$L^2$-Hodge theory on Complete Almost Kähler Manifolds and the Hopf Conjecture

In this article, we develop an $L^2$-Hodge theory on complete $2n$-dimensional almost Kähler 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 $\ker {\mathrm{\Delta }}_{\mathrm{\partial }}\cap \ker {\mathrm{\Delta }}_{\bar{\mathrm{\partial }}}$ 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 Käher 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 Kähler 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 $(-1{)}^{n-p}{\chi }_p(X)\ge 1$ for all $p=0,1,\cdots ,n$, which in particular implies the Hopf conjecture for the Euler number $(-1{)}^n\chi (X)\ge n+1$. This extends a classical result of Gromov [J. Differential Geom., 1991] from the Kähler to the almost Kähler setting.