Equivariant deformations of Hamiltonian stationary Lagrangian submanifolds

TL;DR

This paper proves an equivariant deformation result for Hamiltonian stationary Lagrangian submanifolds in Kahler manifolds, enabling their existence in nearby non-Kahler symplectic manifolds with compatible structures.

Contribution

It introduces a new deformation theorem for Hamiltonian stationary Lagrangians under group actions, extending existence results beyond Kahler manifolds.

Findings

Existence of Hamiltonian stationary Lagrangians in non-Kahler symplectic manifolds.

Deformation results hold under compatible metric and almost complex structure changes.

Applicable to manifolds with isometric Hamiltonian group actions.

Abstract

We prove an equivariant deformation result for Hamiltonian stationary Lagrangian submanifolds of a Kahler manifold, with respect to deformations of its metric and almost complex structure that are compatible with an isometric Hamiltonian group action. This yields existence of Hamiltonian stationary Lagrangian submanifolds in possibly non-Kahler symplectic manifolds whose metric is arbitrarily close to a Kahler metric.