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

Abstract

In this article, we develop an $L^{2}$-Hodge theory on complete $2n$-dimensional almost Kähler manifolds $(X,ω)$. 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 $\ker{Δ_{\partial}}\cap\ker{Δ_{\bar{\partial}}}$ 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 Käher form $ω$. 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 $χ_{y}$-genus satisfies $(-1)^{n-p}χ_{p}(X)\geq1$ for all $p=0,1,\cdots,n$, which in particular implies the Hopf conjecture for the Euler number $(-1)^{n}χ(X)\geq n+1$. This extends a classical result of Gromov [J. Differential Geom., 1991] from the Kähler to the almost Kähler setting.