On $L_2$-cohomology of almost Hermitian manifolds

Abstract

Let $(X,J,\omega ,g)$ be a complete $n$-dimensional Kähler manifold. A Theorem by Gromov \cite{G} states that the if the Kähler form is $d$-bounded, then the space of harmonic $L_2$ forms of degree $k$ is trivial, unless $k=\frac{n}{2}$. Starting with a contact manifold $(M,\alpha )$ 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 $L_2$ $2^{nd}$-cohomology group decomposes as direct sum of the closure of the invariant and anti-invariant $L_2$-cohomology. This generalizes a decomposition theorem by Drǎghici, Li and Zhang \cite{DLZ} for $4$-dimensional closed almost complex manifolds to the $L_2$-setting.

13 pages