L²-Hodge theory on Complete Almost K\"{a}hler Manifolds and the Hopf Conjecture

In this article, we develop an $L^{2}$-Hodge theory on complete $2n$-dimensional almost K\"{a}hler manifolds $(X,\omega)$. In the first part, we establish several identities for various Laplacians, generalized Hodge and Serre dualities, a generalized Hard Lefschetz duality, and a Lefschetz decomposition, all restricted to the space $\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}$ of forms of pure bidegree. In the second part, as applications of these identities, we prove vanishing theorems for $L^{2}$-harmonic $(p,q)$-forms on $X$ under some growth assumptions on the K\"{a}her form $\omega$. We also provide refined $L^{2}$-estimates to sharpen the vanishing theorems in three specific settings. As a final application, the topology of compact almost K\"ahler manifolds with negative sectional curvature is studied. Under a smallness condition on the Nijenhuis tensor depending on the curvature, the authors prove that the Hirzebruch $\chi_{y}$-genus satisfies $(-1)^{n-p}\chi_{p}(X)\geq1$ for all $p=0,1,\cdots,n$, which in particular implies the Hopf conjecture for the Euler number $(-1)^{n}\chi(X)\geq n+1$. This extends a classical result of Gromov [J. Differential Geom., 1991] from the K\"ahler to the almost K\"ahler setting.