On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

TL;DR

This paper proves the existence of Hamiltonian stationary Lagrangian submanifolds in any compact symplectic manifold, extending known examples and providing new constructions with various topologies.

Contribution

It establishes that Hamiltonian stationary Lagrangians, modeled on rigid examples in C^n, exist in all compact symplectic manifolds near any point.

Findings

Existence of Hamiltonian stationary Lagrangians in all compact symplectic manifolds.

Construction of families with various topologies, including tori and other manifolds.

Extension of known examples in C^n to general symplectic manifolds.

Abstract

Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. 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 under Hamiltonian deformations. It is called Hamiltonian stable if in addition the second variation of volume under Hamiltonian deformations is nonnegative. Our main result is that if L is a compact, Hamiltonian stationary Lagrangian in C^n satisfying the extra condition of being Hamiltonian rigid, then for any M,w,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 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 S^{n-1})/{1,-1}, and with other topologies, in every compact symplectic 2n-manifold (M,w) with compatible metric g.