stable\/} if in addition the second variation of $\mathop{\rm Vol}\nolimits_g$ under Hamiltonian deformations is nonnegative. Our main result is that if $L$ is a compact, Hamiltonian stationary Lagrangian in $\mathbin{{\Bbb C}}^n$ which is {\it 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 $p\in M$ and locally modelled on $t L$ for small $t>0$, identifying $M$ near $p$ with $\mathbin{{\Bbb C}}^n$ near $0$. If $L$ is Hamiltonian stable, we can take $L'$ to be Hamiltonian stable. Applying this to known examples $L$ in $\mathbin{{\Bbb C}}^n$ shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to $T^n$, and to $({\cal S}^1\times{\cal S}^{n-1})/\mathbin{{\Bbb Z}}_2$, and with other topologies, in every compact symplectic $2n$-manifold $(M,\omega)$ with compatible metric~$g$.