On L2-cohomology of almost Hermitian manifolds

We prove two results regarding the L2 cohomology of almost-complex manifolds. First we show that there exist complete, d-bounded almost Kähler manifolds of any complex dimension n ≥ 2 such that the space of harmonic 1-forms in L2 has infinite dimension. By contrast a theorem of Gromov [6] states that a complete d-bounded Kähler manifold X has no nontrivial harmonic forms of degree different from n = dimC X. Second 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 [4] for 4-dimensional closed almost complex manifolds to the L2-setting.