On L₂-cohomology of almost Hermitian manifolds

Let $(X,J,ω,g)$ be a complete n-dimensional Kähler manifold. A Theorem by Gromov {G} states that the if the Kähler form is d-bounded, then the space of harmonic L₂ forms of degree k is trivial, unless k=n/2. Starting with a contact manifold $(M,α)$ we show that the same conclusion does not hold in the category of almost Kähler manifolds. Let $(X,J,g)$ be a complete almost Hermitian manifold of dimension four. We prove that the reduced L₂ 2ⁿᵈ-cohomology group decomposes as direct sum of the closure of the invariant and anti-invariant L₂-cohomology. This generalizes a decomposition theorem by Drǎghici, Li and Zhang {DLZ} for $4$-dimensional closed almost complex manifolds to the L₂-setting.