On $L_2$-cohomology of almost Hermitian manifolds

TL;DR

This paper investigates the $L_2$-cohomology of almost Hermitian manifolds, showing a decomposition in four dimensions that extends previous results from closed to non-compact cases, contrasting with known theorems for K"ahler manifolds.

Contribution

It generalizes a decomposition theorem for $L_2$-cohomology in 4-dimensional almost Hermitian manifolds, extending prior results from closed to complete non-compact manifolds.

Findings

Decomposition of reduced $L_2$ second cohomology into invariant and anti-invariant parts.

Contrasts with Gromov's theorem for K"ahler manifolds regarding $d$-bounded forms.

Shows the non-applicability of K"ahler results to almost K"ahler manifolds in the $L_2$ context.

Abstract

Let $(X,J,\omega,g)$ be a complete $n$-dimensional K\"ahler manifold. A Theorem by Gromov \cite{G} states that the if the K\"ahler 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 K\"ahler 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 Dr\v{a}ghici, Li and Zhang \cite{DLZ} for $4$-dimensional closed almost complex manifolds to the $L_2$-setting.