Rigidity properties of Hamiltonian stationary Lagrangian submanifolds

A symplectic manifold (𝛭,𝛚) can always be equipped with a Riemannian metric 𝘨 and an almost complex structure 𝑱 such that these three structures are compatible in the sense that they satisfy 𝛚(𝑿,𝒀) = 𝘨(𝑱𝑿,𝒀) for all tangent vectors 𝑿 and 𝒀. This allows us to study Lagrangian submanifolds from a Riemannian geometric point of view. We consider a specific type of Lagrangian submanifolds called Hamiltonian stationary, which arise as critical points of the volume functional under Hamiltonian variations, and investigate the rigidity of their Riemannian properties. When the ambient space is Kähler-Einstein, Hamiltonian stationary Lagrangians are characterized by the harmonicity of the so-called mean curvature 1-form. This allows us to combine the Bochner formula with certain Liouville-type theorems to establish classification results for Hamiltonian stationary Lagrangians whose Ricci curvature and mean curvature vector field satisfy some natural conditions. For example, we characterize the Hamiltonian stationary Lagrangian surfaces in simply connected complex space forms whose Gaussian curvature is non-negative and whose mean curvature vector field is in some 𝑳ᴾ space or converges to zero at infinity. This characterization tells us, for example, that the only complete, non-compact, connected and oriented Hamiltonian stationary Lagrangian submanifolds of ℂ² whose Gaussian curvature is non-negative and whose mean curvature is converging to zero at infinity are Lagrangian planes. We also study a certain class of immersed Lagrangian tori in ℂⁿ which can be constructed by twisting a simple planar curve, and thus, are usually referred to as twisted tori. Both the Clifford tori and the exotic Chekanov tori, which are the only known examples of Lagrangian tori in ℂ² up to Hamiltonian isotopy, can be constructed in this way. We show that the only Hamiltonian stationary twisted tori in ℂ² are the product ones. In particular, none of the Chekanov tori are Hamiltonian stationary.